{"unique_id": "cad_t1_cad_belt_tangent_0045", "problem_id": "cad_t1_cad_belt_tangent_0045", "domain": "PyGeoX-CAD", "template": "cad_belt_tangent", "tier": 1, "nl_description": "A bracket carries two pulleys. Set up coordinates with the centre of the large pulley at the origin C; the centre of the small pulley is at D, 31.5 mm to the right and 13.6 mm above it. The large pulley is 15.9 mm across and the small one is 7.7 mm across. A straight belt run passes over the tops of both pulleys, touching the large one at T and the small one at V. The belt does not cross over between the two pulleys.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Belt run tangent to two pulleys\n\nOracle formalization for cad_t1_cad_belt_tangent_0045. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(46)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.x - C.x, 31.5, description=\"the small pulley centre is 31.5 to the right of the large one\")\nscene.constraint.eq(D.y - C.y, 13.6, description=\"the small pulley centre is 13.6 above the large one\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 15.9, description=\"the large pulley is 15.9 across\")\nscene.constraint.eq(2 * pulley_D.radius, 7.7, description=\"the small pulley is 7.7 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run passes over the TOP of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [31.5, 13.6], "T": [-2.2564596423286627, 7.623049906864182], "V": [30.407249104029518, 17.291665678166932]}, "circles": {"C": 7.95, "D": 3.85}}, "implicit_facts": ["each pulley radius is half the stated across-size", "'touching' means the belt line is tangent to both circles", "the perpendicular distance from each centre to the belt equals that pulley's radius", "both tangency points are the feet of the perpendiculars from the centres"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "'over the tops' picks the upper of the two external tangents"]} {"unique_id": "cad_t1_cad_belt_tangent_0046", "problem_id": "cad_t1_cad_belt_tangent_0046", "domain": "PyGeoX-CAD", "template": "cad_belt_tangent", "tier": 1, "nl_description": "A bracket carries two pulleys. Set up coordinates with the centre of the large pulley at the origin C; the centre of the small pulley is at D, 29.5 mm to the right and 10.7 mm above it. The large pulley is 21.3 mm across and the small one is 8.4 mm across. A straight belt run passes over the tops of both pulleys, touching the large one at T and the small one at V. The belt does not cross over between the two pulleys.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Belt run tangent to two pulleys\n\nOracle formalization for cad_t1_cad_belt_tangent_0046. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(43)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.x - C.x, 29.5, description=\"the small pulley centre is 29.5 to the right of the large one\")\nscene.constraint.eq(D.y - C.y, 10.7, description=\"the small pulley centre is 10.7 above the large one\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 21.3, description=\"the large pulley is 21.3 across\")\nscene.constraint.eq(2 * pulley_D.radius, 8.4, description=\"the small pulley is 8.4 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run passes over the TOP of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [29.5, 10.7], "T": [-1.4960205412765912, 10.54440242688406], "V": [28.91002006822895, 14.8583558866585]}, "circles": {"C": 10.65, "D": 4.2}}, "implicit_facts": ["each pulley radius is half the stated across-size", "'touching' means the belt line is tangent to both circles", "the perpendicular distance from each centre to the belt equals that pulley's radius", "both tangency points are the feet of the perpendiculars from the centres"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "'over the tops' picks the upper of the two external tangents"]} {"unique_id": "cad_t1_cad_belt_tangent_0047", "problem_id": "cad_t1_cad_belt_tangent_0047", "domain": "PyGeoX-CAD", "template": "cad_belt_tangent", "tier": 1, "nl_description": "A bracket carries two pulleys. Set up coordinates with the centre of the large pulley at the origin C; the centre of the small pulley is at D, 21.3 mm to the right and 8.1 mm above it. The large pulley is 21.1 mm across and the small one is 12.9 mm across. A straight belt run passes over the tops of both pulleys, touching the large one at T and the small one at V. The belt does not cross over between the two pulleys.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Belt run tangent to two pulleys\n\nOracle formalization for cad_t1_cad_belt_tangent_0047. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(33)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.x - C.x, 21.3, description=\"the small pulley centre is 21.3 to the right of the large one\")\nscene.constraint.eq(D.y - C.y, 8.1, description=\"the small pulley centre is 8.1 above the large one\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 21.1, description=\"the large pulley is 21.1 across\")\nscene.constraint.eq(2 * pulley_D.radius, 12.9, description=\"the small pulley is 12.9 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run passes over the TOP of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [21.3, 8.1], "T": [-1.9146012952219775, 10.3748157516331], "V": [20.12946176737614, 14.442896833936825]}, "circles": {"C": 10.55, "D": 6.45}}, "implicit_facts": ["each pulley radius is half the stated across-size", "'touching' means the belt line is tangent to both circles", "the perpendicular distance from each centre to the belt equals that pulley's radius", "both tangency points are the feet of the perpendiculars from the centres"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "'over the tops' picks the upper of the two external tangents"]} {"unique_id": "cad_t1_cad_belt_tangent_0048", "problem_id": "cad_t1_cad_belt_tangent_0048", "domain": "PyGeoX-CAD", "template": "cad_belt_tangent", "tier": 1, "nl_description": "A bracket carries two pulleys. Set up coordinates with the centre of the large pulley at the origin C; the centre of the small pulley is at D, 21.2 mm to the right and 5.8 mm above it. The large pulley is 15.1 mm across and the small one is 7.5 mm across. A straight belt run passes over the tops of both pulleys, touching the large one at T and the small one at V. The belt does not cross over between the two pulleys.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Belt run tangent to two pulleys\n\nOracle formalization for cad_t1_cad_belt_tangent_0048. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(33)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.x - C.x, 21.2, description=\"the small pulley centre is 21.2 to the right of the large one\")\nscene.constraint.eq(D.y - C.y, 5.8, description=\"the small pulley centre is 5.8 above the large one\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 15.1, description=\"the large pulley is 15.1 across\")\nscene.constraint.eq(2 * pulley_D.radius, 7.5, description=\"the small pulley is 7.5 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run passes over the TOP of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [21.2, 5.8], "T": [-0.7032832299528273, 7.5171731853448165], "V": [20.85068713744065, 9.53369529073418]}, "circles": {"C": 7.55, "D": 3.75}}, "implicit_facts": ["each pulley radius is half the stated across-size", "'touching' means the belt line is tangent to both circles", "the perpendicular distance from each centre to the belt equals that pulley's radius", "both tangency points are the feet of the perpendiculars from the centres"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "'over the tops' picks the upper of the two external tangents"]} {"unique_id": "cad_t1_cad_belt_tangent_0049", "problem_id": "cad_t1_cad_belt_tangent_0049", "domain": "PyGeoX-CAD", "template": "cad_belt_tangent", "tier": 1, "nl_description": "A bracket carries two pulleys. Set up coordinates with the centre of the large pulley at the origin C; the centre of the small pulley is at D, 27.4 mm to the right and 7.1 mm above it. The large pulley is 18.3 mm across and the small one is 12.5 mm across. A straight belt run passes over the tops of both pulleys, touching the large one at T and the small one at V. The belt does not cross over between the two pulleys.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Belt run tangent to two pulleys\n\nOracle formalization for cad_t1_cad_belt_tangent_0049. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(41)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.x - C.x, 27.4, description=\"the small pulley centre is 27.4 to the right of the large one\")\nscene.constraint.eq(D.y - C.y, 7.1, description=\"the small pulley centre is 7.1 above the large one\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 18.3, description=\"the large pulley is 18.3 across\")\nscene.constraint.eq(2 * pulley_D.radius, 12.5, description=\"the small pulley is 12.5 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run passes over the TOP of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [27.4, 7.1], "T": [-1.3756071150562472, 9.046004922893125], "V": [26.460377653650106, 13.278965111265794]}, "circles": {"C": 9.15, "D": 6.25}}, "implicit_facts": ["each pulley radius is half the stated across-size", "'touching' means the belt line is tangent to both circles", "the perpendicular distance from each centre to the belt equals that pulley's radius", "both tangency points are the feet of the perpendiculars from the centres"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "'over the tops' picks the upper of the two external tangents"]} {"unique_id": "cad_t1_cad_belt_tangent_0050", "problem_id": "cad_t1_cad_belt_tangent_0050", "domain": "PyGeoX-CAD", "template": "cad_belt_tangent", "tier": 1, "nl_description": "A bracket carries two pulleys. Set up coordinates with the centre of the large pulley at the origin C; the centre of the small pulley is at D, 22.8 mm to the right and 5.7 mm above it. The large pulley is 19.5 mm across and the small one is 11.6 mm across. A straight belt run passes over the tops of both pulleys, touching the large one at T and the small one at V. The belt does not cross over between the two pulleys.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Belt run tangent to two pulleys\n\nOracle formalization for cad_t1_cad_belt_tangent_0050. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(35)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.x - C.x, 22.8, description=\"the small pulley centre is 22.8 to the right of the large one\")\nscene.constraint.eq(D.y - C.y, 5.7, description=\"the small pulley centre is 5.7 above the large one\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 19.5, description=\"the large pulley is 19.5 across\")\nscene.constraint.eq(2 * pulley_D.radius, 11.6, description=\"the small pulley is 11.6 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run passes over the TOP of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [22.8, 5.7], "T": [-0.7412998537476712, 9.721778362359105], "V": [22.359021625462926, 11.48321174376234]}, "circles": {"C": 9.75, "D": 5.8}}, "implicit_facts": ["each pulley radius is half the stated across-size", "'touching' means the belt line is tangent to both circles", "the perpendicular distance from each centre to the belt equals that pulley's radius", "both tangency points are the feet of the perpendiculars from the centres"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "'over the tops' picks the upper of the two external tangents"]} {"unique_id": "cad_t1_cad_belt_tangent_0051", "problem_id": "cad_t1_cad_belt_tangent_0051", "domain": "PyGeoX-CAD", "template": "cad_belt_tangent", "tier": 1, "nl_description": "A bracket carries two pulleys. Set up coordinates with the centre of the large pulley at the origin C; the centre of the small pulley is at D, 29.1 mm to the right and 5.6 mm above it. The large pulley is 18.2 mm across and the small one is 11.8 mm across. A straight belt run passes over the tops of both pulleys, touching the large one at T and the small one at V. The belt does not cross over between the two pulleys.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Belt run tangent to two pulleys\n\nOracle formalization for cad_t1_cad_belt_tangent_0051. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(43)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.x - C.x, 29.1, description=\"the small pulley centre is 29.1 to the right of the large one\")\nscene.constraint.eq(D.y - C.y, 5.6, description=\"the small pulley centre is 5.6 above the large one\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 18.2, description=\"the large pulley is 18.2 across\")\nscene.constraint.eq(2 * pulley_D.radius, 11.8, description=\"the small pulley is 11.8 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run passes over the TOP of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [29.1, 5.6], "T": [-0.7446426225589549, 9.069482199368855], "V": [28.61720972823101, 11.48021373365673]}, "circles": {"C": 9.1, "D": 5.9}}, "implicit_facts": ["each pulley radius is half the stated across-size", "'touching' means the belt line is tangent to both circles", "the perpendicular distance from each centre to the belt equals that pulley's radius", "both tangency points are the feet of the perpendiculars from the centres"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "'over the tops' picks the upper of the two external tangents"]} {"unique_id": "cad_t1_cad_belt_tangent_0052", "problem_id": "cad_t1_cad_belt_tangent_0052", "domain": "PyGeoX-CAD", "template": "cad_belt_tangent", "tier": 1, "nl_description": "A bracket carries two pulleys. Set up coordinates with the centre of the large pulley at the origin C; the centre of the small pulley is at D, 32.4 mm to the right and 9.9 mm above it. The large pulley is 22.5 mm across and the small one is 7.6 mm across. A straight belt run passes over the tops of both pulleys, touching the large one at T and the small one at V. The belt does not cross over between the two pulleys.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Belt run tangent to two pulleys\n\nOracle formalization for cad_t1_cad_belt_tangent_0052. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(47)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.x - C.x, 32.4, description=\"the small pulley centre is 32.4 to the right of the large one\")\nscene.constraint.eq(D.y - C.y, 9.9, description=\"the small pulley centre is 9.9 above the large one\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 22.5, description=\"the large pulley is 22.5 across\")\nscene.constraint.eq(2 * pulley_D.radius, 7.6, description=\"the small pulley is 7.6 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run passes over the TOP of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [32.4, 9.9], "T": [-0.8410742364592507, 11.21851568295755], "V": [32.11590381346265, 13.68936529735455]}, "circles": {"C": 11.25, "D": 3.8}}, "implicit_facts": ["each pulley radius is half the stated across-size", "'touching' means the belt line is tangent to both circles", "the perpendicular distance from each centre to the belt equals that pulley's radius", "both tangency points are the feet of the perpendiculars from the centres"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "'over the tops' picks the upper of the two external tangents"]} {"unique_id": "cad_t1_cad_gusset_hole_0069", "problem_id": "cad_t1_cad_gusset_hole_0069", "domain": "PyGeoX-CAD", "template": "cad_gusset_hole", "tier": 1, "nl_description": "A triangular gusset plate has two straight edges meeting at a square corner. Set up coordinates with that square corner at the origin G, one edge running from G along the positive x-direction to A, and the other running from G along the positive y-direction to B. Edge GA is 33.8 mm long and edge GB is 22.8 mm long; the third edge of the plate is the straight line from A to B. A hole 6.1 mm across is drilled with its centre at N, 12.8 mm from the square corner and positioned so that it sits the same distance from edge GA as it does from edge GB. The design requires at least 2.5 mm of material to remain between the hole and every edge of the plate.", "answer_schema": {"points": ["A", "B", "G", "N"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Gusset plate with a clearance hole\n\nOracle formalization for cad_t1_cad_gusset_hole_0069. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(49)\n\nG, A, B, N = scene.add.points([\"G\", \"A\", \"B\", \"N\"])\n\nscene.constraint.eq(G.x, 0, description=\"datum: G is the origin, the square corner of the gusset (x)\")\nscene.constraint.eq(G.y, 0, description=\"datum: G is the origin, the square corner of the gusset (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: edge GA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: edge GA runs in the +x direction\")\nscene.constraint.eq(G.distance(A), 33.8, description=\"edge GA is 33.8 long\")\nscene.constraint.eq(G.distance(B), 22.8, description=\"edge GB is 22.8 long\")\nscene.relate.right_angle(A, G, B)\nscene.constraint.gt(B.y, 1.0, description=\"datum: edge GB runs in the +y direction\")\nscene.constraint.eq(N.x, N.y, description=\"the hole centre is the same distance from edge GA as from edge GB\")\nscene.constraint.eq(G.distance(N), 12.8, description=\"the hole centre is 12.8 from the square corner\")\nscene.constraint.geq(N.y, 5.55, description=\"at least 2.5 of material between the hole and edge GA\")\nscene.constraint.geq(N.x, 5.55, description=\"at least 2.5 of material between the hole and edge GB\")\nscene.constraint.geq(cross(A, B, N) / A.distance(B), 5.55, description=\"at least 2.5 of material between the hole and the sloping edge AB\")\n", "possible_solution": {"points": {"G": [0.0, 0.0], "A": [33.8, 0.0], "B": [0.0, 22.8], "N": [9.050966799187808, 9.050966799187808]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'the same distance from both edges' puts the centre on the 45 degree bisector of the square corner", "the clearance is measured from the hole EDGE, so the centre must be at least clearance + hole radius from each edge", "the clearance to the sloping edge needs the perpendicular distance from N to the line AB"], "branch_facts": ["edge GB runs in the +y direction, which fixes which side the plate is on"]} {"unique_id": "cad_t1_cad_gusset_hole_0070", "problem_id": "cad_t1_cad_gusset_hole_0070", "domain": "PyGeoX-CAD", "template": "cad_gusset_hole", "tier": 1, "nl_description": "A triangular gusset plate has two straight edges meeting at a square corner. Set up coordinates with that square corner at the origin G, one edge running from G along the positive x-direction to A, and the other running from G along the positive y-direction to B. Edge GA is 31.5 mm long and edge GB is 19.1 mm long; the third edge of the plate is the straight line from A to B. A hole 6.4 mm across is drilled with its centre at N, 10.5 mm from the square corner and positioned so that it sits the same distance from edge GA as it does from edge GB. The design requires at least 1.9 mm of material to remain between the hole and every edge of the plate.", "answer_schema": {"points": ["A", "B", "G", "N"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Gusset plate with a clearance hole\n\nOracle formalization for cad_t1_cad_gusset_hole_0070. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(46)\n\nG, A, B, N = scene.add.points([\"G\", \"A\", \"B\", \"N\"])\n\nscene.constraint.eq(G.x, 0, description=\"datum: G is the origin, the square corner of the gusset (x)\")\nscene.constraint.eq(G.y, 0, description=\"datum: G is the origin, the square corner of the gusset (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: edge GA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: edge GA runs in the +x direction\")\nscene.constraint.eq(G.distance(A), 31.5, description=\"edge GA is 31.5 long\")\nscene.constraint.eq(G.distance(B), 19.1, description=\"edge GB is 19.1 long\")\nscene.relate.right_angle(A, G, B)\nscene.constraint.gt(B.y, 1.0, description=\"datum: edge GB runs in the +y direction\")\nscene.constraint.eq(N.x, N.y, description=\"the hole centre is the same distance from edge GA as from edge GB\")\nscene.constraint.eq(G.distance(N), 10.5, description=\"the hole centre is 10.5 from the square corner\")\nscene.constraint.geq(N.y, 5.1, description=\"at least 1.9 of material between the hole and edge GA\")\nscene.constraint.geq(N.x, 5.1, description=\"at least 1.9 of material between the hole and edge GB\")\nscene.constraint.geq(cross(A, B, N) / A.distance(B), 5.1, description=\"at least 1.9 of material between the hole and the sloping edge AB\")\n", "possible_solution": {"points": {"G": [0.0, 0.0], "A": [31.5, 0.0], "B": [0.0, 19.1], "N": [7.424621202458749, 7.424621202458749]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'the same distance from both edges' puts the centre on the 45 degree bisector of the square corner", "the clearance is measured from the hole EDGE, so the centre must be at least clearance + hole radius from each edge", "the clearance to the sloping edge needs the perpendicular distance from N to the line AB"], "branch_facts": ["edge GB runs in the +y direction, which fixes which side the plate is on"]} {"unique_id": "cad_t1_cad_gusset_hole_0071", "problem_id": "cad_t1_cad_gusset_hole_0071", "domain": "PyGeoX-CAD", "template": "cad_gusset_hole", "tier": 1, "nl_description": "A triangular gusset plate has two straight edges meeting at a square corner. Set up coordinates with that square corner at the origin G, one edge running from G along the positive x-direction to A, and the other running from G along the positive y-direction to B. Edge GA is 28.5 mm long and edge GB is 25 mm long; the third edge of the plate is the straight line from A to B. A hole 6.7 mm across is drilled with its centre at N, 12.3 mm from the square corner and positioned so that it sits the same distance from edge GA as it does from edge GB. The design requires at least 2.1 mm of material to remain between the hole and every edge of the plate.", "answer_schema": {"points": ["A", "B", "G", "N"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Gusset plate with a clearance hole\n\nOracle formalization for cad_t1_cad_gusset_hole_0071. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(42)\n\nG, A, B, N = scene.add.points([\"G\", \"A\", \"B\", \"N\"])\n\nscene.constraint.eq(G.x, 0, description=\"datum: G is the origin, the square corner of the gusset (x)\")\nscene.constraint.eq(G.y, 0, description=\"datum: G is the origin, the square corner of the gusset (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: edge GA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: edge GA runs in the +x direction\")\nscene.constraint.eq(G.distance(A), 28.5, description=\"edge GA is 28.5 long\")\nscene.constraint.eq(G.distance(B), 25, description=\"edge GB is 25 long\")\nscene.relate.right_angle(A, G, B)\nscene.constraint.gt(B.y, 1.0, description=\"datum: edge GB runs in the +y direction\")\nscene.constraint.eq(N.x, N.y, description=\"the hole centre is the same distance from edge GA as from edge GB\")\nscene.constraint.eq(G.distance(N), 12.3, description=\"the hole centre is 12.3 from the square corner\")\nscene.constraint.geq(N.y, 5.45, description=\"at least 2.1 of material between the hole and edge GA\")\nscene.constraint.geq(N.x, 5.45, description=\"at least 2.1 of material between the hole and edge GB\")\nscene.constraint.geq(cross(A, B, N) / A.distance(B), 5.45, description=\"at least 2.1 of material between the hole and the sloping edge AB\")\n", "possible_solution": {"points": {"G": [0.0, 0.0], "A": [28.5, 0.0], "B": [0.0, 25.0], "N": [8.697413408594535, 8.697413408594535]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'the same distance from both edges' puts the centre on the 45 degree bisector of the square corner", "the clearance is measured from the hole EDGE, so the centre must be at least clearance + hole radius from each edge", "the clearance to the sloping edge needs the perpendicular distance from N to the line AB"], "branch_facts": ["edge GB runs in the +y direction, which fixes which side the plate is on"]} {"unique_id": "cad_t1_cad_gusset_hole_0072", "problem_id": "cad_t1_cad_gusset_hole_0072", "domain": "PyGeoX-CAD", "template": "cad_gusset_hole", "tier": 1, "nl_description": "A triangular gusset plate has two straight edges meeting at a square corner. Set up coordinates with that square corner at the origin G, one edge running from G along the positive x-direction to A, and the other running from G along the positive y-direction to B. Edge GA is 28.9 mm long and edge GB is 25.5 mm long; the third edge of the plate is the straight line from A to B. A hole 5.6 mm across is drilled with its centre at N, 12.8 mm from the square corner and positioned so that it sits the same distance from edge GA as it does from edge GB. The design requires at least 2.5 mm of material to remain between the hole and every edge of the plate.", "answer_schema": {"points": ["A", "B", "G", "N"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Gusset plate with a clearance hole\n\nOracle formalization for cad_t1_cad_gusset_hole_0072. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(43)\n\nG, A, B, N = scene.add.points([\"G\", \"A\", \"B\", \"N\"])\n\nscene.constraint.eq(G.x, 0, description=\"datum: G is the origin, the square corner of the gusset (x)\")\nscene.constraint.eq(G.y, 0, description=\"datum: G is the origin, the square corner of the gusset (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: edge GA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: edge GA runs in the +x direction\")\nscene.constraint.eq(G.distance(A), 28.9, description=\"edge GA is 28.9 long\")\nscene.constraint.eq(G.distance(B), 25.5, description=\"edge GB is 25.5 long\")\nscene.relate.right_angle(A, G, B)\nscene.constraint.gt(B.y, 1.0, description=\"datum: edge GB runs in the +y direction\")\nscene.constraint.eq(N.x, N.y, description=\"the hole centre is the same distance from edge GA as from edge GB\")\nscene.constraint.eq(G.distance(N), 12.8, description=\"the hole centre is 12.8 from the square corner\")\nscene.constraint.geq(N.y, 5.3, description=\"at least 2.5 of material between the hole and edge GA\")\nscene.constraint.geq(N.x, 5.3, description=\"at least 2.5 of material between the hole and edge GB\")\nscene.constraint.geq(cross(A, B, N) / A.distance(B), 5.3, description=\"at least 2.5 of material between the hole and the sloping edge AB\")\n", "possible_solution": {"points": {"G": [0.0, 0.0], "A": [28.9, 0.0], "B": [0.0, 25.5], "N": [9.050966799187808, 9.050966799187808]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'the same distance from both edges' puts the centre on the 45 degree bisector of the square corner", "the clearance is measured from the hole EDGE, so the centre must be at least clearance + hole radius from each edge", "the clearance to the sloping edge needs the perpendicular distance from N to the line AB"], "branch_facts": ["edge GB runs in the +y direction, which fixes which side the plate is on"]} {"unique_id": "cad_t1_cad_gusset_hole_0073", "problem_id": "cad_t1_cad_gusset_hole_0073", "domain": "PyGeoX-CAD", "template": "cad_gusset_hole", "tier": 1, "nl_description": "A triangular gusset plate has two straight edges meeting at a square corner. Set up coordinates with that square corner at the origin G, one edge running from G along the positive x-direction to A, and the other running from G along the positive y-direction to B. Edge GA is 30.9 mm long and edge GB is 20.6 mm long; the third edge of the plate is the straight line from A to B. A hole 6.9 mm across is drilled with its centre at N, 9.5 mm from the square corner and positioned so that it sits the same distance from edge GA as it does from edge GB. The design requires at least 2.6 mm of material to remain between the hole and every edge of the plate.", "answer_schema": {"points": ["A", "B", "G", "N"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Gusset plate with a clearance hole\n\nOracle formalization for cad_t1_cad_gusset_hole_0073. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(45)\n\nG, A, B, N = scene.add.points([\"G\", \"A\", \"B\", \"N\"])\n\nscene.constraint.eq(G.x, 0, description=\"datum: G is the origin, the square corner of the gusset (x)\")\nscene.constraint.eq(G.y, 0, description=\"datum: G is the origin, the square corner of the gusset (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: edge GA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: edge GA runs in the +x direction\")\nscene.constraint.eq(G.distance(A), 30.9, description=\"edge GA is 30.9 long\")\nscene.constraint.eq(G.distance(B), 20.6, description=\"edge GB is 20.6 long\")\nscene.relate.right_angle(A, G, B)\nscene.constraint.gt(B.y, 1.0, description=\"datum: edge GB runs in the +y direction\")\nscene.constraint.eq(N.x, N.y, description=\"the hole centre is the same distance from edge GA as from edge GB\")\nscene.constraint.eq(G.distance(N), 9.5, description=\"the hole centre is 9.5 from the square corner\")\nscene.constraint.geq(N.y, 6.05, description=\"at least 2.6 of material between the hole and edge GA\")\nscene.constraint.geq(N.x, 6.05, description=\"at least 2.6 of material between the hole and edge GB\")\nscene.constraint.geq(cross(A, B, N) / A.distance(B), 6.05, description=\"at least 2.6 of material between the hole and the sloping edge AB\")\n", "possible_solution": {"points": {"G": [0.0, 0.0], "A": [30.9, 0.0], "B": [0.0, 20.6], "N": [6.717514421272201, 6.717514421272201]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'the same distance from both edges' puts the centre on the 45 degree bisector of the square corner", "the clearance is measured from the hole EDGE, so the centre must be at least clearance + hole radius from each edge", "the clearance to the sloping edge needs the perpendicular distance from N to the line AB"], "branch_facts": ["edge GB runs in the +y direction, which fixes which side the plate is on"]} {"unique_id": "cad_t1_cad_gusset_hole_0074", "problem_id": "cad_t1_cad_gusset_hole_0074", "domain": "PyGeoX-CAD", "template": "cad_gusset_hole", "tier": 1, "nl_description": "A triangular gusset plate has two straight edges meeting at a square corner. Set up coordinates with that square corner at the origin G, one edge running from G along the positive x-direction to A, and the other running from G along the positive y-direction to B. Edge GA is 33.5 mm long and edge GB is 21.9 mm long; the third edge of the plate is the straight line from A to B. A hole 5.1 mm across is drilled with its centre at N, 10.6 mm from the square corner and positioned so that it sits the same distance from edge GA as it does from edge GB. The design requires at least 2.6 mm of material to remain between the hole and every edge of the plate.", "answer_schema": {"points": ["A", "B", "G", "N"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Gusset plate with a clearance hole\n\nOracle formalization for cad_t1_cad_gusset_hole_0074. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(49)\n\nG, A, B, N = scene.add.points([\"G\", \"A\", \"B\", \"N\"])\n\nscene.constraint.eq(G.x, 0, description=\"datum: G is the origin, the square corner of the gusset (x)\")\nscene.constraint.eq(G.y, 0, description=\"datum: G is the origin, the square corner of the gusset (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: edge GA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: edge GA runs in the +x direction\")\nscene.constraint.eq(G.distance(A), 33.5, description=\"edge GA is 33.5 long\")\nscene.constraint.eq(G.distance(B), 21.9, description=\"edge GB is 21.9 long\")\nscene.relate.right_angle(A, G, B)\nscene.constraint.gt(B.y, 1.0, description=\"datum: edge GB runs in the +y direction\")\nscene.constraint.eq(N.x, N.y, description=\"the hole centre is the same distance from edge GA as from edge GB\")\nscene.constraint.eq(G.distance(N), 10.6, description=\"the hole centre is 10.6 from the square corner\")\nscene.constraint.geq(N.y, 5.15, description=\"at least 2.6 of material between the hole and edge GA\")\nscene.constraint.geq(N.x, 5.15, description=\"at least 2.6 of material between the hole and edge GB\")\nscene.constraint.geq(cross(A, B, N) / A.distance(B), 5.15, description=\"at least 2.6 of material between the hole and the sloping edge AB\")\n", "possible_solution": {"points": {"G": [0.0, 0.0], "A": [33.5, 0.0], "B": [0.0, 21.9], "N": [7.495331880577403, 7.495331880577403]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'the same distance from both edges' puts the centre on the 45 degree bisector of the square corner", "the clearance is measured from the hole EDGE, so the centre must be at least clearance + hole radius from each edge", "the clearance to the sloping edge needs the perpendicular distance from N to the line AB"], "branch_facts": ["edge GB runs in the +y direction, which fixes which side the plate is on"]} {"unique_id": "cad_t1_cad_gusset_hole_0075", "problem_id": "cad_t1_cad_gusset_hole_0075", "domain": "PyGeoX-CAD", "template": "cad_gusset_hole", "tier": 1, "nl_description": "A triangular gusset plate has two straight edges meeting at a square corner. Set up coordinates with that square corner at the origin G, one edge running from G along the positive x-direction to A, and the other running from G along the positive y-direction to B. Edge GA is 32.8 mm long and edge GB is 18.3 mm long; the third edge of the plate is the straight line from A to B. A hole 5.1 mm across is drilled with its centre at N, 8.7 mm from the square corner and positioned so that it sits the same distance from edge GA as it does from edge GB. The design requires at least 2.7 mm of material to remain between the hole and every edge of the plate.", "answer_schema": {"points": ["A", "B", "G", "N"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Gusset plate with a clearance hole\n\nOracle formalization for cad_t1_cad_gusset_hole_0075. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(48)\n\nG, A, B, N = scene.add.points([\"G\", \"A\", \"B\", \"N\"])\n\nscene.constraint.eq(G.x, 0, description=\"datum: G is the origin, the square corner of the gusset (x)\")\nscene.constraint.eq(G.y, 0, description=\"datum: G is the origin, the square corner of the gusset (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: edge GA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: edge GA runs in the +x direction\")\nscene.constraint.eq(G.distance(A), 32.8, description=\"edge GA is 32.8 long\")\nscene.constraint.eq(G.distance(B), 18.3, description=\"edge GB is 18.3 long\")\nscene.relate.right_angle(A, G, B)\nscene.constraint.gt(B.y, 1.0, description=\"datum: edge GB runs in the +y direction\")\nscene.constraint.eq(N.x, N.y, description=\"the hole centre is the same distance from edge GA as from edge GB\")\nscene.constraint.eq(G.distance(N), 8.7, description=\"the hole centre is 8.7 from the square corner\")\nscene.constraint.geq(N.y, 5.25, description=\"at least 2.7 of material between the hole and edge GA\")\nscene.constraint.geq(N.x, 5.25, description=\"at least 2.7 of material between the hole and edge GB\")\nscene.constraint.geq(cross(A, B, N) / A.distance(B), 5.25, description=\"at least 2.7 of material between the hole and the sloping edge AB\")\n", "possible_solution": {"points": {"G": [0.0, 0.0], "A": [32.8, 0.0], "B": [0.0, 18.3], "N": [6.151828996322963, 6.151828996322963]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'the same distance from both edges' puts the centre on the 45 degree bisector of the square corner", "the clearance is measured from the hole EDGE, so the centre must be at least clearance + hole radius from each edge", "the clearance to the sloping edge needs the perpendicular distance from N to the line AB"], "branch_facts": ["edge GB runs in the +y direction, which fixes which side the plate is on"]} {"unique_id": "cad_t1_cad_gusset_hole_0076", "problem_id": "cad_t1_cad_gusset_hole_0076", "domain": "PyGeoX-CAD", "template": "cad_gusset_hole", "tier": 1, "nl_description": "A triangular gusset plate has two straight edges meeting at a square corner. Set up coordinates with that square corner at the origin G, one edge running from G along the positive x-direction to A, and the other running from G along the positive y-direction to B. Edge GA is 31 mm long and edge GB is 22.6 mm long; the third edge of the plate is the straight line from A to B. A hole 6.3 mm across is drilled with its centre at N, 11.3 mm from the square corner and positioned so that it sits the same distance from edge GA as it does from edge GB. The design requires at least 1.6 mm of material to remain between the hole and every edge of the plate.", "answer_schema": {"points": ["A", "B", "G", "N"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Gusset plate with a clearance hole\n\nOracle formalization for cad_t1_cad_gusset_hole_0076. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(45)\n\nG, A, B, N = scene.add.points([\"G\", \"A\", \"B\", \"N\"])\n\nscene.constraint.eq(G.x, 0, description=\"datum: G is the origin, the square corner of the gusset (x)\")\nscene.constraint.eq(G.y, 0, description=\"datum: G is the origin, the square corner of the gusset (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: edge GA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: edge GA runs in the +x direction\")\nscene.constraint.eq(G.distance(A), 31, description=\"edge GA is 31 long\")\nscene.constraint.eq(G.distance(B), 22.6, description=\"edge GB is 22.6 long\")\nscene.relate.right_angle(A, G, B)\nscene.constraint.gt(B.y, 1.0, description=\"datum: edge GB runs in the +y direction\")\nscene.constraint.eq(N.x, N.y, description=\"the hole centre is the same distance from edge GA as from edge GB\")\nscene.constraint.eq(G.distance(N), 11.3, description=\"the hole centre is 11.3 from the square corner\")\nscene.constraint.geq(N.y, 4.75, description=\"at least 1.6 of material between the hole and edge GA\")\nscene.constraint.geq(N.x, 4.75, description=\"at least 1.6 of material between the hole and edge GB\")\nscene.constraint.geq(cross(A, B, N) / A.distance(B), 4.75, description=\"at least 1.6 of material between the hole and the sloping edge AB\")\n", "possible_solution": {"points": {"G": [0.0, 0.0], "A": [31.0, 0.0], "B": [0.0, 22.6], "N": [7.990306627407987, 7.990306627407987]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'the same distance from both edges' puts the centre on the 45 degree bisector of the square corner", "the clearance is measured from the hole EDGE, so the centre must be at least clearance + hole radius from each edge", "the clearance to the sloping edge needs the perpendicular distance from N to the line AB"], "branch_facts": ["edge GB runs in the +y direction, which fixes which side the plate is on"]} {"unique_id": "cad_t1_cad_three_hole_0085", "problem_id": "cad_t1_cad_three_hole_0085", "domain": "PyGeoX-CAD", "template": "cad_three_hole", "tier": 1, "nl_description": "A rectangular bracket measures 38.4 mm by 24.5 mm. Set up coordinates with the bottom-left corner at the origin P, the 38.4 mm edge running from P along the positive x-direction to Q, and the 24.5 mm edge running from P along the positive y-direction to S; R is the remaining corner. Three holes, each 6.3 mm across, are drilled through it. The centre of the first, N1, is 6.8 mm from edge PS and 5.1 mm from edge PQ. The centre of the second, N2, is 25.3 mm from edge PS and 18.8 mm from edge PQ. The third hole centre N3 is 11 mm from edge PQ, and it must sit the same distance from the first hole as from the second. At least 1.7 mm of material must remain between the third hole and every edge of the bracket.", "answer_schema": {"points": ["N1", "N2", "N3", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Three-hole bracket, third hole equidistant\n\nOracle formalization for cad_t1_cad_three_hole_0085. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(55)\n\nP, Q, R, S, N1, N2, N3 = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"N1\", \"N2\", \"N3\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the bracket (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the bracket (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the bracket lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 38.4, description=\"bracket is 38.4 long along PQ\")\nscene.constraint.eq(P.distance(S), 24.5, description=\"bracket is 24.5 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the bracket lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"bracket corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"bracket corner R closes the rectangle (y)\")\nscene.constraint.eq(N1.x, 6.8, description=\"hole 1 is 6.8 from the datum edge PS\")\nscene.constraint.eq(N1.y, 5.1, description=\"hole 1 is 5.1 from the datum edge PQ\")\nscene.constraint.eq(N2.x, 25.3, description=\"hole 2 is 25.3 from the datum edge PS\")\nscene.constraint.eq(N2.y, 18.8, description=\"hole 2 is 18.8 from the datum edge PQ\")\nscene.constraint.eq(N3.y, 11, description=\"hole 3 is 11 from the datum edge PQ\")\nscene.constraint.eq(N3.distance(N1), N3.distance(N2), description=\"hole 3 is the same distance from hole 1 as from hole 2\")\nscene.constraint.geq(N3.y, 4.85, description=\"at least 1.7 of material between hole 3 and edge PQ\")\nscene.constraint.geq(N3.x, 4.85, description=\"at least 1.7 of material between hole 3 and edge PS\")\nscene.constraint.geq(Q.x - N3.x, 4.85, description=\"at least 1.7 of material between hole 3 and edge QR\")\nscene.constraint.geq(S.y - N3.y, 4.85, description=\"at least 1.7 of material between hole 3 and edge SR\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [38.4, 0.0], "R": [38.4, 24.5], "S": [0.0, 24.5], "N1": [6.8, 5.1], "N2": [25.3, 18.8], "N3": [16.753513513513514, 11.0]}, "circles": {}}, "implicit_facts": ["each hole radius is half the stated across-size", "'the same distance from the first hole as from the second' puts N3 on the perpendicular bisector of N1N2", "combined with the stated height, that pins N3 completely", "edge clearance is measured from the hole edge, so it uses clearance + hole radius from the centre"], "branch_facts": []} {"unique_id": "cad_t1_cad_three_hole_0086", "problem_id": "cad_t1_cad_three_hole_0086", "domain": "PyGeoX-CAD", "template": "cad_three_hole", "tier": 1, "nl_description": "A rectangular bracket measures 45.8 mm by 25.9 mm. Set up coordinates with the bottom-left corner at the origin P, the 45.8 mm edge running from P along the positive x-direction to Q, and the 25.9 mm edge running from P along the positive y-direction to S; R is the remaining corner. Three holes, each 4.9 mm across, are drilled through it. The centre of the first, N1, is 8.8 mm from edge PS and 8.6 mm from edge PQ. The centre of the second, N2, is 24.7 mm from edge PS and 17.5 mm from edge PQ. The third hole centre N3 is 13.7 mm from edge PQ, and it must sit the same distance from the first hole as from the second. At least 1.8 mm of material must remain between the third hole and every edge of the bracket.", "answer_schema": {"points": ["N1", "N2", "N3", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Three-hole bracket, third hole equidistant\n\nOracle formalization for cad_t1_cad_three_hole_0086. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(65)\n\nP, Q, R, S, N1, N2, N3 = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"N1\", \"N2\", \"N3\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the bracket (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the bracket (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the bracket lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 45.8, description=\"bracket is 45.8 long along PQ\")\nscene.constraint.eq(P.distance(S), 25.9, description=\"bracket is 25.9 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the bracket lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"bracket corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"bracket corner R closes the rectangle (y)\")\nscene.constraint.eq(N1.x, 8.8, description=\"hole 1 is 8.8 from the datum edge PS\")\nscene.constraint.eq(N1.y, 8.6, description=\"hole 1 is 8.6 from the datum edge PQ\")\nscene.constraint.eq(N2.x, 24.7, description=\"hole 2 is 24.7 from the datum edge PS\")\nscene.constraint.eq(N2.y, 17.5, description=\"hole 2 is 17.5 from the datum edge PQ\")\nscene.constraint.eq(N3.y, 13.7, description=\"hole 3 is 13.7 from the datum edge PQ\")\nscene.constraint.eq(N3.distance(N1), N3.distance(N2), description=\"hole 3 is the same distance from hole 1 as from hole 2\")\nscene.constraint.geq(N3.y, 4.25, description=\"at least 1.8 of material between hole 3 and edge PQ\")\nscene.constraint.geq(N3.x, 4.25, description=\"at least 1.8 of material between hole 3 and edge PS\")\nscene.constraint.geq(Q.x - N3.x, 4.25, description=\"at least 1.8 of material between hole 3 and edge QR\")\nscene.constraint.geq(S.y - N3.y, 4.25, description=\"at least 1.8 of material between hole 3 and edge SR\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [45.8, 0.0], "R": [45.8, 25.9], "S": [0.0, 25.9], "N1": [8.8, 8.6], "N2": [24.7, 17.5], "N3": [16.38616352201258, 13.7]}, "circles": {}}, "implicit_facts": ["each hole radius is half the stated across-size", "'the same distance from the first hole as from the second' puts N3 on the perpendicular bisector of N1N2", "combined with the stated height, that pins N3 completely", "edge clearance is measured from the hole edge, so it uses clearance + hole radius from the centre"], "branch_facts": []} {"unique_id": "cad_t1_cad_three_hole_0087", "problem_id": "cad_t1_cad_three_hole_0087", "domain": "PyGeoX-CAD", "template": "cad_three_hole", "tier": 1, "nl_description": "A rectangular bracket measures 43.5 mm by 29.2 mm. Set up coordinates with the bottom-left corner at the origin P, the 43.5 mm edge running from P along the positive x-direction to Q, and the 29.2 mm edge running from P along the positive y-direction to S; R is the remaining corner. Three holes, each 6 mm across, are drilled through it. The centre of the first, N1, is 6.5 mm from edge PS and 6.5 mm from edge PQ. The centre of the second, N2, is 28.3 mm from edge PS and 16.8 mm from edge PQ. The third hole centre N3 is 12.4 mm from edge PQ, and it must sit the same distance from the first hole as from the second. At least 1.9 mm of material must remain between the third hole and every edge of the bracket.", "answer_schema": {"points": ["N1", "N2", "N3", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Three-hole bracket, third hole equidistant\n\nOracle formalization for cad_t1_cad_three_hole_0087. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(62)\n\nP, Q, R, S, N1, N2, N3 = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"N1\", \"N2\", \"N3\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the bracket (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the bracket (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the bracket lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 43.5, description=\"bracket is 43.5 long along PQ\")\nscene.constraint.eq(P.distance(S), 29.2, description=\"bracket is 29.2 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the bracket lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"bracket corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"bracket corner R closes the rectangle (y)\")\nscene.constraint.eq(N1.x, 6.5, description=\"hole 1 is 6.5 from the datum edge PS\")\nscene.constraint.eq(N1.y, 6.5, description=\"hole 1 is 6.5 from the datum edge PQ\")\nscene.constraint.eq(N2.x, 28.3, description=\"hole 2 is 28.3 from the datum edge PS\")\nscene.constraint.eq(N2.y, 16.8, description=\"hole 2 is 16.8 from the datum edge PQ\")\nscene.constraint.eq(N3.y, 12.4, description=\"hole 3 is 12.4 from the datum edge PQ\")\nscene.constraint.eq(N3.distance(N1), N3.distance(N2), description=\"hole 3 is the same distance from hole 1 as from hole 2\")\nscene.constraint.geq(N3.y, 4.9, description=\"at least 1.9 of material between hole 3 and edge PQ\")\nscene.constraint.geq(N3.x, 4.9, description=\"at least 1.9 of material between hole 3 and edge PS\")\nscene.constraint.geq(Q.x - N3.x, 4.9, description=\"at least 1.9 of material between hole 3 and edge QR\")\nscene.constraint.geq(S.y - N3.y, 4.9, description=\"at least 1.9 of material between hole 3 and edge SR\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [43.5, 0.0], "R": [43.5, 29.2], "S": [0.0, 29.2], "N1": [6.5, 6.5], "N2": [28.3, 16.8], "N3": [17.045642201834863, 12.4]}, "circles": {}}, "implicit_facts": ["each hole radius is half the stated across-size", "'the same distance from the first hole as from the second' puts N3 on the perpendicular bisector of N1N2", "combined with the stated height, that pins N3 completely", "edge clearance is measured from the hole edge, so it uses clearance + hole radius from the centre"], "branch_facts": []} {"unique_id": "cad_t1_cad_three_hole_0088", "problem_id": "cad_t1_cad_three_hole_0088", "domain": "PyGeoX-CAD", "template": "cad_three_hole", "tier": 1, "nl_description": "A rectangular bracket measures 41.4 mm by 30.4 mm. Set up coordinates with the bottom-left corner at the origin P, the 41.4 mm edge running from P along the positive x-direction to Q, and the 30.4 mm edge running from P along the positive y-direction to S; R is the remaining corner. Three holes, each 4.5 mm across, are drilled through it. The centre of the first, N1, is 10.6 mm from edge PS and 5 mm from edge PQ. The centre of the second, N2, is 28.1 mm from edge PS and 16.2 mm from edge PQ. The third hole centre N3 is 16 mm from edge PQ, and it must sit the same distance from the first hole as from the second. At least 2.3 mm of material must remain between the third hole and every edge of the bracket.", "answer_schema": {"points": ["N1", "N2", "N3", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Three-hole bracket, third hole equidistant\n\nOracle formalization for cad_t1_cad_three_hole_0088. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(59)\n\nP, Q, R, S, N1, N2, N3 = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"N1\", \"N2\", \"N3\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the bracket (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the bracket (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the bracket lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 41.4, description=\"bracket is 41.4 long along PQ\")\nscene.constraint.eq(P.distance(S), 30.4, description=\"bracket is 30.4 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the bracket lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"bracket corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"bracket corner R closes the rectangle (y)\")\nscene.constraint.eq(N1.x, 10.6, description=\"hole 1 is 10.6 from the datum edge PS\")\nscene.constraint.eq(N1.y, 5, description=\"hole 1 is 5 from the datum edge PQ\")\nscene.constraint.eq(N2.x, 28.1, description=\"hole 2 is 28.1 from the datum edge PS\")\nscene.constraint.eq(N2.y, 16.2, description=\"hole 2 is 16.2 from the datum edge PQ\")\nscene.constraint.eq(N3.y, 16, description=\"hole 3 is 16 from the datum edge PQ\")\nscene.constraint.eq(N3.distance(N1), N3.distance(N2), description=\"hole 3 is the same distance from hole 1 as from hole 2\")\nscene.constraint.geq(N3.y, 4.55, description=\"at least 2.3 of material between hole 3 and edge PQ\")\nscene.constraint.geq(N3.x, 4.55, description=\"at least 2.3 of material between hole 3 and edge PS\")\nscene.constraint.geq(Q.x - N3.x, 4.55, description=\"at least 2.3 of material between hole 3 and edge QR\")\nscene.constraint.geq(S.y - N3.y, 4.55, description=\"at least 2.3 of material between hole 3 and edge SR\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [41.4, 0.0], "R": [41.4, 30.4], "S": [0.0, 30.4], "N1": [10.6, 5.0], "N2": [28.1, 16.2], "N3": [15.894000000000002, 16.0]}, "circles": {}}, "implicit_facts": ["each hole radius is half the stated across-size", "'the same distance from the first hole as from the second' puts N3 on the perpendicular bisector of N1N2", "combined with the stated height, that pins N3 completely", "edge clearance is measured from the hole edge, so it uses clearance + hole radius from the centre"], "branch_facts": []} {"unique_id": "cad_t1_cad_three_hole_0089", "problem_id": "cad_t1_cad_three_hole_0089", "domain": "PyGeoX-CAD", "template": "cad_three_hole", "tier": 1, "nl_description": "A rectangular bracket measures 43.3 mm by 26.9 mm. Set up coordinates with the bottom-left corner at the origin P, the 43.3 mm edge running from P along the positive x-direction to Q, and the 26.9 mm edge running from P along the positive y-direction to S; R is the remaining corner. Three holes, each 4.8 mm across, are drilled through it. The centre of the first, N1, is 10.2 mm from edge PS and 5.1 mm from edge PQ. The centre of the second, N2, is 32.4 mm from edge PS and 16.1 mm from edge PQ. The third hole centre N3 is 9.4 mm from edge PQ, and it must sit the same distance from the first hole as from the second. At least 1.8 mm of material must remain between the third hole and every edge of the bracket.", "answer_schema": {"points": ["N1", "N2", "N3", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Three-hole bracket, third hole equidistant\n\nOracle formalization for cad_t1_cad_three_hole_0089. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(61)\n\nP, Q, R, S, N1, N2, N3 = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"N1\", \"N2\", \"N3\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the bracket (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the bracket (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the bracket lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 43.3, description=\"bracket is 43.3 long along PQ\")\nscene.constraint.eq(P.distance(S), 26.9, description=\"bracket is 26.9 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the bracket lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"bracket corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"bracket corner R closes the rectangle (y)\")\nscene.constraint.eq(N1.x, 10.2, description=\"hole 1 is 10.2 from the datum edge PS\")\nscene.constraint.eq(N1.y, 5.1, description=\"hole 1 is 5.1 from the datum edge PQ\")\nscene.constraint.eq(N2.x, 32.4, description=\"hole 2 is 32.4 from the datum edge PS\")\nscene.constraint.eq(N2.y, 16.1, description=\"hole 2 is 16.1 from the datum edge PQ\")\nscene.constraint.eq(N3.y, 9.4, description=\"hole 3 is 9.4 from the datum edge PQ\")\nscene.constraint.eq(N3.distance(N1), N3.distance(N2), description=\"hole 3 is the same distance from hole 1 as from hole 2\")\nscene.constraint.geq(N3.y, 4.2, description=\"at least 1.8 of material between hole 3 and edge PQ\")\nscene.constraint.geq(N3.x, 4.2, description=\"at least 1.8 of material between hole 3 and edge PS\")\nscene.constraint.geq(Q.x - N3.x, 4.2, description=\"at least 1.8 of material between hole 3 and edge QR\")\nscene.constraint.geq(S.y - N3.y, 4.2, description=\"at least 1.8 of material between hole 3 and edge SR\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [43.3, 0.0], "R": [43.3, 26.9], "S": [0.0, 26.9], "N1": [10.2, 5.1], "N2": [32.4, 16.1], "N3": [21.894594594594594, 9.4]}, "circles": {}}, "implicit_facts": ["each hole radius is half the stated across-size", "'the same distance from the first hole as from the second' puts N3 on the perpendicular bisector of N1N2", "combined with the stated height, that pins N3 completely", "edge clearance is measured from the hole edge, so it uses clearance + hole radius from the centre"], "branch_facts": []} {"unique_id": "cad_t1_cad_three_hole_0090", "problem_id": "cad_t1_cad_three_hole_0090", "domain": "PyGeoX-CAD", "template": "cad_three_hole", "tier": 1, "nl_description": "A rectangular bracket measures 43.8 mm by 26.3 mm. Set up coordinates with the bottom-left corner at the origin P, the 43.8 mm edge running from P along the positive x-direction to Q, and the 26.3 mm edge running from P along the positive y-direction to S; R is the remaining corner. Three holes, each 6.4 mm across, are drilled through it. The centre of the first, N1, is 7.9 mm from edge PS and 5.9 mm from edge PQ. The centre of the second, N2, is 31.3 mm from edge PS and 16.9 mm from edge PQ. The third hole centre N3 is 12.8 mm from edge PQ, and it must sit the same distance from the first hole as from the second. At least 2.1 mm of material must remain between the third hole and every edge of the bracket.", "answer_schema": {"points": ["N1", "N2", "N3", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Three-hole bracket, third hole equidistant\n\nOracle formalization for cad_t1_cad_three_hole_0090. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(62)\n\nP, Q, R, S, N1, N2, N3 = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"N1\", \"N2\", \"N3\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the bracket (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the bracket (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the bracket lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 43.8, description=\"bracket is 43.8 long along PQ\")\nscene.constraint.eq(P.distance(S), 26.3, description=\"bracket is 26.3 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the bracket lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"bracket corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"bracket corner R closes the rectangle (y)\")\nscene.constraint.eq(N1.x, 7.9, description=\"hole 1 is 7.9 from the datum edge PS\")\nscene.constraint.eq(N1.y, 5.9, description=\"hole 1 is 5.9 from the datum edge PQ\")\nscene.constraint.eq(N2.x, 31.3, description=\"hole 2 is 31.3 from the datum edge PS\")\nscene.constraint.eq(N2.y, 16.9, description=\"hole 2 is 16.9 from the datum edge PQ\")\nscene.constraint.eq(N3.y, 12.8, description=\"hole 3 is 12.8 from the datum edge PQ\")\nscene.constraint.eq(N3.distance(N1), N3.distance(N2), description=\"hole 3 is the same distance from hole 1 as from hole 2\")\nscene.constraint.geq(N3.y, 5.3, description=\"at least 2.1 of material between hole 3 and edge PQ\")\nscene.constraint.geq(N3.x, 5.3, description=\"at least 2.1 of material between hole 3 and edge PS\")\nscene.constraint.geq(Q.x - N3.x, 5.3, description=\"at least 2.1 of material between hole 3 and edge QR\")\nscene.constraint.geq(S.y - N3.y, 5.3, description=\"at least 2.1 of material between hole 3 and edge SR\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [43.8, 0.0], "R": [43.8, 26.3], "S": [0.0, 26.3], "N1": [7.9, 5.9], "N2": [31.3, 16.9], "N3": [18.941880341880342, 12.8]}, "circles": {}}, "implicit_facts": ["each hole radius is half the stated across-size", "'the same distance from the first hole as from the second' puts N3 on the perpendicular bisector of N1N2", "combined with the stated height, that pins N3 completely", "edge clearance is measured from the hole edge, so it uses clearance + hole radius from the centre"], "branch_facts": []} {"unique_id": "cad_t1_cad_three_hole_0091", "problem_id": "cad_t1_cad_three_hole_0091", "domain": "PyGeoX-CAD", "template": "cad_three_hole", "tier": 1, "nl_description": "A rectangular bracket measures 38.6 mm by 22.9 mm. Set up coordinates with the bottom-left corner at the origin P, the 38.6 mm edge running from P along the positive x-direction to Q, and the 22.9 mm edge running from P along the positive y-direction to S; R is the remaining corner. Three holes, each 6.5 mm across, are drilled through it. The centre of the first, N1, is 11.1 mm from edge PS and 8.5 mm from edge PQ. The centre of the second, N2, is 30.5 mm from edge PS and 14.6 mm from edge PQ. The third hole centre N3 is 9.6 mm from edge PQ, and it must sit the same distance from the first hole as from the second. At least 3 mm of material must remain between the third hole and every edge of the bracket.", "answer_schema": {"points": ["N1", "N2", "N3", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Three-hole bracket, third hole equidistant\n\nOracle formalization for cad_t1_cad_three_hole_0091. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(55)\n\nP, Q, R, S, N1, N2, N3 = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"N1\", \"N2\", \"N3\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the bracket (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the bracket (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the bracket lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 38.6, description=\"bracket is 38.6 long along PQ\")\nscene.constraint.eq(P.distance(S), 22.9, description=\"bracket is 22.9 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the bracket lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"bracket corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"bracket corner R closes the rectangle (y)\")\nscene.constraint.eq(N1.x, 11.1, description=\"hole 1 is 11.1 from the datum edge PS\")\nscene.constraint.eq(N1.y, 8.5, description=\"hole 1 is 8.5 from the datum edge PQ\")\nscene.constraint.eq(N2.x, 30.5, description=\"hole 2 is 30.5 from the datum edge PS\")\nscene.constraint.eq(N2.y, 14.6, description=\"hole 2 is 14.6 from the datum edge PQ\")\nscene.constraint.eq(N3.y, 9.6, description=\"hole 3 is 9.6 from the datum edge PQ\")\nscene.constraint.eq(N3.distance(N1), N3.distance(N2), description=\"hole 3 is the same distance from hole 1 as from hole 2\")\nscene.constraint.geq(N3.y, 6.25, description=\"at least 3 of material between hole 3 and edge PQ\")\nscene.constraint.geq(N3.x, 6.25, description=\"at least 3 of material between hole 3 and edge PS\")\nscene.constraint.geq(Q.x - N3.x, 6.25, description=\"at least 3 of material between hole 3 and edge QR\")\nscene.constraint.geq(S.y - N3.y, 6.25, description=\"at least 3 of material between hole 3 and edge SR\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [38.6, 0.0], "R": [38.6, 22.9], "S": [0.0, 22.9], "N1": [11.1, 8.5], "N2": [30.5, 14.6], "N3": [21.41314432989691, 9.6]}, "circles": {}}, "implicit_facts": ["each hole radius is half the stated across-size", "'the same distance from the first hole as from the second' puts N3 on the perpendicular bisector of N1N2", "combined with the stated height, that pins N3 completely", "edge clearance is measured from the hole edge, so it uses clearance + hole radius from the centre"], "branch_facts": []} {"unique_id": "cad_t1_cad_three_hole_0092", "problem_id": "cad_t1_cad_three_hole_0092", "domain": "PyGeoX-CAD", "template": "cad_three_hole", "tier": 1, "nl_description": "A rectangular bracket measures 44.9 mm by 27.3 mm. Set up coordinates with the bottom-left corner at the origin P, the 44.9 mm edge running from P along the positive x-direction to Q, and the 27.3 mm edge running from P along the positive y-direction to S; R is the remaining corner. Three holes, each 6.7 mm across, are drilled through it. The centre of the first, N1, is 6.9 mm from edge PS and 8.3 mm from edge PQ. The centre of the second, N2, is 27.5 mm from edge PS and 15.7 mm from edge PQ. The third hole centre N3 is 9.9 mm from edge PQ, and it must sit the same distance from the first hole as from the second. At least 2.6 mm of material must remain between the third hole and every edge of the bracket.", "answer_schema": {"points": ["N1", "N2", "N3", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Three-hole bracket, third hole equidistant\n\nOracle formalization for cad_t1_cad_three_hole_0092. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(63)\n\nP, Q, R, S, N1, N2, N3 = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"N1\", \"N2\", \"N3\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the bracket (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the bracket (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the bracket lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 44.9, description=\"bracket is 44.9 long along PQ\")\nscene.constraint.eq(P.distance(S), 27.3, description=\"bracket is 27.3 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the bracket lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"bracket corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"bracket corner R closes the rectangle (y)\")\nscene.constraint.eq(N1.x, 6.9, description=\"hole 1 is 6.9 from the datum edge PS\")\nscene.constraint.eq(N1.y, 8.3, description=\"hole 1 is 8.3 from the datum edge PQ\")\nscene.constraint.eq(N2.x, 27.5, description=\"hole 2 is 27.5 from the datum edge PS\")\nscene.constraint.eq(N2.y, 15.7, description=\"hole 2 is 15.7 from the datum edge PQ\")\nscene.constraint.eq(N3.y, 9.9, description=\"hole 3 is 9.9 from the datum edge PQ\")\nscene.constraint.eq(N3.distance(N1), N3.distance(N2), description=\"hole 3 is the same distance from hole 1 as from hole 2\")\nscene.constraint.geq(N3.y, 5.95, description=\"at least 2.6 of material between hole 3 and edge PQ\")\nscene.constraint.geq(N3.x, 5.95, description=\"at least 2.6 of material between hole 3 and edge PS\")\nscene.constraint.geq(Q.x - N3.x, 5.95, description=\"at least 2.6 of material between hole 3 and edge QR\")\nscene.constraint.geq(S.y - N3.y, 5.95, description=\"at least 2.6 of material between hole 3 and edge SR\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [44.9, 0.0], "R": [44.9, 27.3], "S": [0.0, 27.3], "N1": [6.9, 8.3], "N2": [27.5, 15.7], "N3": [17.954368932038832, 9.9]}, "circles": {}}, "implicit_facts": ["each hole radius is half the stated across-size", "'the same distance from the first hole as from the second' puts N3 on the perpendicular bisector of N1N2", "combined with the stated height, that pins N3 completely", "edge clearance is measured from the hole edge, so it uses clearance + hole radius from the centre"], "branch_facts": []} {"unique_id": "cad_t2_cad_bolt_circle_0019", "problem_id": "cad_t2_cad_bolt_circle_0019", "domain": "PyGeoX-CAD", "template": "cad_bolt_circle", "tier": 2, "nl_description": "A circular flange is drilled with 4 bolt holes spaced evenly around a bolt circle that is 25.2 mm across. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; number the hole centres H1, H2, H3, H4 going anticlockwise. Each bolt hole is 3.6 mm across. A circular boss of radius 3.8 mm sits at the centre of the flange, and the design requires at least 2.3 mm of material to remain between the boss and every bolt hole.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle on a flange\n\nOracle formalization for cad_t2_cad_bolt_circle_0019. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(21)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(2 * bolt_circle.radius, 25.2, description=\"the bolt circle is 25.2 across\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 90, description=\"holes H1 and H2 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 90, description=\"holes H2 and H3 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 90, description=\"holes H3 and H4 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\nscene.constraint.geq(O.distance(H1) - 1.8, 6.1, description=\"at least 2.3 of material between the central boss and hole H1\")\nscene.constraint.geq(O.distance(H2) - 1.8, 6.1, description=\"at least 2.3 of material between the central boss and hole H2\")\nscene.constraint.geq(O.distance(H3) - 1.8, 6.1, description=\"at least 2.3 of material between the central boss and hole H3\")\nscene.constraint.geq(O.distance(H4) - 1.8, 6.1, description=\"at least 2.3 of material between the central boss and hole H4\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [12.6, 0.0], "H2": [7.715274834628325e-16, 12.6], "H3": [-12.6, 1.543054966925665e-15], "H4": [-2.3145824503884972e-15, -12.6]}, "circles": {"O": 12.6}}, "implicit_facts": ["the bolt circle RADIUS is half the stated across-size", "evenly spaced means adjacent hole centres subtend 90 degrees at the centre", "every hole centre is the same distance from the flange centre", "the clearance is measured from the hole EDGE, so it uses the hole radius (half the stated hole size)"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cad_t2_cad_bolt_circle_0020", "problem_id": "cad_t2_cad_bolt_circle_0020", "domain": "PyGeoX-CAD", "template": "cad_bolt_circle", "tier": 2, "nl_description": "A circular flange is drilled with 4 bolt holes spaced evenly around a bolt circle that is 35.4 mm across. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; number the hole centres H1, H2, H3, H4 going anticlockwise. Each bolt hole is 4.7 mm across. A circular boss of radius 4.2 mm sits at the centre of the flange, and the design requires at least 2.2 mm of material to remain between the boss and every bolt hole.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle on a flange\n\nOracle formalization for cad_t2_cad_bolt_circle_0020. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(28)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(2 * bolt_circle.radius, 35.4, description=\"the bolt circle is 35.4 across\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 90, description=\"holes H1 and H2 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 90, description=\"holes H2 and H3 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 90, description=\"holes H3 and H4 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\nscene.constraint.geq(O.distance(H1) - 2.35, 6.4, description=\"at least 2.2 of material between the central boss and hole H1\")\nscene.constraint.geq(O.distance(H2) - 2.35, 6.4, description=\"at least 2.2 of material between the central boss and hole H2\")\nscene.constraint.geq(O.distance(H3) - 2.35, 6.4, description=\"at least 2.2 of material between the central boss and hole H3\")\nscene.constraint.geq(O.distance(H4) - 2.35, 6.4, description=\"at least 2.2 of material between the central boss and hole H4\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [17.7, 0.0], "H2": [1.0838124172454075e-15, 17.7], "H3": [-17.7, 2.167624834490815e-15], "H4": [-3.2514372517362224e-15, -17.7]}, "circles": {"O": 17.7}}, "implicit_facts": ["the bolt circle RADIUS is half the stated across-size", "evenly spaced means adjacent hole centres subtend 90 degrees at the centre", "every hole centre is the same distance from the flange centre", "the clearance is measured from the hole EDGE, so it uses the hole radius (half the stated hole size)"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cad_t2_cad_bolt_circle_0021", "problem_id": "cad_t2_cad_bolt_circle_0021", "domain": "PyGeoX-CAD", "template": "cad_bolt_circle", "tier": 2, "nl_description": "A circular flange is drilled with 4 bolt holes spaced evenly around a bolt circle that is 35.9 mm across. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; number the hole centres H1, H2, H3, H4 going anticlockwise. Each bolt hole is 4.6 mm across. A circular boss of radius 3.2 mm sits at the centre of the flange, and the design requires at least 2.3 mm of material to remain between the boss and every bolt hole.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle on a flange\n\nOracle formalization for cad_t2_cad_bolt_circle_0021. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(28)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(2 * bolt_circle.radius, 35.9, description=\"the bolt circle is 35.9 across\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 90, description=\"holes H1 and H2 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 90, description=\"holes H2 and H3 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 90, description=\"holes H3 and H4 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\nscene.constraint.geq(O.distance(H1) - 2.3, 5.5, description=\"at least 2.3 of material between the central boss and hole H1\")\nscene.constraint.geq(O.distance(H2) - 2.3, 5.5, description=\"at least 2.3 of material between the central boss and hole H2\")\nscene.constraint.geq(O.distance(H3) - 2.3, 5.5, description=\"at least 2.3 of material between the central boss and hole H3\")\nscene.constraint.geq(O.distance(H4) - 2.3, 5.5, description=\"at least 2.3 of material between the central boss and hole H4\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [17.95, 0.0], "H2": [1.0991205022347494e-15, 17.95], "H3": [-17.95, 2.1982410044694987e-15], "H4": [-3.297361506704248e-15, -17.95]}, "circles": {"O": 17.95}}, "implicit_facts": ["the bolt circle RADIUS is half the stated across-size", "evenly spaced means adjacent hole centres subtend 90 degrees at the centre", "every hole centre is the same distance from the flange centre", "the clearance is measured from the hole EDGE, so it uses the hole radius (half the stated hole size)"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cad_t2_cad_bolt_circle_0022", "problem_id": "cad_t2_cad_bolt_circle_0022", "domain": "PyGeoX-CAD", "template": "cad_bolt_circle", "tier": 2, "nl_description": "A circular flange is drilled with 6 bolt holes spaced evenly around a bolt circle that is 31.7 mm across. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; number the hole centres H1, H2, H3, H4, H5, H6 going anticlockwise. Each bolt hole is 4.3 mm across. A circular boss of radius 5.9 mm sits at the centre of the flange, and the design requires at least 2.9 mm of material to remain between the boss and every bolt hole.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "H5", "H6", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle on a flange\n\nOracle formalization for cad_t2_cad_bolt_circle_0022. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(26)\n\nO, H1, H2, H3, H4, H5, H6 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\", \"H5\", \"H6\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(2 * bolt_circle.radius, 31.7, description=\"the bolt circle is 31.7 across\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.relate.point_lies_on(H5, bolt_circle)\nscene.relate.point_lies_on(H6, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 60, description=\"holes H1 and H2 are adjacent on an evenly spaced set of 6\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 60, description=\"holes H2 and H3 are adjacent on an evenly spaced set of 6\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 60, description=\"holes H3 and H4 are adjacent on an evenly spaced set of 6\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\nscene.constraint.eq(scene.add.angle(H4, O, H5).value, 60, description=\"holes H4 and H5 are adjacent on an evenly spaced set of 6\")\nscene.constraint.gt(cross(O, H4, H5), 0.05, description=\"the holes are numbered anticlockwise, so H5 follows H4\")\nscene.constraint.eq(scene.add.angle(H5, O, H6).value, 60, description=\"holes H5 and H6 are adjacent on an evenly spaced set of 6\")\nscene.constraint.gt(cross(O, H5, H6), 0.05, description=\"the holes are numbered anticlockwise, so H6 follows H5\")\nscene.constraint.geq(O.distance(H1) - 2.15, 8.8, description=\"at least 2.9 of material between the central boss and hole H1\")\nscene.constraint.geq(O.distance(H2) - 2.15, 8.8, description=\"at least 2.9 of material between the central boss and hole H2\")\nscene.constraint.geq(O.distance(H3) - 2.15, 8.8, description=\"at least 2.9 of material between the central boss and hole H3\")\nscene.constraint.geq(O.distance(H4) - 2.15, 8.8, description=\"at least 2.9 of material between the central boss and hole H4\")\nscene.constraint.geq(O.distance(H5) - 2.15, 8.8, description=\"at least 2.9 of material between the central boss and hole H5\")\nscene.constraint.geq(O.distance(H6) - 2.15, 8.8, description=\"at least 2.9 of material between the central boss and hole H6\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [15.85, 0.0], "H2": [7.925000000000002, 13.72650264998335], "H3": [-7.924999999999996, 13.726502649983352], "H4": [-15.85, 1.9410651766485547e-15], "H5": [-7.925000000000007, -13.726502649983349], "H6": [7.925000000000002, -13.72650264998335]}, "circles": {"O": 15.85}}, "implicit_facts": ["the bolt circle RADIUS is half the stated across-size", "evenly spaced means adjacent hole centres subtend 60 degrees at the centre", "every hole centre is the same distance from the flange centre", "the clearance is measured from the hole EDGE, so it uses the hole radius (half the stated hole size)"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cad_t2_cad_bolt_circle_0023", "problem_id": "cad_t2_cad_bolt_circle_0023", "domain": "PyGeoX-CAD", "template": "cad_bolt_circle", "tier": 2, "nl_description": "A circular flange is drilled with 4 bolt holes spaced evenly around a bolt circle that is 23.5 mm across. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; number the hole centres H1, H2, H3, H4 going anticlockwise. Each bolt hole is 4.4 mm across. A circular boss of radius 4.5 mm sits at the centre of the flange, and the design requires at least 2.9 mm of material to remain between the boss and every bolt hole.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle on a flange\n\nOracle formalization for cad_t2_cad_bolt_circle_0023. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(20)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(2 * bolt_circle.radius, 23.5, description=\"the bolt circle is 23.5 across\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 90, description=\"holes H1 and H2 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 90, description=\"holes H2 and H3 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 90, description=\"holes H3 and H4 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\nscene.constraint.geq(O.distance(H1) - 2.2, 7.4, description=\"at least 2.9 of material between the central boss and hole H1\")\nscene.constraint.geq(O.distance(H2) - 2.2, 7.4, description=\"at least 2.9 of material between the central boss and hole H2\")\nscene.constraint.geq(O.distance(H3) - 2.2, 7.4, description=\"at least 2.9 of material between the central boss and hole H3\")\nscene.constraint.geq(O.distance(H4) - 2.2, 7.4, description=\"at least 2.9 of material between the central boss and hole H4\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [11.75, 0.0], "H2": [7.1947999449907e-16, 11.75], "H3": [-11.75, 1.43895998899814e-15], "H4": [-2.1584399834972097e-15, -11.75]}, "circles": {"O": 11.75}}, "implicit_facts": ["the bolt circle RADIUS is half the stated across-size", "evenly spaced means adjacent hole centres subtend 90 degrees at the centre", "every hole centre is the same distance from the flange centre", "the clearance is measured from the hole EDGE, so it uses the hole radius (half the stated hole size)"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cad_t2_cad_bolt_circle_0024", "problem_id": "cad_t2_cad_bolt_circle_0024", "domain": "PyGeoX-CAD", "template": "cad_bolt_circle", "tier": 2, "nl_description": "A circular flange is drilled with 5 bolt holes spaced evenly around a bolt circle that is 28 mm across. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; number the hole centres H1, H2, H3, H4, H5 going anticlockwise. Each bolt hole is 3.5 mm across. A circular boss of radius 5.6 mm sits at the centre of the flange, and the design requires at least 2.8 mm of material to remain between the boss and every bolt hole.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "H5", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle on a flange\n\nOracle formalization for cad_t2_cad_bolt_circle_0024. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(23)\n\nO, H1, H2, H3, H4, H5 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\", \"H5\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(2 * bolt_circle.radius, 28, description=\"the bolt circle is 28 across\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.relate.point_lies_on(H5, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 72, description=\"holes H1 and H2 are adjacent on an evenly spaced set of 5\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 72, description=\"holes H2 and H3 are adjacent on an evenly spaced set of 5\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 72, description=\"holes H3 and H4 are adjacent on an evenly spaced set of 5\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\nscene.constraint.eq(scene.add.angle(H4, O, H5).value, 72, description=\"holes H4 and H5 are adjacent on an evenly spaced set of 5\")\nscene.constraint.gt(cross(O, H4, H5), 0.05, description=\"the holes are numbered anticlockwise, so H5 follows H4\")\nscene.constraint.geq(O.distance(H1) - 1.75, 8.4, description=\"at least 2.8 of material between the central boss and hole H1\")\nscene.constraint.geq(O.distance(H2) - 1.75, 8.4, description=\"at least 2.8 of material between the central boss and hole H2\")\nscene.constraint.geq(O.distance(H3) - 1.75, 8.4, description=\"at least 2.8 of material between the central boss and hole H3\")\nscene.constraint.geq(O.distance(H4) - 1.75, 8.4, description=\"at least 2.8 of material between the central boss and hole H4\")\nscene.constraint.geq(O.distance(H5) - 1.75, 8.4, description=\"at least 2.8 of material between the central boss and hole H5\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [14.0, 0.0], "H2": [4.326237921249264, 13.31479122813215], "H3": [-11.326237921249263, 8.228993532094625], "H4": [-11.326237921249264, -8.228993532094622], "H5": [4.326237921249261, -13.314791228132151]}, "circles": {"O": 14.0}}, "implicit_facts": ["the bolt circle RADIUS is half the stated across-size", "evenly spaced means adjacent hole centres subtend 72 degrees at the centre", "every hole centre is the same distance from the flange centre", "the clearance is measured from the hole EDGE, so it uses the hole radius (half the stated hole size)"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cad_t2_cad_bolt_circle_0025", "problem_id": "cad_t2_cad_bolt_circle_0025", "domain": "PyGeoX-CAD", "template": "cad_bolt_circle", "tier": 2, "nl_description": "A circular flange is drilled with 4 bolt holes spaced evenly around a bolt circle that is 25.7 mm across. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; number the hole centres H1, H2, H3, H4 going anticlockwise. Each bolt hole is 4.6 mm across. A circular boss of radius 4.1 mm sits at the centre of the flange, and the design requires at least 3 mm of material to remain between the boss and every bolt hole.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle on a flange\n\nOracle formalization for cad_t2_cad_bolt_circle_0025. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(22)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(2 * bolt_circle.radius, 25.7, description=\"the bolt circle is 25.7 across\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 90, description=\"holes H1 and H2 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 90, description=\"holes H2 and H3 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 90, description=\"holes H3 and H4 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\nscene.constraint.geq(O.distance(H1) - 2.3, 7.1, description=\"at least 3 of material between the central boss and hole H1\")\nscene.constraint.geq(O.distance(H2) - 2.3, 7.1, description=\"at least 3 of material between the central boss and hole H2\")\nscene.constraint.geq(O.distance(H3) - 2.3, 7.1, description=\"at least 3 of material between the central boss and hole H3\")\nscene.constraint.geq(O.distance(H4) - 2.3, 7.1, description=\"at least 3 of material between the central boss and hole H4\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [12.85, 0.0], "H2": [7.868355684521744e-16, 12.85], "H3": [-12.85, 1.5736711369043489e-15], "H4": [-2.360506705356523e-15, -12.85]}, "circles": {"O": 12.85}}, "implicit_facts": ["the bolt circle RADIUS is half the stated across-size", "evenly spaced means adjacent hole centres subtend 90 degrees at the centre", "every hole centre is the same distance from the flange centre", "the clearance is measured from the hole EDGE, so it uses the hole radius (half the stated hole size)"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cad_t2_cad_bolt_circle_0026", "problem_id": "cad_t2_cad_bolt_circle_0026", "domain": "PyGeoX-CAD", "template": "cad_bolt_circle", "tier": 2, "nl_description": "A circular flange is drilled with 4 bolt holes spaced evenly around a bolt circle that is 30.2 mm across. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; number the hole centres H1, H2, H3, H4 going anticlockwise. Each bolt hole is 3.9 mm across. A circular boss of radius 4.8 mm sits at the centre of the flange, and the design requires at least 2.3 mm of material to remain between the boss and every bolt hole.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle on a flange\n\nOracle formalization for cad_t2_cad_bolt_circle_0026. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(25)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(2 * bolt_circle.radius, 30.2, description=\"the bolt circle is 30.2 across\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 90, description=\"holes H1 and H2 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 90, description=\"holes H2 and H3 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 90, description=\"holes H3 and H4 are adjacent on an evenly spaced set of 4\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\nscene.constraint.geq(O.distance(H1) - 1.95, 7.1, description=\"at least 2.3 of material between the central boss and hole H1\")\nscene.constraint.geq(O.distance(H2) - 1.95, 7.1, description=\"at least 2.3 of material between the central boss and hole H2\")\nscene.constraint.geq(O.distance(H3) - 1.95, 7.1, description=\"at least 2.3 of material between the central boss and hole H3\")\nscene.constraint.geq(O.distance(H4) - 1.95, 7.1, description=\"at least 2.3 of material between the central boss and hole H4\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [15.1, 0.0], "H2": [9.246083333562517e-16, 15.1], "H3": [-15.1, 1.8492166667125033e-15], "H4": [-2.773825000068755e-15, -15.1]}, "circles": {"O": 15.1}}, "implicit_facts": ["the bolt circle RADIUS is half the stated across-size", "evenly spaced means adjacent hole centres subtend 90 degrees at the centre", "every hole centre is the same distance from the flange centre", "the clearance is measured from the hole EDGE, so it uses the hole radius (half the stated hole size)"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cad_t2_cad_bolt_circle_0027", "problem_id": "cad_t2_cad_bolt_circle_0027", "domain": "PyGeoX-CAD", "template": "cad_bolt_circle", "tier": 2, "nl_description": "A circular flange is drilled with 5 bolt holes spaced evenly around a bolt circle that is 23.9 mm across. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; number the hole centres H1, H2, H3, H4, H5 going anticlockwise. Each bolt hole is 3.1 mm across. A circular boss of radius 3.4 mm sits at the centre of the flange, and the design requires at least 1.7 mm of material to remain between the boss and every bolt hole.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "H5", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle on a flange\n\nOracle formalization for cad_t2_cad_bolt_circle_0027. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(21)\n\nO, H1, H2, H3, H4, H5 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\", \"H5\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(2 * bolt_circle.radius, 23.9, description=\"the bolt circle is 23.9 across\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.relate.point_lies_on(H5, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 72, description=\"holes H1 and H2 are adjacent on an evenly spaced set of 5\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 72, description=\"holes H2 and H3 are adjacent on an evenly spaced set of 5\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 72, description=\"holes H3 and H4 are adjacent on an evenly spaced set of 5\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\nscene.constraint.eq(scene.add.angle(H4, O, H5).value, 72, description=\"holes H4 and H5 are adjacent on an evenly spaced set of 5\")\nscene.constraint.gt(cross(O, H4, H5), 0.05, description=\"the holes are numbered anticlockwise, so H5 follows H4\")\nscene.constraint.geq(O.distance(H1) - 1.55, 5.1, description=\"at least 1.7 of material between the central boss and hole H1\")\nscene.constraint.geq(O.distance(H2) - 1.55, 5.1, description=\"at least 1.7 of material between the central boss and hole H2\")\nscene.constraint.geq(O.distance(H3) - 1.55, 5.1, description=\"at least 1.7 of material between the central boss and hole H3\")\nscene.constraint.geq(O.distance(H4) - 1.55, 5.1, description=\"at least 1.7 of material between the central boss and hole H4\")\nscene.constraint.geq(O.distance(H5) - 1.55, 5.1, description=\"at least 1.7 of material between the central boss and hole H5\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [11.95, 0.0], "H2": [3.6927530827806216, 11.365125369727084], "H3": [-9.66775308278062, 7.024033764895055], "H4": [-9.66775308278062, -7.024033764895052], "H5": [3.692753082780619, -11.365125369727085]}, "circles": {"O": 11.95}}, "implicit_facts": ["the bolt circle RADIUS is half the stated across-size", "evenly spaced means adjacent hole centres subtend 72 degrees at the centre", "every hole centre is the same distance from the flange centre", "the clearance is measured from the hole EDGE, so it uses the hole radius (half the stated hole size)"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cad_t2_cad_chamfer_corner_0093", "problem_id": "cad_t2_cad_chamfer_corner_0093", "domain": "PyGeoX-CAD", "template": "cad_chamfer_corner", "tier": 2, "nl_description": "A rectangular plate measures 35.3 mm by 31.5 mm. Set up coordinates with the bottom-left corner at the origin P, the 35.3 mm edge running from P along the positive x-direction to Q, and the 31.5 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut that takes 8.6 mm off along each of the two edges that meet there: the cut starts at G1, which lies on edge QR, and finishes at G2, which lies on edge SR - both of them on the plate, between the corners named. A hole 4.4 mm across is drilled with its centre N at 16.1 mm from edge PS and 12.4 mm from edge PQ. At least 2.8 mm of material must remain between the hole and the chamfer face, and between the hole and the two datum edges.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Chamfered corner with a clearance hole\n\nOracle formalization for cad_t2_cad_chamfer_corner_0093. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(51)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 35.3, description=\"plate is 35.3 long along PQ\")\nscene.constraint.eq(P.distance(S), 31.5, description=\"plate is 31.5 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 8.6, description=\"the chamfer takes 8.6 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 8.6, description=\"the chamfer takes 8.6 off along edge SR\")\nscene.constraint.eq(N.x, 16.1, description=\"the hole centre is 16.1 from the datum edge PS\")\nscene.constraint.eq(N.y, 12.4, description=\"the hole centre is 12.4 from the datum edge PQ\")\nscene.constraint.geq(cross(G1, G2, N) / G1.distance(G2), 5, description=\"at least 2.8 of material between the hole and the chamfer face\")\nscene.constraint.geq(N.x, 5, description=\"at least 2.8 of material between the hole and edge PS\")\nscene.constraint.geq(N.y, 5, description=\"at least 2.8 of material between the hole and edge PQ\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [35.3, 0.0], "R": [35.3, 31.5], "S": [0.0, 31.5], "G1": [35.3, 22.9], "G2": [26.699999999999996, 31.5], "N": [16.1, 12.4]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "the chamfer endpoints are each g from the corner R, measured ALONG the edges", "the clearance to the chamfer is the perpendicular distance from N to the line G1G2", "edge clearance is measured from the hole edge, so it uses clearance + hole radius"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cad_t2_cad_chamfer_corner_0094", "problem_id": "cad_t2_cad_chamfer_corner_0094", "domain": "PyGeoX-CAD", "template": "cad_chamfer_corner", "tier": 2, "nl_description": "A rectangular plate measures 42.9 mm by 24.7 mm. Set up coordinates with the bottom-left corner at the origin P, the 42.9 mm edge running from P along the positive x-direction to Q, and the 24.7 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut that takes 9.2 mm off along each of the two edges that meet there: the cut starts at G1, which lies on edge QR, and finishes at G2, which lies on edge SR - both of them on the plate, between the corners named. A hole 5.7 mm across is drilled with its centre N at 16.2 mm from edge PS and 9.1 mm from edge PQ. At least 2.7 mm of material must remain between the hole and the chamfer face, and between the hole and the two datum edges.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Chamfered corner with a clearance hole\n\nOracle formalization for cad_t2_cad_chamfer_corner_0094. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(61)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 42.9, description=\"plate is 42.9 long along PQ\")\nscene.constraint.eq(P.distance(S), 24.7, description=\"plate is 24.7 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 9.2, description=\"the chamfer takes 9.2 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 9.2, description=\"the chamfer takes 9.2 off along edge SR\")\nscene.constraint.eq(N.x, 16.2, description=\"the hole centre is 16.2 from the datum edge PS\")\nscene.constraint.eq(N.y, 9.1, description=\"the hole centre is 9.1 from the datum edge PQ\")\nscene.constraint.geq(cross(G1, G2, N) / G1.distance(G2), 5.55, description=\"at least 2.7 of material between the hole and the chamfer face\")\nscene.constraint.geq(N.x, 5.55, description=\"at least 2.7 of material between the hole and edge PS\")\nscene.constraint.geq(N.y, 5.55, description=\"at least 2.7 of material between the hole and edge PQ\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [42.9, 0.0], "R": [42.9, 24.7], "S": [0.0, 24.7], "G1": [42.9, 15.5], "G2": [33.7, 24.7], "N": [16.2, 9.1]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "the chamfer endpoints are each g from the corner R, measured ALONG the edges", "the clearance to the chamfer is the perpendicular distance from N to the line G1G2", "edge clearance is measured from the hole edge, so it uses clearance + hole radius"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cad_t2_cad_chamfer_corner_0095", "problem_id": "cad_t2_cad_chamfer_corner_0095", "domain": "PyGeoX-CAD", "template": "cad_chamfer_corner", "tier": 2, "nl_description": "A rectangular plate measures 39.3 mm by 31.3 mm. Set up coordinates with the bottom-left corner at the origin P, the 39.3 mm edge running from P along the positive x-direction to Q, and the 31.3 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut that takes 12.1 mm off along each of the two edges that meet there: the cut starts at G1, which lies on edge QR, and finishes at G2, which lies on edge SR - both of them on the plate, between the corners named. A hole 4.3 mm across is drilled with its centre N at 17.1 mm from edge PS and 10.5 mm from edge PQ. At least 2.6 mm of material must remain between the hole and the chamfer face, and between the hole and the two datum edges.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Chamfered corner with a clearance hole\n\nOracle formalization for cad_t2_cad_chamfer_corner_0095. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(56)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 39.3, description=\"plate is 39.3 long along PQ\")\nscene.constraint.eq(P.distance(S), 31.3, description=\"plate is 31.3 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 12.1, description=\"the chamfer takes 12.1 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 12.1, description=\"the chamfer takes 12.1 off along edge SR\")\nscene.constraint.eq(N.x, 17.1, description=\"the hole centre is 17.1 from the datum edge PS\")\nscene.constraint.eq(N.y, 10.5, description=\"the hole centre is 10.5 from the datum edge PQ\")\nscene.constraint.geq(cross(G1, G2, N) / G1.distance(G2), 4.75, description=\"at least 2.6 of material between the hole and the chamfer face\")\nscene.constraint.geq(N.x, 4.75, description=\"at least 2.6 of material between the hole and edge PS\")\nscene.constraint.geq(N.y, 4.75, description=\"at least 2.6 of material between the hole and edge PQ\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [39.3, 0.0], "R": [39.3, 31.3], "S": [0.0, 31.3], "G1": [39.3, 19.200000000000003], "G2": [27.199999999999996, 31.3], "N": [17.1, 10.5]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "the chamfer endpoints are each g from the corner R, measured ALONG the edges", "the clearance to the chamfer is the perpendicular distance from N to the line G1G2", "edge clearance is measured from the hole edge, so it uses clearance + hole radius"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cad_t2_cad_chamfer_corner_0096", "problem_id": "cad_t2_cad_chamfer_corner_0096", "domain": "PyGeoX-CAD", "template": "cad_chamfer_corner", "tier": 2, "nl_description": "A rectangular plate measures 33.6 mm by 25.1 mm. Set up coordinates with the bottom-left corner at the origin P, the 33.6 mm edge running from P along the positive x-direction to Q, and the 25.1 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut that takes 12.6 mm off along each of the two edges that meet there: the cut starts at G1, which lies on edge QR, and finishes at G2, which lies on edge SR - both of them on the plate, between the corners named. A hole 4 mm across is drilled with its centre N at 15.3 mm from edge PS and 13.7 mm from edge PQ. At least 1.6 mm of material must remain between the hole and the chamfer face, and between the hole and the two datum edges.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Chamfered corner with a clearance hole\n\nOracle formalization for cad_t2_cad_chamfer_corner_0096. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(49)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 33.6, description=\"plate is 33.6 long along PQ\")\nscene.constraint.eq(P.distance(S), 25.1, description=\"plate is 25.1 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 12.6, description=\"the chamfer takes 12.6 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 12.6, description=\"the chamfer takes 12.6 off along edge SR\")\nscene.constraint.eq(N.x, 15.3, description=\"the hole centre is 15.3 from the datum edge PS\")\nscene.constraint.eq(N.y, 13.7, description=\"the hole centre is 13.7 from the datum edge PQ\")\nscene.constraint.geq(cross(G1, G2, N) / G1.distance(G2), 3.6, description=\"at least 1.6 of material between the hole and the chamfer face\")\nscene.constraint.geq(N.x, 3.6, description=\"at least 1.6 of material between the hole and edge PS\")\nscene.constraint.geq(N.y, 3.6, description=\"at least 1.6 of material between the hole and edge PQ\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [33.6, 0.0], "R": [33.6, 25.1], "S": [0.0, 25.1], "G1": [33.6, 12.500000000000002], "G2": [21.0, 25.1], "N": [15.3, 13.7]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "the chamfer endpoints are each g from the corner R, measured ALONG the edges", "the clearance to the chamfer is the perpendicular distance from N to the line G1G2", "edge clearance is measured from the hole edge, so it uses clearance + hole radius"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cad_t2_cad_chamfer_corner_0097", "problem_id": "cad_t2_cad_chamfer_corner_0097", "domain": "PyGeoX-CAD", "template": "cad_chamfer_corner", "tier": 2, "nl_description": "A rectangular plate measures 41.2 mm by 23.3 mm. Set up coordinates with the bottom-left corner at the origin P, the 41.2 mm edge running from P along the positive x-direction to Q, and the 23.3 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut that takes 9 mm off along each of the two edges that meet there: the cut starts at G1, which lies on edge QR, and finishes at G2, which lies on edge SR - both of them on the plate, between the corners named. A hole 6.4 mm across is drilled with its centre N at 17.3 mm from edge PS and 10.3 mm from edge PQ. At least 2.1 mm of material must remain between the hole and the chamfer face, and between the hole and the two datum edges.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Chamfered corner with a clearance hole\n\nOracle formalization for cad_t2_cad_chamfer_corner_0097. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(59)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 41.2, description=\"plate is 41.2 long along PQ\")\nscene.constraint.eq(P.distance(S), 23.3, description=\"plate is 23.3 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 9, description=\"the chamfer takes 9 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 9, description=\"the chamfer takes 9 off along edge SR\")\nscene.constraint.eq(N.x, 17.3, description=\"the hole centre is 17.3 from the datum edge PS\")\nscene.constraint.eq(N.y, 10.3, description=\"the hole centre is 10.3 from the datum edge PQ\")\nscene.constraint.geq(cross(G1, G2, N) / G1.distance(G2), 5.3, description=\"at least 2.1 of material between the hole and the chamfer face\")\nscene.constraint.geq(N.x, 5.3, description=\"at least 2.1 of material between the hole and edge PS\")\nscene.constraint.geq(N.y, 5.3, description=\"at least 2.1 of material between the hole and edge PQ\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [41.2, 0.0], "R": [41.2, 23.3], "S": [0.0, 23.3], "G1": [41.2, 14.3], "G2": [32.2, 23.3], "N": [17.3, 10.3]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "the chamfer endpoints are each g from the corner R, measured ALONG the edges", "the clearance to the chamfer is the perpendicular distance from N to the line G1G2", "edge clearance is measured from the hole edge, so it uses clearance + hole radius"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cad_t2_cad_chamfer_corner_0098", "problem_id": "cad_t2_cad_chamfer_corner_0098", "domain": "PyGeoX-CAD", "template": "cad_chamfer_corner", "tier": 2, "nl_description": "A rectangular plate measures 32.6 mm by 26.7 mm. Set up coordinates with the bottom-left corner at the origin P, the 32.6 mm edge running from P along the positive x-direction to Q, and the 26.7 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut that takes 12.7 mm off along each of the two edges that meet there: the cut starts at G1, which lies on edge QR, and finishes at G2, which lies on edge SR - both of them on the plate, between the corners named. A hole 5.3 mm across is drilled with its centre N at 15.4 mm from edge PS and 11.5 mm from edge PQ. At least 2.4 mm of material must remain between the hole and the chamfer face, and between the hole and the two datum edges.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Chamfered corner with a clearance hole\n\nOracle formalization for cad_t2_cad_chamfer_corner_0098. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(47)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 32.6, description=\"plate is 32.6 long along PQ\")\nscene.constraint.eq(P.distance(S), 26.7, description=\"plate is 26.7 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 12.7, description=\"the chamfer takes 12.7 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 12.7, description=\"the chamfer takes 12.7 off along edge SR\")\nscene.constraint.eq(N.x, 15.4, description=\"the hole centre is 15.4 from the datum edge PS\")\nscene.constraint.eq(N.y, 11.5, description=\"the hole centre is 11.5 from the datum edge PQ\")\nscene.constraint.geq(cross(G1, G2, N) / G1.distance(G2), 5.05, description=\"at least 2.4 of material between the hole and the chamfer face\")\nscene.constraint.geq(N.x, 5.05, description=\"at least 2.4 of material between the hole and edge PS\")\nscene.constraint.geq(N.y, 5.05, description=\"at least 2.4 of material between the hole and edge PQ\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [32.6, 0.0], "R": [32.6, 26.7], "S": [0.0, 26.7], "G1": [32.6, 14.0], "G2": [19.900000000000002, 26.7], "N": [15.4, 11.5]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "the chamfer endpoints are each g from the corner R, measured ALONG the edges", "the clearance to the chamfer is the perpendicular distance from N to the line G1G2", "edge clearance is measured from the hole edge, so it uses clearance + hole radius"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cad_t2_cad_chamfer_corner_0099", "problem_id": "cad_t2_cad_chamfer_corner_0099", "domain": "PyGeoX-CAD", "template": "cad_chamfer_corner", "tier": 2, "nl_description": "A rectangular plate measures 41.8 mm by 24.7 mm. Set up coordinates with the bottom-left corner at the origin P, the 41.8 mm edge running from P along the positive x-direction to Q, and the 24.7 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut that takes 8.3 mm off along each of the two edges that meet there: the cut starts at G1, which lies on edge QR, and finishes at G2, which lies on edge SR - both of them on the plate, between the corners named. A hole 4.8 mm across is drilled with its centre N at 15.8 mm from edge PS and 10.1 mm from edge PQ. At least 3 mm of material must remain between the hole and the chamfer face, and between the hole and the two datum edges.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Chamfered corner with a clearance hole\n\nOracle formalization for cad_t2_cad_chamfer_corner_0099. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(59)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 41.8, description=\"plate is 41.8 long along PQ\")\nscene.constraint.eq(P.distance(S), 24.7, description=\"plate is 24.7 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 8.3, description=\"the chamfer takes 8.3 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 8.3, description=\"the chamfer takes 8.3 off along edge SR\")\nscene.constraint.eq(N.x, 15.8, description=\"the hole centre is 15.8 from the datum edge PS\")\nscene.constraint.eq(N.y, 10.1, description=\"the hole centre is 10.1 from the datum edge PQ\")\nscene.constraint.geq(cross(G1, G2, N) / G1.distance(G2), 5.4, description=\"at least 3 of material between the hole and the chamfer face\")\nscene.constraint.geq(N.x, 5.4, description=\"at least 3 of material between the hole and edge PS\")\nscene.constraint.geq(N.y, 5.4, description=\"at least 3 of material between the hole and edge PQ\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [41.8, 0.0], "R": [41.8, 24.7], "S": [0.0, 24.7], "G1": [41.8, 16.4], "G2": [33.5, 24.7], "N": [15.8, 10.1]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "the chamfer endpoints are each g from the corner R, measured ALONG the edges", "the clearance to the chamfer is the perpendicular distance from N to the line G1G2", "edge clearance is measured from the hole edge, so it uses clearance + hole radius"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cad_t2_cad_chamfer_corner_0100", "problem_id": "cad_t2_cad_chamfer_corner_0100", "domain": "PyGeoX-CAD", "template": "cad_chamfer_corner", "tier": 2, "nl_description": "A rectangular plate measures 40.7 mm by 25.5 mm. Set up coordinates with the bottom-left corner at the origin P, the 40.7 mm edge running from P along the positive x-direction to Q, and the 25.5 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut that takes 8.3 mm off along each of the two edges that meet there: the cut starts at G1, which lies on edge QR, and finishes at G2, which lies on edge SR - both of them on the plate, between the corners named. A hole 6.1 mm across is drilled with its centre N at 14.4 mm from edge PS and 10.7 mm from edge PQ. At least 1.9 mm of material must remain between the hole and the chamfer face, and between the hole and the two datum edges.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Chamfered corner with a clearance hole\n\nOracle formalization for cad_t2_cad_chamfer_corner_0100. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(58)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 40.7, description=\"plate is 40.7 long along PQ\")\nscene.constraint.eq(P.distance(S), 25.5, description=\"plate is 25.5 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 8.3, description=\"the chamfer takes 8.3 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 8.3, description=\"the chamfer takes 8.3 off along edge SR\")\nscene.constraint.eq(N.x, 14.4, description=\"the hole centre is 14.4 from the datum edge PS\")\nscene.constraint.eq(N.y, 10.7, description=\"the hole centre is 10.7 from the datum edge PQ\")\nscene.constraint.geq(cross(G1, G2, N) / G1.distance(G2), 4.95, description=\"at least 1.9 of material between the hole and the chamfer face\")\nscene.constraint.geq(N.x, 4.95, description=\"at least 1.9 of material between the hole and edge PS\")\nscene.constraint.geq(N.y, 4.95, description=\"at least 1.9 of material between the hole and edge PQ\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [40.7, 0.0], "R": [40.7, 25.5], "S": [0.0, 25.5], "G1": [40.7, 17.2], "G2": [32.400000000000006, 25.5], "N": [14.4, 10.7]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "the chamfer endpoints are each g from the corner R, measured ALONG the edges", "the clearance to the chamfer is the perpendicular distance from N to the line G1G2", "edge clearance is measured from the hole edge, so it uses clearance + hole radius"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cad_t2_cad_dogbone_link_0028", "problem_id": "cad_t2_cad_dogbone_link_0028", "domain": "PyGeoX-CAD", "template": "cad_dogbone_link", "tier": 2, "nl_description": "A flat link is made from two round ends joined by a straight-sided waist. Set up coordinates with the centre of the large round end at the origin C, and the centre of the small round end at D on the positive x-axis. The large end is 14.3 mm across, the small end is 8.8 mm across, and the two centres are 30.2 mm apart. The waist has two straight flanks. The upper flank runs smoothly into both round ends, touching the large end at T and the small end at V; the lower flank does the same, touching the large end at W and the small end at X. Neither flank crosses over between the two ends - each one stays on its own side of the link.", "answer_schema": {"points": ["C", "D", "T", "V", "W", "X"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Dogbone link with tangent flanks\n\nOracle formalization for cad_t2_cad_dogbone_link_0028. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(45)\n\nC, D, T, V, W, X = scene.add.points([\"C\", \"D\", \"T\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large round end (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large round end (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small end lies in the +x direction from C\")\nscene.constraint.eq(C.distance(D), 30.2, description=\"the two round ends are 30.2 apart between centres\")\nend_C = scene.add.circle(C)\nend_D = scene.add.circle(D)\nscene.constraint.eq(2 * end_C.radius, 14.3, description=\"the large round end is 14.3 across\")\nscene.constraint.eq(2 * end_D.radius, 8.8, description=\"the small round end is 8.8 across\")\nflank_upper = scene.add.line(T, V)\nflank_lower = scene.add.line(W, X)\nscene.relate.tangent_to_circle(flank_upper, end_C, T)\nscene.relate.tangent_to_circle(flank_upper, end_D, V)\nscene.relate.tangent_to_circle(flank_lower, end_C, W)\nscene.relate.tangent_to_circle(flank_lower, end_D, X)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper flank\")\nscene.constraint.lt(cross(C, D, W), -0.05, description=\"W is on the lower flank\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the upper flank does not cross between the ends: it touches both on the same side\")\nscene.constraint.gt(dot(C, W, D, X), 0.05, description=\"the lower flank does not cross between the ends: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [30.2, 0.0], "T": [0.6510761589403974, 7.120294926143117], "V": [30.600662251655628, 4.38171995454961], "W": [0.6510761589403974, -7.120294926143117], "X": [30.600662251655628, -4.38171995454961]}, "circles": {"C": 7.15, "D": 4.4}}, "implicit_facts": ["each end radius is half the stated across-size", "'runs smoothly into' means each flank is tangent to both end circles", "because the two ends have different diameters the flanks are NOT parallel to the line of centres", "each tangency point is the foot of the perpendicular from that end's centre onto the flank"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "T/V is the upper flank and W/X the lower one"]} {"unique_id": "cad_t2_cad_dogbone_link_0029", "problem_id": "cad_t2_cad_dogbone_link_0029", "domain": "PyGeoX-CAD", "template": "cad_dogbone_link", "tier": 2, "nl_description": "A flat link is made from two round ends joined by a straight-sided waist. Set up coordinates with the centre of the large round end at the origin C, and the centre of the small round end at D on the positive x-axis. The large end is 19.8 mm across, the small end is 9.5 mm across, and the two centres are 33.3 mm apart. The waist has two straight flanks. The upper flank runs smoothly into both round ends, touching the large end at T and the small end at V; the lower flank does the same, touching the large end at W and the small end at X. Neither flank crosses over between the two ends - each one stays on its own side of the link.", "answer_schema": {"points": ["C", "D", "T", "V", "W", "X"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Dogbone link with tangent flanks\n\nOracle formalization for cad_t2_cad_dogbone_link_0029. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(49)\n\nC, D, T, V, W, X = scene.add.points([\"C\", \"D\", \"T\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large round end (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large round end (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small end lies in the +x direction from C\")\nscene.constraint.eq(C.distance(D), 33.3, description=\"the two round ends are 33.3 apart between centres\")\nend_C = scene.add.circle(C)\nend_D = scene.add.circle(D)\nscene.constraint.eq(2 * end_C.radius, 19.8, description=\"the large round end is 19.8 across\")\nscene.constraint.eq(2 * end_D.radius, 9.5, description=\"the small round end is 9.5 across\")\nflank_upper = scene.add.line(T, V)\nflank_lower = scene.add.line(W, X)\nscene.relate.tangent_to_circle(flank_upper, end_C, T)\nscene.relate.tangent_to_circle(flank_upper, end_D, V)\nscene.relate.tangent_to_circle(flank_lower, end_C, W)\nscene.relate.tangent_to_circle(flank_lower, end_D, X)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper flank\")\nscene.constraint.lt(cross(C, D, W), -0.05, description=\"W is on the lower flank\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the upper flank does not cross between the ends: it touches both on the same side\")\nscene.constraint.gt(dot(C, W, D, X), 0.05, description=\"the lower flank does not cross between the ends: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [33.3, 0.0], "T": [1.5310810810810813, 9.78088905586581], "V": [34.0346096096096, 4.692850809632586], "W": [1.5310810810810813, -9.78088905586581], "X": [34.0346096096096, -4.692850809632586]}, "circles": {"C": 9.9, "D": 4.75}}, "implicit_facts": ["each end radius is half the stated across-size", "'runs smoothly into' means each flank is tangent to both end circles", "because the two ends have different diameters the flanks are NOT parallel to the line of centres", "each tangency point is the foot of the perpendicular from that end's centre onto the flank"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "T/V is the upper flank and W/X the lower one"]} {"unique_id": "cad_t2_cad_dogbone_link_0030", "problem_id": "cad_t2_cad_dogbone_link_0030", "domain": "PyGeoX-CAD", "template": "cad_dogbone_link", "tier": 2, "nl_description": "A flat link is made from two round ends joined by a straight-sided waist. Set up coordinates with the centre of the large round end at the origin C, and the centre of the small round end at D on the positive x-axis. The large end is 14.1 mm across, the small end is 9.5 mm across, and the two centres are 36.8 mm apart. The waist has two straight flanks. The upper flank runs smoothly into both round ends, touching the large end at T and the small end at V; the lower flank does the same, touching the large end at W and the small end at X. Neither flank crosses over between the two ends - each one stays on its own side of the link.", "answer_schema": {"points": ["C", "D", "T", "V", "W", "X"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Dogbone link with tangent flanks\n\nOracle formalization for cad_t2_cad_dogbone_link_0030. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(53)\n\nC, D, T, V, W, X = scene.add.points([\"C\", \"D\", \"T\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large round end (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large round end (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small end lies in the +x direction from C\")\nscene.constraint.eq(C.distance(D), 36.8, description=\"the two round ends are 36.8 apart between centres\")\nend_C = scene.add.circle(C)\nend_D = scene.add.circle(D)\nscene.constraint.eq(2 * end_C.radius, 14.1, description=\"the large round end is 14.1 across\")\nscene.constraint.eq(2 * end_D.radius, 9.5, description=\"the small round end is 9.5 across\")\nflank_upper = scene.add.line(T, V)\nflank_lower = scene.add.line(W, X)\nscene.relate.tangent_to_circle(flank_upper, end_C, T)\nscene.relate.tangent_to_circle(flank_upper, end_D, V)\nscene.relate.tangent_to_circle(flank_lower, end_C, W)\nscene.relate.tangent_to_circle(flank_lower, end_D, X)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper flank\")\nscene.constraint.lt(cross(C, D, W), -0.05, description=\"W is on the lower flank\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the upper flank does not cross between the ends: it touches both on the same side\")\nscene.constraint.gt(dot(C, W, D, X), 0.05, description=\"the lower flank does not cross between the ends: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [36.8, 0.0], "T": [0.440625, 7.036216995614547], "V": [37.096875, 4.740713578605545], "W": [0.440625, -7.036216995614547], "X": [37.096875, -4.740713578605545]}, "circles": {"C": 7.05, "D": 4.75}}, "implicit_facts": ["each end radius is half the stated across-size", "'runs smoothly into' means each flank is tangent to both end circles", "because the two ends have different diameters the flanks are NOT parallel to the line of centres", "each tangency point is the foot of the perpendicular from that end's centre onto the flank"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "T/V is the upper flank and W/X the lower one"]} {"unique_id": "cad_t2_cad_dogbone_link_0031", "problem_id": "cad_t2_cad_dogbone_link_0031", "domain": "PyGeoX-CAD", "template": "cad_dogbone_link", "tier": 2, "nl_description": "A flat link is made from two round ends joined by a straight-sided waist. Set up coordinates with the centre of the large round end at the origin C, and the centre of the small round end at D on the positive x-axis. The large end is 17.7 mm across, the small end is 9.6 mm across, and the two centres are 28.9 mm apart. The waist has two straight flanks. The upper flank runs smoothly into both round ends, touching the large end at T and the small end at V; the lower flank does the same, touching the large end at W and the small end at X. Neither flank crosses over between the two ends - each one stays on its own side of the link.", "answer_schema": {"points": ["C", "D", "T", "V", "W", "X"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Dogbone link with tangent flanks\n\nOracle formalization for cad_t2_cad_dogbone_link_0031. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(43)\n\nC, D, T, V, W, X = scene.add.points([\"C\", \"D\", \"T\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large round end (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large round end (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small end lies in the +x direction from C\")\nscene.constraint.eq(C.distance(D), 28.9, description=\"the two round ends are 28.9 apart between centres\")\nend_C = scene.add.circle(C)\nend_D = scene.add.circle(D)\nscene.constraint.eq(2 * end_C.radius, 17.7, description=\"the large round end is 17.7 across\")\nscene.constraint.eq(2 * end_D.radius, 9.6, description=\"the small round end is 9.6 across\")\nflank_upper = scene.add.line(T, V)\nflank_lower = scene.add.line(W, X)\nscene.relate.tangent_to_circle(flank_upper, end_C, T)\nscene.relate.tangent_to_circle(flank_upper, end_D, V)\nscene.relate.tangent_to_circle(flank_lower, end_C, W)\nscene.relate.tangent_to_circle(flank_lower, end_D, X)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper flank\")\nscene.constraint.lt(cross(C, D, W), -0.05, description=\"W is on the lower flank\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the upper flank does not cross between the ends: it touches both on the same side\")\nscene.constraint.gt(dot(C, W, D, X), 0.05, description=\"the lower flank does not cross between the ends: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [28.9, 0.0], "T": [1.2402249134948098, 8.762667525585277], "V": [29.57266435986159, 4.752633234215743], "W": [1.2402249134948098, -8.762667525585277], "X": [29.57266435986159, -4.752633234215743]}, "circles": {"C": 8.85, "D": 4.8}}, "implicit_facts": ["each end radius is half the stated across-size", "'runs smoothly into' means each flank is tangent to both end circles", "because the two ends have different diameters the flanks are NOT parallel to the line of centres", "each tangency point is the foot of the perpendicular from that end's centre onto the flank"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "T/V is the upper flank and W/X the lower one"]} {"unique_id": "cad_t2_cad_dogbone_link_0032", "problem_id": "cad_t2_cad_dogbone_link_0032", "domain": "PyGeoX-CAD", "template": "cad_dogbone_link", "tier": 2, "nl_description": "A flat link is made from two round ends joined by a straight-sided waist. Set up coordinates with the centre of the large round end at the origin C, and the centre of the small round end at D on the positive x-axis. The large end is 16.8 mm across, the small end is 7.4 mm across, and the two centres are 31.4 mm apart. The waist has two straight flanks. The upper flank runs smoothly into both round ends, touching the large end at T and the small end at V; the lower flank does the same, touching the large end at W and the small end at X. Neither flank crosses over between the two ends - each one stays on its own side of the link.", "answer_schema": {"points": ["C", "D", "T", "V", "W", "X"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Dogbone link with tangent flanks\n\nOracle formalization for cad_t2_cad_dogbone_link_0032. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(47)\n\nC, D, T, V, W, X = scene.add.points([\"C\", \"D\", \"T\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large round end (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large round end (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small end lies in the +x direction from C\")\nscene.constraint.eq(C.distance(D), 31.4, description=\"the two round ends are 31.4 apart between centres\")\nend_C = scene.add.circle(C)\nend_D = scene.add.circle(D)\nscene.constraint.eq(2 * end_C.radius, 16.8, description=\"the large round end is 16.8 across\")\nscene.constraint.eq(2 * end_D.radius, 7.4, description=\"the small round end is 7.4 across\")\nflank_upper = scene.add.line(T, V)\nflank_lower = scene.add.line(W, X)\nscene.relate.tangent_to_circle(flank_upper, end_C, T)\nscene.relate.tangent_to_circle(flank_upper, end_D, V)\nscene.relate.tangent_to_circle(flank_lower, end_C, W)\nscene.relate.tangent_to_circle(flank_lower, end_D, X)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper flank\")\nscene.constraint.lt(cross(C, D, W), -0.05, description=\"W is on the lower flank\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the upper flank does not cross between the ends: it touches both on the same side\")\nscene.constraint.gt(dot(C, W, D, X), 0.05, description=\"the lower flank does not cross between the ends: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [31.4, 0.0], "T": [1.2573248407643314, 8.305367797081413], "V": [31.953821656050955, 3.658316767762051], "W": [1.2573248407643314, -8.305367797081413], "X": [31.953821656050955, -3.658316767762051]}, "circles": {"C": 8.4, "D": 3.7}}, "implicit_facts": ["each end radius is half the stated across-size", "'runs smoothly into' means each flank is tangent to both end circles", "because the two ends have different diameters the flanks are NOT parallel to the line of centres", "each tangency point is the foot of the perpendicular from that end's centre onto the flank"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "T/V is the upper flank and W/X the lower one"]} {"unique_id": "cad_t2_cad_dogbone_link_0033", "problem_id": "cad_t2_cad_dogbone_link_0033", "domain": "PyGeoX-CAD", "template": "cad_dogbone_link", "tier": 2, "nl_description": "A flat link is made from two round ends joined by a straight-sided waist. Set up coordinates with the centre of the large round end at the origin C, and the centre of the small round end at D on the positive x-axis. The large end is 20 mm across, the small end is 9.7 mm across, and the two centres are 30.5 mm apart. The waist has two straight flanks. The upper flank runs smoothly into both round ends, touching the large end at T and the small end at V; the lower flank does the same, touching the large end at W and the small end at X. Neither flank crosses over between the two ends - each one stays on its own side of the link.", "answer_schema": {"points": ["C", "D", "T", "V", "W", "X"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Dogbone link with tangent flanks\n\nOracle formalization for cad_t2_cad_dogbone_link_0033. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(46)\n\nC, D, T, V, W, X = scene.add.points([\"C\", \"D\", \"T\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large round end (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large round end (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small end lies in the +x direction from C\")\nscene.constraint.eq(C.distance(D), 30.5, description=\"the two round ends are 30.5 apart between centres\")\nend_C = scene.add.circle(C)\nend_D = scene.add.circle(D)\nscene.constraint.eq(2 * end_C.radius, 20, description=\"the large round end is 20 across\")\nscene.constraint.eq(2 * end_D.radius, 9.7, description=\"the small round end is 9.7 across\")\nflank_upper = scene.add.line(T, V)\nflank_lower = scene.add.line(W, X)\nscene.relate.tangent_to_circle(flank_upper, end_C, T)\nscene.relate.tangent_to_circle(flank_upper, end_D, V)\nscene.relate.tangent_to_circle(flank_lower, end_C, W)\nscene.relate.tangent_to_circle(flank_lower, end_D, X)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper flank\")\nscene.constraint.lt(cross(C, D, W), -0.05, description=\"W is on the lower flank\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the upper flank does not cross between the ends: it touches both on the same side\")\nscene.constraint.gt(dot(C, W, D, X), 0.05, description=\"the lower flank does not cross between the ends: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [30.5, 0.0], "T": [1.6885245901639345, 9.856413379541856], "V": [31.318934426229507, 4.7803604890777995], "W": [1.6885245901639345, -9.856413379541856], "X": [31.318934426229507, -4.7803604890777995]}, "circles": {"C": 10.0, "D": 4.85}}, "implicit_facts": ["each end radius is half the stated across-size", "'runs smoothly into' means each flank is tangent to both end circles", "because the two ends have different diameters the flanks are NOT parallel to the line of centres", "each tangency point is the foot of the perpendicular from that end's centre onto the flank"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "T/V is the upper flank and W/X the lower one"]} {"unique_id": "cad_t2_cad_dogbone_link_0034", "problem_id": "cad_t2_cad_dogbone_link_0034", "domain": "PyGeoX-CAD", "template": "cad_dogbone_link", "tier": 2, "nl_description": "A flat link is made from two round ends joined by a straight-sided waist. Set up coordinates with the centre of the large round end at the origin C, and the centre of the small round end at D on the positive x-axis. The large end is 18 mm across, the small end is 6 mm across, and the two centres are 37.9 mm apart. The waist has two straight flanks. The upper flank runs smoothly into both round ends, touching the large end at T and the small end at V; the lower flank does the same, touching the large end at W and the small end at X. Neither flank crosses over between the two ends - each one stays on its own side of the link.", "answer_schema": {"points": ["C", "D", "T", "V", "W", "X"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Dogbone link with tangent flanks\n\nOracle formalization for cad_t2_cad_dogbone_link_0034. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(55)\n\nC, D, T, V, W, X = scene.add.points([\"C\", \"D\", \"T\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large round end (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large round end (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small end lies in the +x direction from C\")\nscene.constraint.eq(C.distance(D), 37.9, description=\"the two round ends are 37.9 apart between centres\")\nend_C = scene.add.circle(C)\nend_D = scene.add.circle(D)\nscene.constraint.eq(2 * end_C.radius, 18, description=\"the large round end is 18 across\")\nscene.constraint.eq(2 * end_D.radius, 6, description=\"the small round end is 6 across\")\nflank_upper = scene.add.line(T, V)\nflank_lower = scene.add.line(W, X)\nscene.relate.tangent_to_circle(flank_upper, end_C, T)\nscene.relate.tangent_to_circle(flank_upper, end_D, V)\nscene.relate.tangent_to_circle(flank_lower, end_C, W)\nscene.relate.tangent_to_circle(flank_lower, end_D, X)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper flank\")\nscene.constraint.lt(cross(C, D, W), -0.05, description=\"W is on the lower flank\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the upper flank does not cross between the ends: it touches both on the same side\")\nscene.constraint.gt(dot(C, W, D, X), 0.05, description=\"the lower flank does not cross between the ends: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [37.9, 0.0], "T": [1.424802110817942, 8.886503189950968], "V": [38.374934036939315, 2.962167729983656], "W": [1.424802110817942, -8.886503189950968], "X": [38.374934036939315, -2.962167729983656]}, "circles": {"C": 9.0, "D": 3.0}}, "implicit_facts": ["each end radius is half the stated across-size", "'runs smoothly into' means each flank is tangent to both end circles", "because the two ends have different diameters the flanks are NOT parallel to the line of centres", "each tangency point is the foot of the perpendicular from that end's centre onto the flank"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "T/V is the upper flank and W/X the lower one"]} {"unique_id": "cad_t2_cad_dogbone_link_0035", "problem_id": "cad_t2_cad_dogbone_link_0035", "domain": "PyGeoX-CAD", "template": "cad_dogbone_link", "tier": 2, "nl_description": "A flat link is made from two round ends joined by a straight-sided waist. Set up coordinates with the centre of the large round end at the origin C, and the centre of the small round end at D on the positive x-axis. The large end is 18.3 mm across, the small end is 10 mm across, and the two centres are 29.9 mm apart. The waist has two straight flanks. The upper flank runs smoothly into both round ends, touching the large end at T and the small end at V; the lower flank does the same, touching the large end at W and the small end at X. Neither flank crosses over between the two ends - each one stays on its own side of the link.", "answer_schema": {"points": ["C", "D", "T", "V", "W", "X"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Dogbone link with tangent flanks\n\nOracle formalization for cad_t2_cad_dogbone_link_0035. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(45)\n\nC, D, T, V, W, X = scene.add.points([\"C\", \"D\", \"T\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large round end (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large round end (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small end lies in the +x direction from C\")\nscene.constraint.eq(C.distance(D), 29.9, description=\"the two round ends are 29.9 apart between centres\")\nend_C = scene.add.circle(C)\nend_D = scene.add.circle(D)\nscene.constraint.eq(2 * end_C.radius, 18.3, description=\"the large round end is 18.3 across\")\nscene.constraint.eq(2 * end_D.radius, 10, description=\"the small round end is 10 across\")\nflank_upper = scene.add.line(T, V)\nflank_lower = scene.add.line(W, X)\nscene.relate.tangent_to_circle(flank_upper, end_C, T)\nscene.relate.tangent_to_circle(flank_upper, end_D, V)\nscene.relate.tangent_to_circle(flank_lower, end_C, W)\nscene.relate.tangent_to_circle(flank_lower, end_D, X)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper flank\")\nscene.constraint.lt(cross(C, D, W), -0.05, description=\"W is on the lower flank\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the upper flank does not cross between the ends: it touches both on the same side\")\nscene.constraint.gt(dot(C, W, D, X), 0.05, description=\"the lower flank does not cross between the ends: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [29.9, 0.0], "T": [1.2699832775919735, 9.061437108684073], "V": [30.593979933110365, 4.95160497742299], "W": [1.2699832775919735, -9.061437108684073], "X": [30.593979933110365, -4.95160497742299]}, "circles": {"C": 9.15, "D": 5.0}}, "implicit_facts": ["each end radius is half the stated across-size", "'runs smoothly into' means each flank is tangent to both end circles", "because the two ends have different diameters the flanks are NOT parallel to the line of centres", "each tangency point is the foot of the perpendicular from that end's centre onto the flank"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "T/V is the upper flank and W/X the lower one"]} {"unique_id": "cad_t2_cad_dogbone_link_0036", "problem_id": "cad_t2_cad_dogbone_link_0036", "domain": "PyGeoX-CAD", "template": "cad_dogbone_link", "tier": 2, "nl_description": "A flat link is made from two round ends joined by a straight-sided waist. Set up coordinates with the centre of the large round end at the origin C, and the centre of the small round end at D on the positive x-axis. The large end is 16.9 mm across, the small end is 11 mm across, and the two centres are 33.6 mm apart. The waist has two straight flanks. The upper flank runs smoothly into both round ends, touching the large end at T and the small end at V; the lower flank does the same, touching the large end at W and the small end at X. Neither flank crosses over between the two ends - each one stays on its own side of the link.", "answer_schema": {"points": ["C", "D", "T", "V", "W", "X"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Dogbone link with tangent flanks\n\nOracle formalization for cad_t2_cad_dogbone_link_0036. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(49)\n\nC, D, T, V, W, X = scene.add.points([\"C\", \"D\", \"T\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large round end (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large round end (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small end lies in the +x direction from C\")\nscene.constraint.eq(C.distance(D), 33.6, description=\"the two round ends are 33.6 apart between centres\")\nend_C = scene.add.circle(C)\nend_D = scene.add.circle(D)\nscene.constraint.eq(2 * end_C.radius, 16.9, description=\"the large round end is 16.9 across\")\nscene.constraint.eq(2 * end_D.radius, 11, description=\"the small round end is 11 across\")\nflank_upper = scene.add.line(T, V)\nflank_lower = scene.add.line(W, X)\nscene.relate.tangent_to_circle(flank_upper, end_C, T)\nscene.relate.tangent_to_circle(flank_upper, end_D, V)\nscene.relate.tangent_to_circle(flank_lower, end_C, W)\nscene.relate.tangent_to_circle(flank_lower, end_D, X)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper flank\")\nscene.constraint.lt(cross(C, D, W), -0.05, description=\"W is on the lower flank\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the upper flank does not cross between the ends: it touches both on the same side\")\nscene.constraint.gt(dot(C, W, D, X), 0.05, description=\"the lower flank does not cross between the ends: it touches both on the same side\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [33.6, 0.0], "T": [0.7418898809523807, 8.417368912227884], "V": [34.08288690476191, 5.478760830444186], "W": [0.7418898809523807, -8.417368912227884], "X": [34.08288690476191, -5.478760830444186]}, "circles": {"C": 8.45, "D": 5.5}}, "implicit_facts": ["each end radius is half the stated across-size", "'runs smoothly into' means each flank is tangent to both end circles", "because the two ends have different diameters the flanks are NOT parallel to the line of centres", "each tangency point is the foot of the perpendicular from that end's centre onto the flank"], "branch_facts": ["'does not cross over' selects the external common tangent, not the internal one", "T/V is the upper flank and W/X the lower one"]} {"unique_id": "cad_t2_cad_fillet_bracket_0001", "problem_id": "cad_t2_cad_fillet_bracket_0001", "domain": "PyGeoX-CAD", "template": "cad_fillet_bracket", "tier": 2, "nl_description": "An L-shaped bracket is cut from 5.9 mm flat plate. Set up coordinates with the outside corner of the L at the origin O, the bottom outer edge running from O along the positive x-direction to A, and the upright outer edge running from O along the positive y-direction to B. The bottom outer edge OA is 25.2 mm long and the upright outer edge OB is 25 mm long. Both arms are 5.9 mm thick. The inner face of the bottom arm is the straight edge from the inner corner K out to E at the end of the bottom arm; the inner face of the upright is the straight edge from K up to G at the top of the upright. K is the point where those two inner faces would meet. To relieve the stress raiser there, the inner corner is blended with a 4.7 mm radius round centred at F. The round is cut into the material on the inside of the corner, and it runs smoothly into both inner faces with no step or kink, meeting the bottom inner face at T and the upright inner face at U.", "answer_schema": {"points": ["A", "B", "E", "F", "G", "K", "O", "T", "U"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Filleted L-bracket inner corner\n\nOracle formalization for cad_t2_cad_fillet_bracket_0001. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(38)\n\nO, A, B, K, E, G, F, T, U = scene.add.points([\"O\", \"A\", \"B\", \"K\", \"E\", \"G\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the outside corner of the L (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the outside corner of the L (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: the bottom outer edge OA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: the bottom arm runs in the +x direction\")\nscene.constraint.eq(B.x, 0, description=\"datum: the upright outer edge OB runs along the y-axis\")\nscene.constraint.gt(B.y, 1.0, description=\"datum: the upright arm runs in the +y direction\")\nscene.constraint.eq(O.distance(A), 25.2, description=\"bottom outer edge OA is 25.2 long\")\nscene.constraint.eq(O.distance(B), 25, description=\"upright outer edge OB is 25 long\")\nscene.constraint.eq(K.y, 5.9, description=\"the bottom arm is 5.9 thick, so the inner corner K sits 5.9 above edge OA\")\nscene.constraint.eq(K.x, 5.9, description=\"the upright arm is 5.9 thick, so the inner corner K sits 5.9 in from edge OB\")\nscene.constraint.eq(E.y, K.y, description=\"inner face KE is parallel to the bottom outer edge\")\nscene.constraint.eq(E.x, A.x, description=\"inner face KE ends flush with the end of the bottom arm\")\nscene.constraint.eq(G.x, K.x, description=\"inner face KG is parallel to the upright outer edge\")\nscene.constraint.eq(G.y, B.y, description=\"inner face KG ends flush with the top of the upright\")\nface_h = scene.add.line_segment(K, E)\nface_v = scene.add.line_segment(K, G)\nfillet = scene.add.circle(F)\nscene.constraint.eq(fillet.radius, 4.7, description=\"the blend round has radius 4.7\")\nscene.relate.tangent_to_circle(face_h, fillet, T)\nscene.relate.tangent_to_circle(face_v, fillet, U)\nscene.constraint.gt(cross(K, E, F), 0.05, description=\"the round is cut into the material, on the inner side of face KE\")\nscene.constraint.gt(cross(G, K, F), 0.05, description=\"the round is cut into the material, on the inner side of face KG\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A": [25.2, 0.0], "B": [0.0, 25.0], "K": [5.9, 5.9], "E": [25.2, 5.9], "G": [5.9, 25.0], "F": [10.600000000000001, 10.600000000000001], "T": [10.600000000000001, 5.9], "U": [5.9, 10.600000000000001]}, "circles": {"F": 4.7}}, "implicit_facts": ["'runs smoothly into' means the round is TANGENT to both inner faces", "tangency puts F exactly r from each inner face, i.e. on the bisector of the inner corner", "T and U are the feet of the perpendiculars from F onto the two faces", "the inner faces are offset from the outer edges by the arm thickness"], "branch_facts": ["the round is on the material side of each face, not the outside"]} {"unique_id": "cad_t2_cad_fillet_bracket_0002", "problem_id": "cad_t2_cad_fillet_bracket_0002", "domain": "PyGeoX-CAD", "template": "cad_fillet_bracket", "tier": 2, "nl_description": "An L-shaped bracket is cut from 4.8 mm flat plate. Set up coordinates with the outside corner of the L at the origin O, the bottom outer edge running from O along the positive x-direction to A, and the upright outer edge running from O along the positive y-direction to B. The bottom outer edge OA is 32.5 mm long and the upright outer edge OB is 19.8 mm long. Both arms are 4.8 mm thick. The inner face of the bottom arm is the straight edge from the inner corner K out to E at the end of the bottom arm; the inner face of the upright is the straight edge from K up to G at the top of the upright. K is the point where those two inner faces would meet. To relieve the stress raiser there, the inner corner is blended with a 2.2 mm radius round centred at F. The round is cut into the material on the inside of the corner, and it runs smoothly into both inner faces with no step or kink, meeting the bottom inner face at T and the upright inner face at U.", "answer_schema": {"points": ["A", "B", "E", "F", "G", "K", "O", "T", "U"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Filleted L-bracket inner corner\n\nOracle formalization for cad_t2_cad_fillet_bracket_0002. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(47)\n\nO, A, B, K, E, G, F, T, U = scene.add.points([\"O\", \"A\", \"B\", \"K\", \"E\", \"G\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the outside corner of the L (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the outside corner of the L (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: the bottom outer edge OA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: the bottom arm runs in the +x direction\")\nscene.constraint.eq(B.x, 0, description=\"datum: the upright outer edge OB runs along the y-axis\")\nscene.constraint.gt(B.y, 1.0, description=\"datum: the upright arm runs in the +y direction\")\nscene.constraint.eq(O.distance(A), 32.5, description=\"bottom outer edge OA is 32.5 long\")\nscene.constraint.eq(O.distance(B), 19.8, description=\"upright outer edge OB is 19.8 long\")\nscene.constraint.eq(K.y, 4.8, description=\"the bottom arm is 4.8 thick, so the inner corner K sits 4.8 above edge OA\")\nscene.constraint.eq(K.x, 4.8, description=\"the upright arm is 4.8 thick, so the inner corner K sits 4.8 in from edge OB\")\nscene.constraint.eq(E.y, K.y, description=\"inner face KE is parallel to the bottom outer edge\")\nscene.constraint.eq(E.x, A.x, description=\"inner face KE ends flush with the end of the bottom arm\")\nscene.constraint.eq(G.x, K.x, description=\"inner face KG is parallel to the upright outer edge\")\nscene.constraint.eq(G.y, B.y, description=\"inner face KG ends flush with the top of the upright\")\nface_h = scene.add.line_segment(K, E)\nface_v = scene.add.line_segment(K, G)\nfillet = scene.add.circle(F)\nscene.constraint.eq(fillet.radius, 2.2, description=\"the blend round has radius 2.2\")\nscene.relate.tangent_to_circle(face_h, fillet, T)\nscene.relate.tangent_to_circle(face_v, fillet, U)\nscene.constraint.gt(cross(K, E, F), 0.05, description=\"the round is cut into the material, on the inner side of face KE\")\nscene.constraint.gt(cross(G, K, F), 0.05, description=\"the round is cut into the material, on the inner side of face KG\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A": [32.5, 0.0], "B": [0.0, 19.8], "K": [4.8, 4.8], "E": [32.5, 4.8], "G": [4.8, 19.8], "F": [7.0, 7.0], "T": [7.0, 4.8], "U": [4.8, 7.0]}, "circles": {"F": 2.2}}, "implicit_facts": ["'runs smoothly into' means the round is TANGENT to both inner faces", "tangency puts F exactly r from each inner face, i.e. on the bisector of the inner corner", "T and U are the feet of the perpendiculars from F onto the two faces", "the inner faces are offset from the outer edges by the arm thickness"], "branch_facts": ["the round is on the material side of each face, not the outside"]} {"unique_id": "cad_t2_cad_fillet_bracket_0003", "problem_id": "cad_t2_cad_fillet_bracket_0003", "domain": "PyGeoX-CAD", "template": "cad_fillet_bracket", "tier": 2, "nl_description": "An L-shaped bracket is cut from 7 mm flat plate. Set up coordinates with the outside corner of the L at the origin O, the bottom outer edge running from O along the positive x-direction to A, and the upright outer edge running from O along the positive y-direction to B. The bottom outer edge OA is 28.3 mm long and the upright outer edge OB is 27.9 mm long. Both arms are 7 mm thick. The inner face of the bottom arm is the straight edge from the inner corner K out to E at the end of the bottom arm; the inner face of the upright is the straight edge from K up to G at the top of the upright. K is the point where those two inner faces would meet. To relieve the stress raiser there, the inner corner is blended with a 4.8 mm radius round centred at F. The round is cut into the material on the inside of the corner, and it runs smoothly into both inner faces with no step or kink, meeting the bottom inner face at T and the upright inner face at U.", "answer_schema": {"points": ["A", "B", "E", "F", "G", "K", "O", "T", "U"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Filleted L-bracket inner corner\n\nOracle formalization for cad_t2_cad_fillet_bracket_0003. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(42)\n\nO, A, B, K, E, G, F, T, U = scene.add.points([\"O\", \"A\", \"B\", \"K\", \"E\", \"G\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the outside corner of the L (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the outside corner of the L (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: the bottom outer edge OA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: the bottom arm runs in the +x direction\")\nscene.constraint.eq(B.x, 0, description=\"datum: the upright outer edge OB runs along the y-axis\")\nscene.constraint.gt(B.y, 1.0, description=\"datum: the upright arm runs in the +y direction\")\nscene.constraint.eq(O.distance(A), 28.3, description=\"bottom outer edge OA is 28.3 long\")\nscene.constraint.eq(O.distance(B), 27.9, description=\"upright outer edge OB is 27.9 long\")\nscene.constraint.eq(K.y, 7, description=\"the bottom arm is 7 thick, so the inner corner K sits 7 above edge OA\")\nscene.constraint.eq(K.x, 7, description=\"the upright arm is 7 thick, so the inner corner K sits 7 in from edge OB\")\nscene.constraint.eq(E.y, K.y, description=\"inner face KE is parallel to the bottom outer edge\")\nscene.constraint.eq(E.x, A.x, description=\"inner face KE ends flush with the end of the bottom arm\")\nscene.constraint.eq(G.x, K.x, description=\"inner face KG is parallel to the upright outer edge\")\nscene.constraint.eq(G.y, B.y, description=\"inner face KG ends flush with the top of the upright\")\nface_h = scene.add.line_segment(K, E)\nface_v = scene.add.line_segment(K, G)\nfillet = scene.add.circle(F)\nscene.constraint.eq(fillet.radius, 4.8, description=\"the blend round has radius 4.8\")\nscene.relate.tangent_to_circle(face_h, fillet, T)\nscene.relate.tangent_to_circle(face_v, fillet, U)\nscene.constraint.gt(cross(K, E, F), 0.05, description=\"the round is cut into the material, on the inner side of face KE\")\nscene.constraint.gt(cross(G, K, F), 0.05, description=\"the round is cut into the material, on the inner side of face KG\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A": [28.3, 0.0], "B": [0.0, 27.9], "K": [7.0, 7.0], "E": [28.3, 7.0], "G": [7.0, 27.9], "F": [11.8, 11.8], "T": [11.8, 7.0], "U": [7.0, 11.8]}, "circles": {"F": 4.8}}, "implicit_facts": ["'runs smoothly into' means the round is TANGENT to both inner faces", "tangency puts F exactly r from each inner face, i.e. on the bisector of the inner corner", "T and U are the feet of the perpendiculars from F onto the two faces", "the inner faces are offset from the outer edges by the arm thickness"], "branch_facts": ["the round is on the material side of each face, not the outside"]} {"unique_id": "cad_t2_cad_fillet_bracket_0004", "problem_id": "cad_t2_cad_fillet_bracket_0004", "domain": "PyGeoX-CAD", "template": "cad_fillet_bracket", "tier": 2, "nl_description": "An L-shaped bracket is cut from 5.6 mm flat plate. Set up coordinates with the outside corner of the L at the origin O, the bottom outer edge running from O along the positive x-direction to A, and the upright outer edge running from O along the positive y-direction to B. The bottom outer edge OA is 28.8 mm long and the upright outer edge OB is 18.5 mm long. Both arms are 5.6 mm thick. The inner face of the bottom arm is the straight edge from the inner corner K out to E at the end of the bottom arm; the inner face of the upright is the straight edge from K up to G at the top of the upright. K is the point where those two inner faces would meet. To relieve the stress raiser there, the inner corner is blended with a 2.5 mm radius round centred at F. The round is cut into the material on the inside of the corner, and it runs smoothly into both inner faces with no step or kink, meeting the bottom inner face at T and the upright inner face at U.", "answer_schema": {"points": ["A", "B", "E", "F", "G", "K", "O", "T", "U"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Filleted L-bracket inner corner\n\nOracle formalization for cad_t2_cad_fillet_bracket_0004. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(42)\n\nO, A, B, K, E, G, F, T, U = scene.add.points([\"O\", \"A\", \"B\", \"K\", \"E\", \"G\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the outside corner of the L (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the outside corner of the L (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: the bottom outer edge OA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: the bottom arm runs in the +x direction\")\nscene.constraint.eq(B.x, 0, description=\"datum: the upright outer edge OB runs along the y-axis\")\nscene.constraint.gt(B.y, 1.0, description=\"datum: the upright arm runs in the +y direction\")\nscene.constraint.eq(O.distance(A), 28.8, description=\"bottom outer edge OA is 28.8 long\")\nscene.constraint.eq(O.distance(B), 18.5, description=\"upright outer edge OB is 18.5 long\")\nscene.constraint.eq(K.y, 5.6, description=\"the bottom arm is 5.6 thick, so the inner corner K sits 5.6 above edge OA\")\nscene.constraint.eq(K.x, 5.6, description=\"the upright arm is 5.6 thick, so the inner corner K sits 5.6 in from edge OB\")\nscene.constraint.eq(E.y, K.y, description=\"inner face KE is parallel to the bottom outer edge\")\nscene.constraint.eq(E.x, A.x, description=\"inner face KE ends flush with the end of the bottom arm\")\nscene.constraint.eq(G.x, K.x, description=\"inner face KG is parallel to the upright outer edge\")\nscene.constraint.eq(G.y, B.y, description=\"inner face KG ends flush with the top of the upright\")\nface_h = scene.add.line_segment(K, E)\nface_v = scene.add.line_segment(K, G)\nfillet = scene.add.circle(F)\nscene.constraint.eq(fillet.radius, 2.5, description=\"the blend round has radius 2.5\")\nscene.relate.tangent_to_circle(face_h, fillet, T)\nscene.relate.tangent_to_circle(face_v, fillet, U)\nscene.constraint.gt(cross(K, E, F), 0.05, description=\"the round is cut into the material, on the inner side of face KE\")\nscene.constraint.gt(cross(G, K, F), 0.05, description=\"the round is cut into the material, on the inner side of face KG\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A": [28.8, 0.0], "B": [0.0, 18.5], "K": [5.6, 5.6], "E": [28.8, 5.6], "G": [5.6, 18.5], "F": [8.1, 8.1], "T": [8.1, 5.6], "U": [5.6, 8.1]}, "circles": {"F": 2.5}}, "implicit_facts": ["'runs smoothly into' means the round is TANGENT to both inner faces", "tangency puts F exactly r from each inner face, i.e. on the bisector of the inner corner", "T and U are the feet of the perpendiculars from F onto the two faces", "the inner faces are offset from the outer edges by the arm thickness"], "branch_facts": ["the round is on the material side of each face, not the outside"]} {"unique_id": "cad_t2_cad_fillet_bracket_0005", "problem_id": "cad_t2_cad_fillet_bracket_0005", "domain": "PyGeoX-CAD", "template": "cad_fillet_bracket", "tier": 2, "nl_description": "An L-shaped bracket is cut from 4.5 mm flat plate. Set up coordinates with the outside corner of the L at the origin O, the bottom outer edge running from O along the positive x-direction to A, and the upright outer edge running from O along the positive y-direction to B. The bottom outer edge OA is 34.7 mm long and the upright outer edge OB is 19.6 mm long. Both arms are 4.5 mm thick. The inner face of the bottom arm is the straight edge from the inner corner K out to E at the end of the bottom arm; the inner face of the upright is the straight edge from K up to G at the top of the upright. K is the point where those two inner faces would meet. To relieve the stress raiser there, the inner corner is blended with a 2.6 mm radius round centred at F. The round is cut into the material on the inside of the corner, and it runs smoothly into both inner faces with no step or kink, meeting the bottom inner face at T and the upright inner face at U.", "answer_schema": {"points": ["A", "B", "E", "F", "G", "K", "O", "T", "U"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Filleted L-bracket inner corner\n\nOracle formalization for cad_t2_cad_fillet_bracket_0005. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(50)\n\nO, A, B, K, E, G, F, T, U = scene.add.points([\"O\", \"A\", \"B\", \"K\", \"E\", \"G\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the outside corner of the L (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the outside corner of the L (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: the bottom outer edge OA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: the bottom arm runs in the +x direction\")\nscene.constraint.eq(B.x, 0, description=\"datum: the upright outer edge OB runs along the y-axis\")\nscene.constraint.gt(B.y, 1.0, description=\"datum: the upright arm runs in the +y direction\")\nscene.constraint.eq(O.distance(A), 34.7, description=\"bottom outer edge OA is 34.7 long\")\nscene.constraint.eq(O.distance(B), 19.6, description=\"upright outer edge OB is 19.6 long\")\nscene.constraint.eq(K.y, 4.5, description=\"the bottom arm is 4.5 thick, so the inner corner K sits 4.5 above edge OA\")\nscene.constraint.eq(K.x, 4.5, description=\"the upright arm is 4.5 thick, so the inner corner K sits 4.5 in from edge OB\")\nscene.constraint.eq(E.y, K.y, description=\"inner face KE is parallel to the bottom outer edge\")\nscene.constraint.eq(E.x, A.x, description=\"inner face KE ends flush with the end of the bottom arm\")\nscene.constraint.eq(G.x, K.x, description=\"inner face KG is parallel to the upright outer edge\")\nscene.constraint.eq(G.y, B.y, description=\"inner face KG ends flush with the top of the upright\")\nface_h = scene.add.line_segment(K, E)\nface_v = scene.add.line_segment(K, G)\nfillet = scene.add.circle(F)\nscene.constraint.eq(fillet.radius, 2.6, description=\"the blend round has radius 2.6\")\nscene.relate.tangent_to_circle(face_h, fillet, T)\nscene.relate.tangent_to_circle(face_v, fillet, U)\nscene.constraint.gt(cross(K, E, F), 0.05, description=\"the round is cut into the material, on the inner side of face KE\")\nscene.constraint.gt(cross(G, K, F), 0.05, description=\"the round is cut into the material, on the inner side of face KG\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A": [34.7, 0.0], "B": [0.0, 19.6], "K": [4.5, 4.5], "E": [34.7, 4.5], "G": [4.5, 19.6], "F": [7.1, 7.1], "T": [7.1, 4.5], "U": [4.5, 7.1]}, "circles": {"F": 2.6}}, "implicit_facts": ["'runs smoothly into' means the round is TANGENT to both inner faces", "tangency puts F exactly r from each inner face, i.e. on the bisector of the inner corner", "T and U are the feet of the perpendiculars from F onto the two faces", "the inner faces are offset from the outer edges by the arm thickness"], "branch_facts": ["the round is on the material side of each face, not the outside"]} {"unique_id": "cad_t2_cad_fillet_bracket_0006", "problem_id": "cad_t2_cad_fillet_bracket_0006", "domain": "PyGeoX-CAD", "template": "cad_fillet_bracket", "tier": 2, "nl_description": "An L-shaped bracket is cut from 5.1 mm flat plate. Set up coordinates with the outside corner of the L at the origin O, the bottom outer edge running from O along the positive x-direction to A, and the upright outer edge running from O along the positive y-direction to B. The bottom outer edge OA is 33.9 mm long and the upright outer edge OB is 19 mm long. Both arms are 5.1 mm thick. The inner face of the bottom arm is the straight edge from the inner corner K out to E at the end of the bottom arm; the inner face of the upright is the straight edge from K up to G at the top of the upright. K is the point where those two inner faces would meet. To relieve the stress raiser there, the inner corner is blended with a 3.8 mm radius round centred at F. The round is cut into the material on the inside of the corner, and it runs smoothly into both inner faces with no step or kink, meeting the bottom inner face at T and the upright inner face at U.", "answer_schema": {"points": ["A", "B", "E", "F", "G", "K", "O", "T", "U"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Filleted L-bracket inner corner\n\nOracle formalization for cad_t2_cad_fillet_bracket_0006. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(49)\n\nO, A, B, K, E, G, F, T, U = scene.add.points([\"O\", \"A\", \"B\", \"K\", \"E\", \"G\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the outside corner of the L (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the outside corner of the L (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: the bottom outer edge OA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: the bottom arm runs in the +x direction\")\nscene.constraint.eq(B.x, 0, description=\"datum: the upright outer edge OB runs along the y-axis\")\nscene.constraint.gt(B.y, 1.0, description=\"datum: the upright arm runs in the +y direction\")\nscene.constraint.eq(O.distance(A), 33.9, description=\"bottom outer edge OA is 33.9 long\")\nscene.constraint.eq(O.distance(B), 19, description=\"upright outer edge OB is 19 long\")\nscene.constraint.eq(K.y, 5.1, description=\"the bottom arm is 5.1 thick, so the inner corner K sits 5.1 above edge OA\")\nscene.constraint.eq(K.x, 5.1, description=\"the upright arm is 5.1 thick, so the inner corner K sits 5.1 in from edge OB\")\nscene.constraint.eq(E.y, K.y, description=\"inner face KE is parallel to the bottom outer edge\")\nscene.constraint.eq(E.x, A.x, description=\"inner face KE ends flush with the end of the bottom arm\")\nscene.constraint.eq(G.x, K.x, description=\"inner face KG is parallel to the upright outer edge\")\nscene.constraint.eq(G.y, B.y, description=\"inner face KG ends flush with the top of the upright\")\nface_h = scene.add.line_segment(K, E)\nface_v = scene.add.line_segment(K, G)\nfillet = scene.add.circle(F)\nscene.constraint.eq(fillet.radius, 3.8, description=\"the blend round has radius 3.8\")\nscene.relate.tangent_to_circle(face_h, fillet, T)\nscene.relate.tangent_to_circle(face_v, fillet, U)\nscene.constraint.gt(cross(K, E, F), 0.05, description=\"the round is cut into the material, on the inner side of face KE\")\nscene.constraint.gt(cross(G, K, F), 0.05, description=\"the round is cut into the material, on the inner side of face KG\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A": [33.9, 0.0], "B": [0.0, 19.0], "K": [5.1, 5.1], "E": [33.9, 5.1], "G": [5.1, 19.0], "F": [8.899999999999999, 8.899999999999999], "T": [8.899999999999999, 5.1], "U": [5.1, 8.899999999999999]}, "circles": {"F": 3.8}}, "implicit_facts": ["'runs smoothly into' means the round is TANGENT to both inner faces", "tangency puts F exactly r from each inner face, i.e. on the bisector of the inner corner", "T and U are the feet of the perpendiculars from F onto the two faces", "the inner faces are offset from the outer edges by the arm thickness"], "branch_facts": ["the round is on the material side of each face, not the outside"]} {"unique_id": "cad_t2_cad_fillet_bracket_0007", "problem_id": "cad_t2_cad_fillet_bracket_0007", "domain": "PyGeoX-CAD", "template": "cad_fillet_bracket", "tier": 2, "nl_description": "An L-shaped bracket is cut from 4.5 mm flat plate. Set up coordinates with the outside corner of the L at the origin O, the bottom outer edge running from O along the positive x-direction to A, and the upright outer edge running from O along the positive y-direction to B. The bottom outer edge OA is 26 mm long and the upright outer edge OB is 25.2 mm long. Both arms are 4.5 mm thick. The inner face of the bottom arm is the straight edge from the inner corner K out to E at the end of the bottom arm; the inner face of the upright is the straight edge from K up to G at the top of the upright. K is the point where those two inner faces would meet. To relieve the stress raiser there, the inner corner is blended with a 3.3 mm radius round centred at F. The round is cut into the material on the inside of the corner, and it runs smoothly into both inner faces with no step or kink, meeting the bottom inner face at T and the upright inner face at U.", "answer_schema": {"points": ["A", "B", "E", "F", "G", "K", "O", "T", "U"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Filleted L-bracket inner corner\n\nOracle formalization for cad_t2_cad_fillet_bracket_0007. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(39)\n\nO, A, B, K, E, G, F, T, U = scene.add.points([\"O\", \"A\", \"B\", \"K\", \"E\", \"G\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the outside corner of the L (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the outside corner of the L (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: the bottom outer edge OA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: the bottom arm runs in the +x direction\")\nscene.constraint.eq(B.x, 0, description=\"datum: the upright outer edge OB runs along the y-axis\")\nscene.constraint.gt(B.y, 1.0, description=\"datum: the upright arm runs in the +y direction\")\nscene.constraint.eq(O.distance(A), 26, description=\"bottom outer edge OA is 26 long\")\nscene.constraint.eq(O.distance(B), 25.2, description=\"upright outer edge OB is 25.2 long\")\nscene.constraint.eq(K.y, 4.5, description=\"the bottom arm is 4.5 thick, so the inner corner K sits 4.5 above edge OA\")\nscene.constraint.eq(K.x, 4.5, description=\"the upright arm is 4.5 thick, so the inner corner K sits 4.5 in from edge OB\")\nscene.constraint.eq(E.y, K.y, description=\"inner face KE is parallel to the bottom outer edge\")\nscene.constraint.eq(E.x, A.x, description=\"inner face KE ends flush with the end of the bottom arm\")\nscene.constraint.eq(G.x, K.x, description=\"inner face KG is parallel to the upright outer edge\")\nscene.constraint.eq(G.y, B.y, description=\"inner face KG ends flush with the top of the upright\")\nface_h = scene.add.line_segment(K, E)\nface_v = scene.add.line_segment(K, G)\nfillet = scene.add.circle(F)\nscene.constraint.eq(fillet.radius, 3.3, description=\"the blend round has radius 3.3\")\nscene.relate.tangent_to_circle(face_h, fillet, T)\nscene.relate.tangent_to_circle(face_v, fillet, U)\nscene.constraint.gt(cross(K, E, F), 0.05, description=\"the round is cut into the material, on the inner side of face KE\")\nscene.constraint.gt(cross(G, K, F), 0.05, description=\"the round is cut into the material, on the inner side of face KG\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A": [26.0, 0.0], "B": [0.0, 25.2], "K": [4.5, 4.5], "E": [26.0, 4.5], "G": [4.5, 25.2], "F": [7.8, 7.8], "T": [7.8, 4.5], "U": [4.5, 7.8]}, "circles": {"F": 3.3}}, "implicit_facts": ["'runs smoothly into' means the round is TANGENT to both inner faces", "tangency puts F exactly r from each inner face, i.e. on the bisector of the inner corner", "T and U are the feet of the perpendiculars from F onto the two faces", "the inner faces are offset from the outer edges by the arm thickness"], "branch_facts": ["the round is on the material side of each face, not the outside"]} {"unique_id": "cad_t2_cad_fillet_bracket_0008", "problem_id": "cad_t2_cad_fillet_bracket_0008", "domain": "PyGeoX-CAD", "template": "cad_fillet_bracket", "tier": 2, "nl_description": "An L-shaped bracket is cut from 5.7 mm flat plate. Set up coordinates with the outside corner of the L at the origin O, the bottom outer edge running from O along the positive x-direction to A, and the upright outer edge running from O along the positive y-direction to B. The bottom outer edge OA is 24.2 mm long and the upright outer edge OB is 24 mm long. Both arms are 5.7 mm thick. The inner face of the bottom arm is the straight edge from the inner corner K out to E at the end of the bottom arm; the inner face of the upright is the straight edge from K up to G at the top of the upright. K is the point where those two inner faces would meet. To relieve the stress raiser there, the inner corner is blended with a 3.5 mm radius round centred at F. The round is cut into the material on the inside of the corner, and it runs smoothly into both inner faces with no step or kink, meeting the bottom inner face at T and the upright inner face at U.", "answer_schema": {"points": ["A", "B", "E", "F", "G", "K", "O", "T", "U"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Filleted L-bracket inner corner\n\nOracle formalization for cad_t2_cad_fillet_bracket_0008. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(36)\n\nO, A, B, K, E, G, F, T, U = scene.add.points([\"O\", \"A\", \"B\", \"K\", \"E\", \"G\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the outside corner of the L (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the outside corner of the L (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: the bottom outer edge OA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: the bottom arm runs in the +x direction\")\nscene.constraint.eq(B.x, 0, description=\"datum: the upright outer edge OB runs along the y-axis\")\nscene.constraint.gt(B.y, 1.0, description=\"datum: the upright arm runs in the +y direction\")\nscene.constraint.eq(O.distance(A), 24.2, description=\"bottom outer edge OA is 24.2 long\")\nscene.constraint.eq(O.distance(B), 24, description=\"upright outer edge OB is 24 long\")\nscene.constraint.eq(K.y, 5.7, description=\"the bottom arm is 5.7 thick, so the inner corner K sits 5.7 above edge OA\")\nscene.constraint.eq(K.x, 5.7, description=\"the upright arm is 5.7 thick, so the inner corner K sits 5.7 in from edge OB\")\nscene.constraint.eq(E.y, K.y, description=\"inner face KE is parallel to the bottom outer edge\")\nscene.constraint.eq(E.x, A.x, description=\"inner face KE ends flush with the end of the bottom arm\")\nscene.constraint.eq(G.x, K.x, description=\"inner face KG is parallel to the upright outer edge\")\nscene.constraint.eq(G.y, B.y, description=\"inner face KG ends flush with the top of the upright\")\nface_h = scene.add.line_segment(K, E)\nface_v = scene.add.line_segment(K, G)\nfillet = scene.add.circle(F)\nscene.constraint.eq(fillet.radius, 3.5, description=\"the blend round has radius 3.5\")\nscene.relate.tangent_to_circle(face_h, fillet, T)\nscene.relate.tangent_to_circle(face_v, fillet, U)\nscene.constraint.gt(cross(K, E, F), 0.05, description=\"the round is cut into the material, on the inner side of face KE\")\nscene.constraint.gt(cross(G, K, F), 0.05, description=\"the round is cut into the material, on the inner side of face KG\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A": [24.2, 0.0], "B": [0.0, 24.0], "K": [5.7, 5.7], "E": [24.2, 5.7], "G": [5.7, 24.0], "F": [9.2, 9.2], "T": [9.2, 5.7], "U": [5.7, 9.2]}, "circles": {"F": 3.5}}, "implicit_facts": ["'runs smoothly into' means the round is TANGENT to both inner faces", "tangency puts F exactly r from each inner face, i.e. on the bisector of the inner corner", "T and U are the feet of the perpendiculars from F onto the two faces", "the inner faces are offset from the outer edges by the arm thickness"], "branch_facts": ["the round is on the material side of each face, not the outside"]} {"unique_id": "cad_t2_cad_fillet_bracket_0009", "problem_id": "cad_t2_cad_fillet_bracket_0009", "domain": "PyGeoX-CAD", "template": "cad_fillet_bracket", "tier": 2, "nl_description": "An L-shaped bracket is cut from 4.1 mm flat plate. Set up coordinates with the outside corner of the L at the origin O, the bottom outer edge running from O along the positive x-direction to A, and the upright outer edge running from O along the positive y-direction to B. The bottom outer edge OA is 28.2 mm long and the upright outer edge OB is 26.5 mm long. Both arms are 4.1 mm thick. The inner face of the bottom arm is the straight edge from the inner corner K out to E at the end of the bottom arm; the inner face of the upright is the straight edge from K up to G at the top of the upright. K is the point where those two inner faces would meet. To relieve the stress raiser there, the inner corner is blended with a 2.9 mm radius round centred at F. The round is cut into the material on the inside of the corner, and it runs smoothly into both inner faces with no step or kink, meeting the bottom inner face at T and the upright inner face at U.", "answer_schema": {"points": ["A", "B", "E", "F", "G", "K", "O", "T", "U"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Filleted L-bracket inner corner\n\nOracle formalization for cad_t2_cad_fillet_bracket_0009. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(42)\n\nO, A, B, K, E, G, F, T, U = scene.add.points([\"O\", \"A\", \"B\", \"K\", \"E\", \"G\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the outside corner of the L (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the outside corner of the L (y)\")\nscene.constraint.eq(A.y, 0, description=\"datum: the bottom outer edge OA runs along the x-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: the bottom arm runs in the +x direction\")\nscene.constraint.eq(B.x, 0, description=\"datum: the upright outer edge OB runs along the y-axis\")\nscene.constraint.gt(B.y, 1.0, description=\"datum: the upright arm runs in the +y direction\")\nscene.constraint.eq(O.distance(A), 28.2, description=\"bottom outer edge OA is 28.2 long\")\nscene.constraint.eq(O.distance(B), 26.5, description=\"upright outer edge OB is 26.5 long\")\nscene.constraint.eq(K.y, 4.1, description=\"the bottom arm is 4.1 thick, so the inner corner K sits 4.1 above edge OA\")\nscene.constraint.eq(K.x, 4.1, description=\"the upright arm is 4.1 thick, so the inner corner K sits 4.1 in from edge OB\")\nscene.constraint.eq(E.y, K.y, description=\"inner face KE is parallel to the bottom outer edge\")\nscene.constraint.eq(E.x, A.x, description=\"inner face KE ends flush with the end of the bottom arm\")\nscene.constraint.eq(G.x, K.x, description=\"inner face KG is parallel to the upright outer edge\")\nscene.constraint.eq(G.y, B.y, description=\"inner face KG ends flush with the top of the upright\")\nface_h = scene.add.line_segment(K, E)\nface_v = scene.add.line_segment(K, G)\nfillet = scene.add.circle(F)\nscene.constraint.eq(fillet.radius, 2.9, description=\"the blend round has radius 2.9\")\nscene.relate.tangent_to_circle(face_h, fillet, T)\nscene.relate.tangent_to_circle(face_v, fillet, U)\nscene.constraint.gt(cross(K, E, F), 0.05, description=\"the round is cut into the material, on the inner side of face KE\")\nscene.constraint.gt(cross(G, K, F), 0.05, description=\"the round is cut into the material, on the inner side of face KG\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A": [28.2, 0.0], "B": [0.0, 26.5], "K": [4.1, 4.1], "E": [28.2, 4.1], "G": [4.1, 26.5], "F": [7.0, 7.0], "T": [7.0, 4.1], "U": [4.1, 7.0]}, "circles": {"F": 2.9}}, "implicit_facts": ["'runs smoothly into' means the round is TANGENT to both inner faces", "tangency puts F exactly r from each inner face, i.e. on the bisector of the inner corner", "T and U are the feet of the perpendiculars from F onto the two faces", "the inner faces are offset from the outer edges by the arm thickness"], "branch_facts": ["the round is on the material side of each face, not the outside"]} {"unique_id": "cad_t2_cad_keyway_0037", "problem_id": "cad_t2_cad_keyway_0037", "domain": "PyGeoX-CAD", "template": "cad_keyway", "tier": 2, "nl_description": "A shaft cross-section is a circle 26.1 mm across. A keyway is machined into it: a square-sided notch 8.1 mm wide with a flat bottom. Set up coordinates with the shaft centre at the origin O and the keyway cut symmetrically about the positive y-direction, so that the keyway centreline is the y-axis. The flat bottom of the keyway runs from its left corner K to its right corner L. The two side walls run from those corners out to the shaft surface, breaking out at M above K and at N above L. S is the point on the shaft surface directly opposite the keyway. Measured from the flat bottom of the keyway across to the far side of the shaft at S, the distance is 21.8 mm.", "answer_schema": {"points": ["K", "L", "M", "N", "O", "S"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Keyway in a shaft cross-section\n\nOracle formalization for cad_t2_cad_keyway_0037. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(22)\n\nO, S, K, L, M, N = scene.add.points([\"O\", \"S\", \"K\", \"L\", \"M\", \"N\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the shaft centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the shaft centre (y)\")\nscene.constraint.eq(K.x + L.x, 0, description=\"datum: the keyway is cut symmetrically about the y-axis\")\nscene.constraint.eq(L.x - K.x, 8.1, description=\"the keyway is 8.1 wide between its two side walls\")\nscene.constraint.gt(L.x, 0.05, description=\"L is the right-hand bottom corner and K the left-hand one\")\nscene.constraint.eq(K.y, L.y, description=\"the bottom of the keyway is flat and square to the keyway centreline\")\nshaft = scene.add.circle(O)\nscene.constraint.eq(2 * shaft.radius, 26.1, description=\"the shaft is 26.1 across\")\nscene.relate.point_lies_on(M, shaft)\nscene.relate.point_lies_on(N, shaft)\nscene.constraint.eq(M.x, K.x, description=\"the left side wall runs straight up from K, parallel to the keyway centreline, and breaks out at M on the shaft surface\")\nscene.constraint.eq(N.x, L.x, description=\"the right side wall runs straight up from L and breaks out at N on the shaft surface\")\nscene.constraint.gt(M.y - K.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.constraint.gt(N.y - L.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.relate.point_lies_on(S, shaft)\nscene.constraint.eq(S.x, O.x, description=\"S is the point on the shaft surface directly opposite the keyway\")\nscene.constraint.lt(S.y, -0.05, description=\"S is on the far side of the shaft from the keyway\")\nscene.constraint.eq(K.y - S.y, 21.8, description=\"from the flat bottom of the keyway across to the far side of the shaft measures 21.8\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -13.05], "K": [-4.05, 8.75], "L": [4.05, 8.75], "M": [-4.05, 12.4056438768812], "N": [4.05, 12.4056438768812]}, "circles": {"O": 13.05}}, "implicit_facts": ["the shaft radius is half the stated across-size", "the two side walls sit at +/- half the keyway width either side of the centreline", "the side walls are parallel to the keyway centreline and the flat bottom is square to it", "S is diametrically opposite, so the across measurement equals the bottom height plus the radius", "the keyway depth below the shaft surface is the shaft size minus the across measurement"], "branch_facts": ["the side walls run UP from the flat bottom to the surface, not down"]} {"unique_id": "cad_t2_cad_keyway_0038", "problem_id": "cad_t2_cad_keyway_0038", "domain": "PyGeoX-CAD", "template": "cad_keyway", "tier": 2, "nl_description": "A shaft cross-section is a circle 22.3 mm across. A keyway is machined into it: a square-sided notch 7.3 mm wide with a flat bottom. Set up coordinates with the shaft centre at the origin O and the keyway cut symmetrically about the positive y-direction, so that the keyway centreline is the y-axis. The flat bottom of the keyway runs from its left corner K to its right corner L. The two side walls run from those corners out to the shaft surface, breaking out at M above K and at N above L. S is the point on the shaft surface directly opposite the keyway. Measured from the flat bottom of the keyway across to the far side of the shaft at S, the distance is 19.7 mm.", "answer_schema": {"points": ["K", "L", "M", "N", "O", "S"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Keyway in a shaft cross-section\n\nOracle formalization for cad_t2_cad_keyway_0038. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(19)\n\nO, S, K, L, M, N = scene.add.points([\"O\", \"S\", \"K\", \"L\", \"M\", \"N\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the shaft centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the shaft centre (y)\")\nscene.constraint.eq(K.x + L.x, 0, description=\"datum: the keyway is cut symmetrically about the y-axis\")\nscene.constraint.eq(L.x - K.x, 7.3, description=\"the keyway is 7.3 wide between its two side walls\")\nscene.constraint.gt(L.x, 0.05, description=\"L is the right-hand bottom corner and K the left-hand one\")\nscene.constraint.eq(K.y, L.y, description=\"the bottom of the keyway is flat and square to the keyway centreline\")\nshaft = scene.add.circle(O)\nscene.constraint.eq(2 * shaft.radius, 22.3, description=\"the shaft is 22.3 across\")\nscene.relate.point_lies_on(M, shaft)\nscene.relate.point_lies_on(N, shaft)\nscene.constraint.eq(M.x, K.x, description=\"the left side wall runs straight up from K, parallel to the keyway centreline, and breaks out at M on the shaft surface\")\nscene.constraint.eq(N.x, L.x, description=\"the right side wall runs straight up from L and breaks out at N on the shaft surface\")\nscene.constraint.gt(M.y - K.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.constraint.gt(N.y - L.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.relate.point_lies_on(S, shaft)\nscene.constraint.eq(S.x, O.x, description=\"S is the point on the shaft surface directly opposite the keyway\")\nscene.constraint.lt(S.y, -0.05, description=\"S is on the far side of the shaft from the keyway\")\nscene.constraint.eq(K.y - S.y, 19.7, description=\"from the flat bottom of the keyway across to the far side of the shaft measures 19.7\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -11.15], "K": [-3.65, 8.55], "L": [3.65, 8.55], "M": [-3.65, 10.535653752852738], "N": [3.65, 10.535653752852738]}, "circles": {"O": 11.15}}, "implicit_facts": ["the shaft radius is half the stated across-size", "the two side walls sit at +/- half the keyway width either side of the centreline", "the side walls are parallel to the keyway centreline and the flat bottom is square to it", "S is diametrically opposite, so the across measurement equals the bottom height plus the radius", "the keyway depth below the shaft surface is the shaft size minus the across measurement"], "branch_facts": ["the side walls run UP from the flat bottom to the surface, not down"]} {"unique_id": "cad_t2_cad_keyway_0039", "problem_id": "cad_t2_cad_keyway_0039", "domain": "PyGeoX-CAD", "template": "cad_keyway", "tier": 2, "nl_description": "A shaft cross-section is a circle 25.8 mm across. A keyway is machined into it: a square-sided notch 6.8 mm wide with a flat bottom. Set up coordinates with the shaft centre at the origin O and the keyway cut symmetrically about the positive y-direction, so that the keyway centreline is the y-axis. The flat bottom of the keyway runs from its left corner K to its right corner L. The two side walls run from those corners out to the shaft surface, breaking out at M above K and at N above L. S is the point on the shaft surface directly opposite the keyway. Measured from the flat bottom of the keyway across to the far side of the shaft at S, the distance is 22.6 mm.", "answer_schema": {"points": ["K", "L", "M", "N", "O", "S"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Keyway in a shaft cross-section\n\nOracle formalization for cad_t2_cad_keyway_0039. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(22)\n\nO, S, K, L, M, N = scene.add.points([\"O\", \"S\", \"K\", \"L\", \"M\", \"N\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the shaft centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the shaft centre (y)\")\nscene.constraint.eq(K.x + L.x, 0, description=\"datum: the keyway is cut symmetrically about the y-axis\")\nscene.constraint.eq(L.x - K.x, 6.8, description=\"the keyway is 6.8 wide between its two side walls\")\nscene.constraint.gt(L.x, 0.05, description=\"L is the right-hand bottom corner and K the left-hand one\")\nscene.constraint.eq(K.y, L.y, description=\"the bottom of the keyway is flat and square to the keyway centreline\")\nshaft = scene.add.circle(O)\nscene.constraint.eq(2 * shaft.radius, 25.8, description=\"the shaft is 25.8 across\")\nscene.relate.point_lies_on(M, shaft)\nscene.relate.point_lies_on(N, shaft)\nscene.constraint.eq(M.x, K.x, description=\"the left side wall runs straight up from K, parallel to the keyway centreline, and breaks out at M on the shaft surface\")\nscene.constraint.eq(N.x, L.x, description=\"the right side wall runs straight up from L and breaks out at N on the shaft surface\")\nscene.constraint.gt(M.y - K.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.constraint.gt(N.y - L.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.relate.point_lies_on(S, shaft)\nscene.constraint.eq(S.x, O.x, description=\"S is the point on the shaft surface directly opposite the keyway\")\nscene.constraint.lt(S.y, -0.05, description=\"S is on the far side of the shaft from the keyway\")\nscene.constraint.eq(K.y - S.y, 22.6, description=\"from the flat bottom of the keyway across to the far side of the shaft measures 22.6\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -12.9], "K": [-3.4, 9.7], "L": [3.4, 9.7], "M": [-3.4, 12.443873994861889], "N": [3.4, 12.443873994861889]}, "circles": {"O": 12.9}}, "implicit_facts": ["the shaft radius is half the stated across-size", "the two side walls sit at +/- half the keyway width either side of the centreline", "the side walls are parallel to the keyway centreline and the flat bottom is square to it", "S is diametrically opposite, so the across measurement equals the bottom height plus the radius", "the keyway depth below the shaft surface is the shaft size minus the across measurement"], "branch_facts": ["the side walls run UP from the flat bottom to the surface, not down"]} {"unique_id": "cad_t2_cad_keyway_0040", "problem_id": "cad_t2_cad_keyway_0040", "domain": "PyGeoX-CAD", "template": "cad_keyway", "tier": 2, "nl_description": "A shaft cross-section is a circle 21 mm across. A keyway is machined into it: a square-sided notch 6.1 mm wide with a flat bottom. Set up coordinates with the shaft centre at the origin O and the keyway cut symmetrically about the positive y-direction, so that the keyway centreline is the y-axis. The flat bottom of the keyway runs from its left corner K to its right corner L. The two side walls run from those corners out to the shaft surface, breaking out at M above K and at N above L. S is the point on the shaft surface directly opposite the keyway. Measured from the flat bottom of the keyway across to the far side of the shaft at S, the distance is 18 mm.", "answer_schema": {"points": ["K", "L", "M", "N", "O", "S"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Keyway in a shaft cross-section\n\nOracle formalization for cad_t2_cad_keyway_0040. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(19)\n\nO, S, K, L, M, N = scene.add.points([\"O\", \"S\", \"K\", \"L\", \"M\", \"N\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the shaft centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the shaft centre (y)\")\nscene.constraint.eq(K.x + L.x, 0, description=\"datum: the keyway is cut symmetrically about the y-axis\")\nscene.constraint.eq(L.x - K.x, 6.1, description=\"the keyway is 6.1 wide between its two side walls\")\nscene.constraint.gt(L.x, 0.05, description=\"L is the right-hand bottom corner and K the left-hand one\")\nscene.constraint.eq(K.y, L.y, description=\"the bottom of the keyway is flat and square to the keyway centreline\")\nshaft = scene.add.circle(O)\nscene.constraint.eq(2 * shaft.radius, 21, description=\"the shaft is 21 across\")\nscene.relate.point_lies_on(M, shaft)\nscene.relate.point_lies_on(N, shaft)\nscene.constraint.eq(M.x, K.x, description=\"the left side wall runs straight up from K, parallel to the keyway centreline, and breaks out at M on the shaft surface\")\nscene.constraint.eq(N.x, L.x, description=\"the right side wall runs straight up from L and breaks out at N on the shaft surface\")\nscene.constraint.gt(M.y - K.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.constraint.gt(N.y - L.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.relate.point_lies_on(S, shaft)\nscene.constraint.eq(S.x, O.x, description=\"S is the point on the shaft surface directly opposite the keyway\")\nscene.constraint.lt(S.y, -0.05, description=\"S is on the far side of the shaft from the keyway\")\nscene.constraint.eq(K.y - S.y, 18, description=\"from the flat bottom of the keyway across to the far side of the shaft measures 18\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -10.5], "K": [-3.05, 7.5], "L": [3.05, 7.5], "M": [-3.05, 10.047263308981208], "N": [3.05, 10.047263308981208]}, "circles": {"O": 10.5}}, "implicit_facts": ["the shaft radius is half the stated across-size", "the two side walls sit at +/- half the keyway width either side of the centreline", "the side walls are parallel to the keyway centreline and the flat bottom is square to it", "S is diametrically opposite, so the across measurement equals the bottom height plus the radius", "the keyway depth below the shaft surface is the shaft size minus the across measurement"], "branch_facts": ["the side walls run UP from the flat bottom to the surface, not down"]} {"unique_id": "cad_t2_cad_keyway_0041", "problem_id": "cad_t2_cad_keyway_0041", "domain": "PyGeoX-CAD", "template": "cad_keyway", "tier": 2, "nl_description": "A shaft cross-section is a circle 22.1 mm across. A keyway is machined into it: a square-sided notch 7.5 mm wide with a flat bottom. Set up coordinates with the shaft centre at the origin O and the keyway cut symmetrically about the positive y-direction, so that the keyway centreline is the y-axis. The flat bottom of the keyway runs from its left corner K to its right corner L. The two side walls run from those corners out to the shaft surface, breaking out at M above K and at N above L. S is the point on the shaft surface directly opposite the keyway. Measured from the flat bottom of the keyway across to the far side of the shaft at S, the distance is 18.7 mm.", "answer_schema": {"points": ["K", "L", "M", "N", "O", "S"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Keyway in a shaft cross-section\n\nOracle formalization for cad_t2_cad_keyway_0041. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(19)\n\nO, S, K, L, M, N = scene.add.points([\"O\", \"S\", \"K\", \"L\", \"M\", \"N\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the shaft centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the shaft centre (y)\")\nscene.constraint.eq(K.x + L.x, 0, description=\"datum: the keyway is cut symmetrically about the y-axis\")\nscene.constraint.eq(L.x - K.x, 7.5, description=\"the keyway is 7.5 wide between its two side walls\")\nscene.constraint.gt(L.x, 0.05, description=\"L is the right-hand bottom corner and K the left-hand one\")\nscene.constraint.eq(K.y, L.y, description=\"the bottom of the keyway is flat and square to the keyway centreline\")\nshaft = scene.add.circle(O)\nscene.constraint.eq(2 * shaft.radius, 22.1, description=\"the shaft is 22.1 across\")\nscene.relate.point_lies_on(M, shaft)\nscene.relate.point_lies_on(N, shaft)\nscene.constraint.eq(M.x, K.x, description=\"the left side wall runs straight up from K, parallel to the keyway centreline, and breaks out at M on the shaft surface\")\nscene.constraint.eq(N.x, L.x, description=\"the right side wall runs straight up from L and breaks out at N on the shaft surface\")\nscene.constraint.gt(M.y - K.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.constraint.gt(N.y - L.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.relate.point_lies_on(S, shaft)\nscene.constraint.eq(S.x, O.x, description=\"S is the point on the shaft surface directly opposite the keyway\")\nscene.constraint.lt(S.y, -0.05, description=\"S is on the far side of the shaft from the keyway\")\nscene.constraint.eq(K.y - S.y, 18.7, description=\"from the flat bottom of the keyway across to the far side of the shaft measures 18.7\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -11.05], "K": [-3.75, 7.65], "L": [3.75, 7.65], "M": [-3.75, 10.394229168149028], "N": [3.75, 10.394229168149028]}, "circles": {"O": 11.05}}, "implicit_facts": ["the shaft radius is half the stated across-size", "the two side walls sit at +/- half the keyway width either side of the centreline", "the side walls are parallel to the keyway centreline and the flat bottom is square to it", "S is diametrically opposite, so the across measurement equals the bottom height plus the radius", "the keyway depth below the shaft surface is the shaft size minus the across measurement"], "branch_facts": ["the side walls run UP from the flat bottom to the surface, not down"]} {"unique_id": "cad_t2_cad_keyway_0042", "problem_id": "cad_t2_cad_keyway_0042", "domain": "PyGeoX-CAD", "template": "cad_keyway", "tier": 2, "nl_description": "A shaft cross-section is a circle 23.5 mm across. A keyway is machined into it: a square-sided notch 8.2 mm wide with a flat bottom. Set up coordinates with the shaft centre at the origin O and the keyway cut symmetrically about the positive y-direction, so that the keyway centreline is the y-axis. The flat bottom of the keyway runs from its left corner K to its right corner L. The two side walls run from those corners out to the shaft surface, breaking out at M above K and at N above L. S is the point on the shaft surface directly opposite the keyway. Measured from the flat bottom of the keyway across to the far side of the shaft at S, the distance is 21 mm.", "answer_schema": {"points": ["K", "L", "M", "N", "O", "S"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Keyway in a shaft cross-section\n\nOracle formalization for cad_t2_cad_keyway_0042. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(20)\n\nO, S, K, L, M, N = scene.add.points([\"O\", \"S\", \"K\", \"L\", \"M\", \"N\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the shaft centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the shaft centre (y)\")\nscene.constraint.eq(K.x + L.x, 0, description=\"datum: the keyway is cut symmetrically about the y-axis\")\nscene.constraint.eq(L.x - K.x, 8.2, description=\"the keyway is 8.2 wide between its two side walls\")\nscene.constraint.gt(L.x, 0.05, description=\"L is the right-hand bottom corner and K the left-hand one\")\nscene.constraint.eq(K.y, L.y, description=\"the bottom of the keyway is flat and square to the keyway centreline\")\nshaft = scene.add.circle(O)\nscene.constraint.eq(2 * shaft.radius, 23.5, description=\"the shaft is 23.5 across\")\nscene.relate.point_lies_on(M, shaft)\nscene.relate.point_lies_on(N, shaft)\nscene.constraint.eq(M.x, K.x, description=\"the left side wall runs straight up from K, parallel to the keyway centreline, and breaks out at M on the shaft surface\")\nscene.constraint.eq(N.x, L.x, description=\"the right side wall runs straight up from L and breaks out at N on the shaft surface\")\nscene.constraint.gt(M.y - K.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.constraint.gt(N.y - L.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.relate.point_lies_on(S, shaft)\nscene.constraint.eq(S.x, O.x, description=\"S is the point on the shaft surface directly opposite the keyway\")\nscene.constraint.lt(S.y, -0.05, description=\"S is on the far side of the shaft from the keyway\")\nscene.constraint.eq(K.y - S.y, 21, description=\"from the flat bottom of the keyway across to the far side of the shaft measures 21\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -11.75], "K": [-4.1, 9.25], "L": [4.1, 9.25], "M": [-4.1, 11.011471291339772], "N": [4.1, 11.011471291339772]}, "circles": {"O": 11.75}}, "implicit_facts": ["the shaft radius is half the stated across-size", "the two side walls sit at +/- half the keyway width either side of the centreline", "the side walls are parallel to the keyway centreline and the flat bottom is square to it", "S is diametrically opposite, so the across measurement equals the bottom height plus the radius", "the keyway depth below the shaft surface is the shaft size minus the across measurement"], "branch_facts": ["the side walls run UP from the flat bottom to the surface, not down"]} {"unique_id": "cad_t2_cad_keyway_0043", "problem_id": "cad_t2_cad_keyway_0043", "domain": "PyGeoX-CAD", "template": "cad_keyway", "tier": 2, "nl_description": "A shaft cross-section is a circle 33.7 mm across. A keyway is machined into it: a square-sided notch 5.9 mm wide with a flat bottom. Set up coordinates with the shaft centre at the origin O and the keyway cut symmetrically about the positive y-direction, so that the keyway centreline is the y-axis. The flat bottom of the keyway runs from its left corner K to its right corner L. The two side walls run from those corners out to the shaft surface, breaking out at M above K and at N above L. S is the point on the shaft surface directly opposite the keyway. Measured from the flat bottom of the keyway across to the far side of the shaft at S, the distance is 30.7 mm.", "answer_schema": {"points": ["K", "L", "M", "N", "O", "S"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Keyway in a shaft cross-section\n\nOracle formalization for cad_t2_cad_keyway_0043. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(27)\n\nO, S, K, L, M, N = scene.add.points([\"O\", \"S\", \"K\", \"L\", \"M\", \"N\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the shaft centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the shaft centre (y)\")\nscene.constraint.eq(K.x + L.x, 0, description=\"datum: the keyway is cut symmetrically about the y-axis\")\nscene.constraint.eq(L.x - K.x, 5.9, description=\"the keyway is 5.9 wide between its two side walls\")\nscene.constraint.gt(L.x, 0.05, description=\"L is the right-hand bottom corner and K the left-hand one\")\nscene.constraint.eq(K.y, L.y, description=\"the bottom of the keyway is flat and square to the keyway centreline\")\nshaft = scene.add.circle(O)\nscene.constraint.eq(2 * shaft.radius, 33.7, description=\"the shaft is 33.7 across\")\nscene.relate.point_lies_on(M, shaft)\nscene.relate.point_lies_on(N, shaft)\nscene.constraint.eq(M.x, K.x, description=\"the left side wall runs straight up from K, parallel to the keyway centreline, and breaks out at M on the shaft surface\")\nscene.constraint.eq(N.x, L.x, description=\"the right side wall runs straight up from L and breaks out at N on the shaft surface\")\nscene.constraint.gt(M.y - K.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.constraint.gt(N.y - L.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.relate.point_lies_on(S, shaft)\nscene.constraint.eq(S.x, O.x, description=\"S is the point on the shaft surface directly opposite the keyway\")\nscene.constraint.lt(S.y, -0.05, description=\"S is on the far side of the shaft from the keyway\")\nscene.constraint.eq(K.y - S.y, 30.7, description=\"from the flat bottom of the keyway across to the far side of the shaft measures 30.7\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -16.85], "K": [-2.95, 13.850000000000001], "L": [2.95, 13.850000000000001], "M": [-2.95, 16.58975587523819], "N": [2.95, 16.58975587523819]}, "circles": {"O": 16.85}}, "implicit_facts": ["the shaft radius is half the stated across-size", "the two side walls sit at +/- half the keyway width either side of the centreline", "the side walls are parallel to the keyway centreline and the flat bottom is square to it", "S is diametrically opposite, so the across measurement equals the bottom height plus the radius", "the keyway depth below the shaft surface is the shaft size minus the across measurement"], "branch_facts": ["the side walls run UP from the flat bottom to the surface, not down"]} {"unique_id": "cad_t2_cad_keyway_0044", "problem_id": "cad_t2_cad_keyway_0044", "domain": "PyGeoX-CAD", "template": "cad_keyway", "tier": 2, "nl_description": "A shaft cross-section is a circle 26.4 mm across. A keyway is machined into it: a square-sided notch 5.2 mm wide with a flat bottom. Set up coordinates with the shaft centre at the origin O and the keyway cut symmetrically about the positive y-direction, so that the keyway centreline is the y-axis. The flat bottom of the keyway runs from its left corner K to its right corner L. The two side walls run from those corners out to the shaft surface, breaking out at M above K and at N above L. S is the point on the shaft surface directly opposite the keyway. Measured from the flat bottom of the keyway across to the far side of the shaft at S, the distance is 23.8 mm.", "answer_schema": {"points": ["K", "L", "M", "N", "O", "S"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Keyway in a shaft cross-section\n\nOracle formalization for cad_t2_cad_keyway_0044. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(22)\n\nO, S, K, L, M, N = scene.add.points([\"O\", \"S\", \"K\", \"L\", \"M\", \"N\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the shaft centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the shaft centre (y)\")\nscene.constraint.eq(K.x + L.x, 0, description=\"datum: the keyway is cut symmetrically about the y-axis\")\nscene.constraint.eq(L.x - K.x, 5.2, description=\"the keyway is 5.2 wide between its two side walls\")\nscene.constraint.gt(L.x, 0.05, description=\"L is the right-hand bottom corner and K the left-hand one\")\nscene.constraint.eq(K.y, L.y, description=\"the bottom of the keyway is flat and square to the keyway centreline\")\nshaft = scene.add.circle(O)\nscene.constraint.eq(2 * shaft.radius, 26.4, description=\"the shaft is 26.4 across\")\nscene.relate.point_lies_on(M, shaft)\nscene.relate.point_lies_on(N, shaft)\nscene.constraint.eq(M.x, K.x, description=\"the left side wall runs straight up from K, parallel to the keyway centreline, and breaks out at M on the shaft surface\")\nscene.constraint.eq(N.x, L.x, description=\"the right side wall runs straight up from L and breaks out at N on the shaft surface\")\nscene.constraint.gt(M.y - K.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.constraint.gt(N.y - L.y, 0.05, description=\"the side walls run upward from the flat bottom to the shaft surface\")\nscene.relate.point_lies_on(S, shaft)\nscene.constraint.eq(S.x, O.x, description=\"S is the point on the shaft surface directly opposite the keyway\")\nscene.constraint.lt(S.y, -0.05, description=\"S is on the far side of the shaft from the keyway\")\nscene.constraint.eq(K.y - S.y, 23.8, description=\"from the flat bottom of the keyway across to the far side of the shaft measures 23.8\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -13.2], "K": [-2.6, 10.6], "L": [2.6, 10.6], "M": [-2.6, 12.941406415069421], "N": [2.6, 12.941406415069421]}, "circles": {"O": 13.2}}, "implicit_facts": ["the shaft radius is half the stated across-size", "the two side walls sit at +/- half the keyway width either side of the centreline", "the side walls are parallel to the keyway centreline and the flat bottom is square to it", "S is diametrically opposite, so the across measurement equals the bottom height plus the radius", "the keyway depth below the shaft surface is the shaft size minus the across measurement"], "branch_facts": ["the side walls run UP from the flat bottom to the surface, not down"]} {"unique_id": "cad_t2_cad_rounded_slot_0010", "problem_id": "cad_t2_cad_rounded_slot_0010", "domain": "PyGeoX-CAD", "template": "cad_rounded_slot", "tier": 2, "nl_description": "A rectangular cover plate measures 30.1 mm by 21.3 mm. Set up coordinates with the plate's bottom-left corner at the origin P, the 30.1 mm edge running from P along the positive x-direction to Q, and the 21.3 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot is machined along the plate's long centreline. The slot is 6.8 mm wide and both of its ends are rounded off as semicircles. The two end centres are C at the left-hand end and D at the right-hand end; they are 14.7 mm apart and are placed symmetrically about the middle of the plate. The upper straight side of the slot runs smoothly into both rounded ends, touching the left end at T and the right end at V.", "answer_schema": {"points": ["C", "D", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Rounded-end slot in a cover plate\n\nOracle formalization for cad_t2_cad_rounded_slot_0010. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(44)\n\nP, Q, R, S, C, D, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 30.1, description=\"plate is 30.1 long along PQ\")\nscene.constraint.eq(P.distance(S), 21.3, description=\"plate is 21.3 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(2 * C.y, P.y + S.y, description=\"slot end centre C lies midway between the two long edges\")\nscene.constraint.eq(2 * D.y, P.y + S.y, description=\"slot end centre D lies midway between the two long edges\")\nscene.constraint.eq(C.distance(D), 14.7, description=\"the two slot end centres are 14.7 apart\")\nscene.constraint.eq(C.x + D.x, P.x + Q.x, description=\"the two slot end centres are placed symmetrically about the middle of the plate\")\nscene.constraint.gt(D.x - C.x, 0.05, description=\"C is the left-hand end of the slot and D the right-hand end\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 6.8, description=\"the slot is 6.8 wide and its left end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 6.8, description=\"the slot is 6.8 wide and its right end is a semicircle spanning that width\")\nupper_side = scene.add.line(T, V)\nscene.relate.tangent_to_circle(upper_side, arc_C, T)\nscene.relate.tangent_to_circle(upper_side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper side of the slot\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the upper side of the slot\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [30.1, 0.0], "R": [30.1, 21.3], "S": [0.0, 21.3], "C": [7.700000000000001, 10.65], "D": [22.4, 10.65], "T": [7.700000000000001, 14.05], "V": [22.4, 14.05]}, "circles": {"C": 3.4, "D": 3.4}}, "implicit_facts": ["each end arc has radius = HALF the stated slot width", "both end centres lie on the plate's long centreline", "symmetric placement puts the centres at +/- d/2 from the plate centre", "'runs smoothly into' means the straight side is tangent to both end arcs"], "branch_facts": ["the side named is the UPPER one, not the lower"]} {"unique_id": "cad_t2_cad_rounded_slot_0011", "problem_id": "cad_t2_cad_rounded_slot_0011", "domain": "PyGeoX-CAD", "template": "cad_rounded_slot", "tier": 2, "nl_description": "A rectangular cover plate measures 36.3 mm by 19.3 mm. Set up coordinates with the plate's bottom-left corner at the origin P, the 36.3 mm edge running from P along the positive x-direction to Q, and the 19.3 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot is machined along the plate's long centreline. The slot is 8.2 mm wide and both of its ends are rounded off as semicircles. The two end centres are C at the left-hand end and D at the right-hand end; they are 16.5 mm apart and are placed symmetrically about the middle of the plate. The upper straight side of the slot runs smoothly into both rounded ends, touching the left end at T and the right end at V.", "answer_schema": {"points": ["C", "D", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Rounded-end slot in a cover plate\n\nOracle formalization for cad_t2_cad_rounded_slot_0011. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(52)\n\nP, Q, R, S, C, D, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 36.3, description=\"plate is 36.3 long along PQ\")\nscene.constraint.eq(P.distance(S), 19.3, description=\"plate is 19.3 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(2 * C.y, P.y + S.y, description=\"slot end centre C lies midway between the two long edges\")\nscene.constraint.eq(2 * D.y, P.y + S.y, description=\"slot end centre D lies midway between the two long edges\")\nscene.constraint.eq(C.distance(D), 16.5, description=\"the two slot end centres are 16.5 apart\")\nscene.constraint.eq(C.x + D.x, P.x + Q.x, description=\"the two slot end centres are placed symmetrically about the middle of the plate\")\nscene.constraint.gt(D.x - C.x, 0.05, description=\"C is the left-hand end of the slot and D the right-hand end\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 8.2, description=\"the slot is 8.2 wide and its left end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 8.2, description=\"the slot is 8.2 wide and its right end is a semicircle spanning that width\")\nupper_side = scene.add.line(T, V)\nscene.relate.tangent_to_circle(upper_side, arc_C, T)\nscene.relate.tangent_to_circle(upper_side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper side of the slot\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the upper side of the slot\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [36.3, 0.0], "R": [36.3, 19.3], "S": [0.0, 19.3], "C": [9.899999999999999, 9.65], "D": [26.4, 9.65], "T": [9.899999999999999, 13.75], "V": [26.4, 13.75]}, "circles": {"C": 4.1, "D": 4.1}}, "implicit_facts": ["each end arc has radius = HALF the stated slot width", "both end centres lie on the plate's long centreline", "symmetric placement puts the centres at +/- d/2 from the plate centre", "'runs smoothly into' means the straight side is tangent to both end arcs"], "branch_facts": ["the side named is the UPPER one, not the lower"]} {"unique_id": "cad_t2_cad_rounded_slot_0012", "problem_id": "cad_t2_cad_rounded_slot_0012", "domain": "PyGeoX-CAD", "template": "cad_rounded_slot", "tier": 2, "nl_description": "A rectangular cover plate measures 33.3 mm by 16.9 mm. Set up coordinates with the plate's bottom-left corner at the origin P, the 33.3 mm edge running from P along the positive x-direction to Q, and the 16.9 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot is machined along the plate's long centreline. The slot is 7.9 mm wide and both of its ends are rounded off as semicircles. The two end centres are C at the left-hand end and D at the right-hand end; they are 14 mm apart and are placed symmetrically about the middle of the plate. The upper straight side of the slot runs smoothly into both rounded ends, touching the left end at T and the right end at V.", "answer_schema": {"points": ["C", "D", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Rounded-end slot in a cover plate\n\nOracle formalization for cad_t2_cad_rounded_slot_0012. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(48)\n\nP, Q, R, S, C, D, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 33.3, description=\"plate is 33.3 long along PQ\")\nscene.constraint.eq(P.distance(S), 16.9, description=\"plate is 16.9 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(2 * C.y, P.y + S.y, description=\"slot end centre C lies midway between the two long edges\")\nscene.constraint.eq(2 * D.y, P.y + S.y, description=\"slot end centre D lies midway between the two long edges\")\nscene.constraint.eq(C.distance(D), 14, description=\"the two slot end centres are 14 apart\")\nscene.constraint.eq(C.x + D.x, P.x + Q.x, description=\"the two slot end centres are placed symmetrically about the middle of the plate\")\nscene.constraint.gt(D.x - C.x, 0.05, description=\"C is the left-hand end of the slot and D the right-hand end\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 7.9, description=\"the slot is 7.9 wide and its left end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 7.9, description=\"the slot is 7.9 wide and its right end is a semicircle spanning that width\")\nupper_side = scene.add.line(T, V)\nscene.relate.tangent_to_circle(upper_side, arc_C, T)\nscene.relate.tangent_to_circle(upper_side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper side of the slot\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the upper side of the slot\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [33.3, 0.0], "R": [33.3, 16.9], "S": [0.0, 16.9], "C": [9.649999999999999, 8.45], "D": [23.65, 8.45], "T": [9.649999999999999, 12.399999999999999], "V": [23.65, 12.399999999999999]}, "circles": {"C": 3.95, "D": 3.95}}, "implicit_facts": ["each end arc has radius = HALF the stated slot width", "both end centres lie on the plate's long centreline", "symmetric placement puts the centres at +/- d/2 from the plate centre", "'runs smoothly into' means the straight side is tangent to both end arcs"], "branch_facts": ["the side named is the UPPER one, not the lower"]} {"unique_id": "cad_t2_cad_rounded_slot_0013", "problem_id": "cad_t2_cad_rounded_slot_0013", "domain": "PyGeoX-CAD", "template": "cad_rounded_slot", "tier": 2, "nl_description": "A rectangular cover plate measures 31.3 mm by 18.6 mm. Set up coordinates with the plate's bottom-left corner at the origin P, the 31.3 mm edge running from P along the positive x-direction to Q, and the 18.6 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot is machined along the plate's long centreline. The slot is 8.4 mm wide and both of its ends are rounded off as semicircles. The two end centres are C at the left-hand end and D at the right-hand end; they are 14.3 mm apart and are placed symmetrically about the middle of the plate. The upper straight side of the slot runs smoothly into both rounded ends, touching the left end at T and the right end at V.", "answer_schema": {"points": ["C", "D", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Rounded-end slot in a cover plate\n\nOracle formalization for cad_t2_cad_rounded_slot_0013. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(46)\n\nP, Q, R, S, C, D, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 31.3, description=\"plate is 31.3 long along PQ\")\nscene.constraint.eq(P.distance(S), 18.6, description=\"plate is 18.6 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(2 * C.y, P.y + S.y, description=\"slot end centre C lies midway between the two long edges\")\nscene.constraint.eq(2 * D.y, P.y + S.y, description=\"slot end centre D lies midway between the two long edges\")\nscene.constraint.eq(C.distance(D), 14.3, description=\"the two slot end centres are 14.3 apart\")\nscene.constraint.eq(C.x + D.x, P.x + Q.x, description=\"the two slot end centres are placed symmetrically about the middle of the plate\")\nscene.constraint.gt(D.x - C.x, 0.05, description=\"C is the left-hand end of the slot and D the right-hand end\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 8.4, description=\"the slot is 8.4 wide and its left end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 8.4, description=\"the slot is 8.4 wide and its right end is a semicircle spanning that width\")\nupper_side = scene.add.line(T, V)\nscene.relate.tangent_to_circle(upper_side, arc_C, T)\nscene.relate.tangent_to_circle(upper_side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper side of the slot\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the upper side of the slot\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [31.3, 0.0], "R": [31.3, 18.6], "S": [0.0, 18.6], "C": [8.5, 9.3], "D": [22.8, 9.3], "T": [8.5, 13.5], "V": [22.8, 13.5]}, "circles": {"C": 4.2, "D": 4.2}}, "implicit_facts": ["each end arc has radius = HALF the stated slot width", "both end centres lie on the plate's long centreline", "symmetric placement puts the centres at +/- d/2 from the plate centre", "'runs smoothly into' means the straight side is tangent to both end arcs"], "branch_facts": ["the side named is the UPPER one, not the lower"]} {"unique_id": "cad_t2_cad_rounded_slot_0014", "problem_id": "cad_t2_cad_rounded_slot_0014", "domain": "PyGeoX-CAD", "template": "cad_rounded_slot", "tier": 2, "nl_description": "A rectangular cover plate measures 41.7 mm by 17.4 mm. Set up coordinates with the plate's bottom-left corner at the origin P, the 41.7 mm edge running from P along the positive x-direction to Q, and the 17.4 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot is machined along the plate's long centreline. The slot is 6.7 mm wide and both of its ends are rounded off as semicircles. The two end centres are C at the left-hand end and D at the right-hand end; they are 11.9 mm apart and are placed symmetrically about the middle of the plate. The upper straight side of the slot runs smoothly into both rounded ends, touching the left end at T and the right end at V.", "answer_schema": {"points": ["C", "D", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Rounded-end slot in a cover plate\n\nOracle formalization for cad_t2_cad_rounded_slot_0014. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(59)\n\nP, Q, R, S, C, D, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 41.7, description=\"plate is 41.7 long along PQ\")\nscene.constraint.eq(P.distance(S), 17.4, description=\"plate is 17.4 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(2 * C.y, P.y + S.y, description=\"slot end centre C lies midway between the two long edges\")\nscene.constraint.eq(2 * D.y, P.y + S.y, description=\"slot end centre D lies midway between the two long edges\")\nscene.constraint.eq(C.distance(D), 11.9, description=\"the two slot end centres are 11.9 apart\")\nscene.constraint.eq(C.x + D.x, P.x + Q.x, description=\"the two slot end centres are placed symmetrically about the middle of the plate\")\nscene.constraint.gt(D.x - C.x, 0.05, description=\"C is the left-hand end of the slot and D the right-hand end\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 6.7, description=\"the slot is 6.7 wide and its left end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 6.7, description=\"the slot is 6.7 wide and its right end is a semicircle spanning that width\")\nupper_side = scene.add.line(T, V)\nscene.relate.tangent_to_circle(upper_side, arc_C, T)\nscene.relate.tangent_to_circle(upper_side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper side of the slot\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the upper side of the slot\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [41.7, 0.0], "R": [41.7, 17.4], "S": [0.0, 17.4], "C": [14.900000000000002, 8.7], "D": [26.800000000000004, 8.7], "T": [14.900000000000002, 12.049999999999999], "V": [26.800000000000004, 12.049999999999999]}, "circles": {"C": 3.35, "D": 3.35}}, "implicit_facts": ["each end arc has radius = HALF the stated slot width", "both end centres lie on the plate's long centreline", "symmetric placement puts the centres at +/- d/2 from the plate centre", "'runs smoothly into' means the straight side is tangent to both end arcs"], "branch_facts": ["the side named is the UPPER one, not the lower"]} {"unique_id": "cad_t2_cad_rounded_slot_0015", "problem_id": "cad_t2_cad_rounded_slot_0015", "domain": "PyGeoX-CAD", "template": "cad_rounded_slot", "tier": 2, "nl_description": "A rectangular cover plate measures 41.5 mm by 23.4 mm. Set up coordinates with the plate's bottom-left corner at the origin P, the 41.5 mm edge running from P along the positive x-direction to Q, and the 23.4 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot is machined along the plate's long centreline. The slot is 8.5 mm wide and both of its ends are rounded off as semicircles. The two end centres are C at the left-hand end and D at the right-hand end; they are 15.1 mm apart and are placed symmetrically about the middle of the plate. The upper straight side of the slot runs smoothly into both rounded ends, touching the left end at T and the right end at V.", "answer_schema": {"points": ["C", "D", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Rounded-end slot in a cover plate\n\nOracle formalization for cad_t2_cad_rounded_slot_0015. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(59)\n\nP, Q, R, S, C, D, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 41.5, description=\"plate is 41.5 long along PQ\")\nscene.constraint.eq(P.distance(S), 23.4, description=\"plate is 23.4 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(2 * C.y, P.y + S.y, description=\"slot end centre C lies midway between the two long edges\")\nscene.constraint.eq(2 * D.y, P.y + S.y, description=\"slot end centre D lies midway between the two long edges\")\nscene.constraint.eq(C.distance(D), 15.1, description=\"the two slot end centres are 15.1 apart\")\nscene.constraint.eq(C.x + D.x, P.x + Q.x, description=\"the two slot end centres are placed symmetrically about the middle of the plate\")\nscene.constraint.gt(D.x - C.x, 0.05, description=\"C is the left-hand end of the slot and D the right-hand end\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 8.5, description=\"the slot is 8.5 wide and its left end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 8.5, description=\"the slot is 8.5 wide and its right end is a semicircle spanning that width\")\nupper_side = scene.add.line(T, V)\nscene.relate.tangent_to_circle(upper_side, arc_C, T)\nscene.relate.tangent_to_circle(upper_side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper side of the slot\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the upper side of the slot\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [41.5, 0.0], "R": [41.5, 23.4], "S": [0.0, 23.4], "C": [13.2, 11.7], "D": [28.299999999999997, 11.7], "T": [13.2, 15.95], "V": [28.299999999999997, 15.95]}, "circles": {"C": 4.25, "D": 4.25}}, "implicit_facts": ["each end arc has radius = HALF the stated slot width", "both end centres lie on the plate's long centreline", "symmetric placement puts the centres at +/- d/2 from the plate centre", "'runs smoothly into' means the straight side is tangent to both end arcs"], "branch_facts": ["the side named is the UPPER one, not the lower"]} {"unique_id": "cad_t2_cad_rounded_slot_0016", "problem_id": "cad_t2_cad_rounded_slot_0016", "domain": "PyGeoX-CAD", "template": "cad_rounded_slot", "tier": 2, "nl_description": "A rectangular cover plate measures 38.5 mm by 23.7 mm. Set up coordinates with the plate's bottom-left corner at the origin P, the 38.5 mm edge running from P along the positive x-direction to Q, and the 23.7 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot is machined along the plate's long centreline. The slot is 8.5 mm wide and both of its ends are rounded off as semicircles. The two end centres are C at the left-hand end and D at the right-hand end; they are 19.5 mm apart and are placed symmetrically about the middle of the plate. The upper straight side of the slot runs smoothly into both rounded ends, touching the left end at T and the right end at V.", "answer_schema": {"points": ["C", "D", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Rounded-end slot in a cover plate\n\nOracle formalization for cad_t2_cad_rounded_slot_0016. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(55)\n\nP, Q, R, S, C, D, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 38.5, description=\"plate is 38.5 long along PQ\")\nscene.constraint.eq(P.distance(S), 23.7, description=\"plate is 23.7 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(2 * C.y, P.y + S.y, description=\"slot end centre C lies midway between the two long edges\")\nscene.constraint.eq(2 * D.y, P.y + S.y, description=\"slot end centre D lies midway between the two long edges\")\nscene.constraint.eq(C.distance(D), 19.5, description=\"the two slot end centres are 19.5 apart\")\nscene.constraint.eq(C.x + D.x, P.x + Q.x, description=\"the two slot end centres are placed symmetrically about the middle of the plate\")\nscene.constraint.gt(D.x - C.x, 0.05, description=\"C is the left-hand end of the slot and D the right-hand end\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 8.5, description=\"the slot is 8.5 wide and its left end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 8.5, description=\"the slot is 8.5 wide and its right end is a semicircle spanning that width\")\nupper_side = scene.add.line(T, V)\nscene.relate.tangent_to_circle(upper_side, arc_C, T)\nscene.relate.tangent_to_circle(upper_side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper side of the slot\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the upper side of the slot\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [38.5, 0.0], "R": [38.5, 23.7], "S": [0.0, 23.7], "C": [9.5, 11.85], "D": [29.0, 11.85], "T": [9.5, 16.1], "V": [29.0, 16.1]}, "circles": {"C": 4.25, "D": 4.25}}, "implicit_facts": ["each end arc has radius = HALF the stated slot width", "both end centres lie on the plate's long centreline", "symmetric placement puts the centres at +/- d/2 from the plate centre", "'runs smoothly into' means the straight side is tangent to both end arcs"], "branch_facts": ["the side named is the UPPER one, not the lower"]} {"unique_id": "cad_t2_cad_rounded_slot_0017", "problem_id": "cad_t2_cad_rounded_slot_0017", "domain": "PyGeoX-CAD", "template": "cad_rounded_slot", "tier": 2, "nl_description": "A rectangular cover plate measures 42.4 mm by 16.1 mm. Set up coordinates with the plate's bottom-left corner at the origin P, the 42.4 mm edge running from P along the positive x-direction to Q, and the 16.1 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot is machined along the plate's long centreline. The slot is 7.1 mm wide and both of its ends are rounded off as semicircles. The two end centres are C at the left-hand end and D at the right-hand end; they are 16.9 mm apart and are placed symmetrically about the middle of the plate. The upper straight side of the slot runs smoothly into both rounded ends, touching the left end at T and the right end at V.", "answer_schema": {"points": ["C", "D", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Rounded-end slot in a cover plate\n\nOracle formalization for cad_t2_cad_rounded_slot_0017. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(60)\n\nP, Q, R, S, C, D, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 42.4, description=\"plate is 42.4 long along PQ\")\nscene.constraint.eq(P.distance(S), 16.1, description=\"plate is 16.1 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(2 * C.y, P.y + S.y, description=\"slot end centre C lies midway between the two long edges\")\nscene.constraint.eq(2 * D.y, P.y + S.y, description=\"slot end centre D lies midway between the two long edges\")\nscene.constraint.eq(C.distance(D), 16.9, description=\"the two slot end centres are 16.9 apart\")\nscene.constraint.eq(C.x + D.x, P.x + Q.x, description=\"the two slot end centres are placed symmetrically about the middle of the plate\")\nscene.constraint.gt(D.x - C.x, 0.05, description=\"C is the left-hand end of the slot and D the right-hand end\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 7.1, description=\"the slot is 7.1 wide and its left end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 7.1, description=\"the slot is 7.1 wide and its right end is a semicircle spanning that width\")\nupper_side = scene.add.line(T, V)\nscene.relate.tangent_to_circle(upper_side, arc_C, T)\nscene.relate.tangent_to_circle(upper_side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper side of the slot\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the upper side of the slot\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [42.4, 0.0], "R": [42.4, 16.1], "S": [0.0, 16.1], "C": [12.75, 8.05], "D": [29.65, 8.05], "T": [12.75, 11.600000000000001], "V": [29.65, 11.600000000000001]}, "circles": {"C": 3.55, "D": 3.55}}, "implicit_facts": ["each end arc has radius = HALF the stated slot width", "both end centres lie on the plate's long centreline", "symmetric placement puts the centres at +/- d/2 from the plate centre", "'runs smoothly into' means the straight side is tangent to both end arcs"], "branch_facts": ["the side named is the UPPER one, not the lower"]} {"unique_id": "cad_t2_cad_rounded_slot_0018", "problem_id": "cad_t2_cad_rounded_slot_0018", "domain": "PyGeoX-CAD", "template": "cad_rounded_slot", "tier": 2, "nl_description": "A rectangular cover plate measures 42.5 mm by 25.1 mm. Set up coordinates with the plate's bottom-left corner at the origin P, the 42.5 mm edge running from P along the positive x-direction to Q, and the 25.1 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot is machined along the plate's long centreline. The slot is 8.1 mm wide and both of its ends are rounded off as semicircles. The two end centres are C at the left-hand end and D at the right-hand end; they are 17.1 mm apart and are placed symmetrically about the middle of the plate. The upper straight side of the slot runs smoothly into both rounded ends, touching the left end at T and the right end at V.", "answer_schema": {"points": ["C", "D", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Rounded-end slot in a cover plate\n\nOracle formalization for cad_t2_cad_rounded_slot_0018. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(60)\n\nP, Q, R, S, C, D, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 42.5, description=\"plate is 42.5 long along PQ\")\nscene.constraint.eq(P.distance(S), 25.1, description=\"plate is 25.1 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(2 * C.y, P.y + S.y, description=\"slot end centre C lies midway between the two long edges\")\nscene.constraint.eq(2 * D.y, P.y + S.y, description=\"slot end centre D lies midway between the two long edges\")\nscene.constraint.eq(C.distance(D), 17.1, description=\"the two slot end centres are 17.1 apart\")\nscene.constraint.eq(C.x + D.x, P.x + Q.x, description=\"the two slot end centres are placed symmetrically about the middle of the plate\")\nscene.constraint.gt(D.x - C.x, 0.05, description=\"C is the left-hand end of the slot and D the right-hand end\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 8.1, description=\"the slot is 8.1 wide and its left end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 8.1, description=\"the slot is 8.1 wide and its right end is a semicircle spanning that width\")\nupper_side = scene.add.line(T, V)\nscene.relate.tangent_to_circle(upper_side, arc_C, T)\nscene.relate.tangent_to_circle(upper_side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the upper side of the slot\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the upper side of the slot\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [42.5, 0.0], "R": [42.5, 25.1], "S": [0.0, 25.1], "C": [12.7, 12.55], "D": [29.8, 12.55], "T": [12.7, 16.6], "V": [29.8, 16.6]}, "circles": {"C": 4.05, "D": 4.05}}, "implicit_facts": ["each end arc has radius = HALF the stated slot width", "both end centres lie on the plate's long centreline", "symmetric placement puts the centres at +/- d/2 from the plate centre", "'runs smoothly into' means the straight side is tangent to both end arcs"], "branch_facts": ["the side named is the UPPER one, not the lower"]} {"unique_id": "cad_t2_cad_vee_block_0077", "problem_id": "cad_t2_cad_vee_block_0077", "domain": "PyGeoX-CAD", "template": "cad_vee_block", "tier": 2, "nl_description": "A round bar is set down into a vee block and rests against both faces. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 16.1 mm long and the included angle of the vee, measured at the apex between the two faces, is 90 degrees. The bar is 10.7 mm across, its centre is at F, and it seats down inside the vee until it touches the right-hand face at T and the left-hand face at U.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Round bar seated in a vee block\n\nOracle formalization for cad_t2_cad_vee_block_0077. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(20)\n\nZ, A, B, F, T, U = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: face ZA is the right-hand face and ZB the left-hand one\")\nscene.constraint.eq(Z.distance(A), 16.1, description=\"each vee face is 16.1 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 90, description=\"the included angle of the vee is 90 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nbar = scene.add.circle(F)\nscene.constraint.eq(2 * bar.radius, 10.7, description=\"the bar is 10.7 across\")\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.constraint.gt(F.y - Z.y, 0.05, description=\"the bar rests INSIDE the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [11.384419177103418, 11.384419177103418], "B": [-11.384419177103418, 11.384419177103418], "F": [0.0, 7.566042558696058], "T": [3.7830212793480293, 3.7830212793480293], "U": [-3.7830212793480293, 3.7830212793480293]}, "circles": {"F": 5.35}}, "implicit_facts": ["the bar radius is half the stated across-size", "'rests against both faces' means the bar circle is tangent to both faces", "by symmetry the bar centre lies on the bisector of the vee, i.e. the y-axis", "the centre height above the apex is the bar radius divided by the sine of HALF the included angle", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar sits inside the vee, above the apex, not on the far side of it"]} {"unique_id": "cad_t2_cad_vee_block_0078", "problem_id": "cad_t2_cad_vee_block_0078", "domain": "PyGeoX-CAD", "template": "cad_vee_block", "tier": 2, "nl_description": "A round bar is set down into a vee block and rests against both faces. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 17.3 mm long and the included angle of the vee, measured at the apex between the two faces, is 80 degrees. The bar is 14.4 mm across, its centre is at F, and it seats down inside the vee until it touches the right-hand face at T and the left-hand face at U.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Round bar seated in a vee block\n\nOracle formalization for cad_t2_cad_vee_block_0078. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(22)\n\nZ, A, B, F, T, U = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: face ZA is the right-hand face and ZB the left-hand one\")\nscene.constraint.eq(Z.distance(A), 17.3, description=\"each vee face is 17.3 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 80, description=\"the included angle of the vee is 80 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nbar = scene.add.circle(F)\nscene.constraint.eq(2 * bar.radius, 14.4, description=\"the bar is 14.4 across\")\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.constraint.gt(F.y - Z.y, 0.05, description=\"the bar rests INSIDE the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [11.12022564757713, 13.25256886595832], "B": [-11.12022564757713, 13.25256886595832], "F": [0.0, 11.20121155339497], "T": [5.515519990456641, 6.573140763651886], "U": [-5.515519990456641, 6.573140763651886]}, "circles": {"F": 7.2}}, "implicit_facts": ["the bar radius is half the stated across-size", "'rests against both faces' means the bar circle is tangent to both faces", "by symmetry the bar centre lies on the bisector of the vee, i.e. the y-axis", "the centre height above the apex is the bar radius divided by the sine of HALF the included angle", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar sits inside the vee, above the apex, not on the far side of it"]} {"unique_id": "cad_t2_cad_vee_block_0079", "problem_id": "cad_t2_cad_vee_block_0079", "domain": "PyGeoX-CAD", "template": "cad_vee_block", "tier": 2, "nl_description": "A round bar is set down into a vee block and rests against both faces. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 25.3 mm long and the included angle of the vee, measured at the apex between the two faces, is 90 degrees. The bar is 12.8 mm across, its centre is at F, and it seats down inside the vee until it touches the right-hand face at T and the left-hand face at U.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Round bar seated in a vee block\n\nOracle formalization for cad_t2_cad_vee_block_0079. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(28)\n\nZ, A, B, F, T, U = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: face ZA is the right-hand face and ZB the left-hand one\")\nscene.constraint.eq(Z.distance(A), 25.3, description=\"each vee face is 25.3 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 90, description=\"the included angle of the vee is 90 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nbar = scene.add.circle(F)\nscene.constraint.eq(2 * bar.radius, 12.8, description=\"the bar is 12.8 across\")\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.constraint.gt(F.y - Z.y, 0.05, description=\"the bar rests INSIDE the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [17.889801564019653, 17.889801564019653], "B": [-17.889801564019653, 17.889801564019653], "F": [0.0, 9.050966799187808], "T": [4.525483399593905, 4.525483399593905], "U": [-4.525483399593905, 4.525483399593905]}, "circles": {"F": 6.4}}, "implicit_facts": ["the bar radius is half the stated across-size", "'rests against both faces' means the bar circle is tangent to both faces", "by symmetry the bar centre lies on the bisector of the vee, i.e. the y-axis", "the centre height above the apex is the bar radius divided by the sine of HALF the included angle", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar sits inside the vee, above the apex, not on the far side of it"]} {"unique_id": "cad_t2_cad_vee_block_0080", "problem_id": "cad_t2_cad_vee_block_0080", "domain": "PyGeoX-CAD", "template": "cad_vee_block", "tier": 2, "nl_description": "A round bar is set down into a vee block and rests against both faces. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 20.1 mm long and the included angle of the vee, measured at the apex between the two faces, is 60 degrees. The bar is 15.1 mm across, its centre is at F, and it seats down inside the vee until it touches the right-hand face at T and the left-hand face at U.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Round bar seated in a vee block\n\nOracle formalization for cad_t2_cad_vee_block_0080. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(28)\n\nZ, A, B, F, T, U = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: face ZA is the right-hand face and ZB the left-hand one\")\nscene.constraint.eq(Z.distance(A), 20.1, description=\"each vee face is 20.1 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 60, description=\"the included angle of the vee is 60 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nbar = scene.add.circle(F)\nscene.constraint.eq(2 * bar.radius, 15.1, description=\"the bar is 15.1 across\")\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.constraint.gt(F.y - Z.y, 0.05, description=\"the bar rests INSIDE the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [10.049999999999999, 17.407110616067218], "B": [-10.049999999999999, 17.407110616067218], "F": [0.0, 15.100000000000001], "T": [6.538491798572512, 11.325000000000003], "U": [-6.538491798572512, 11.325000000000003]}, "circles": {"F": 7.55}}, "implicit_facts": ["the bar radius is half the stated across-size", "'rests against both faces' means the bar circle is tangent to both faces", "by symmetry the bar centre lies on the bisector of the vee, i.e. the y-axis", "the centre height above the apex is the bar radius divided by the sine of HALF the included angle", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar sits inside the vee, above the apex, not on the far side of it"]} {"unique_id": "cad_t2_cad_vee_block_0081", "problem_id": "cad_t2_cad_vee_block_0081", "domain": "PyGeoX-CAD", "template": "cad_vee_block", "tier": 2, "nl_description": "A round bar is set down into a vee block and rests against both faces. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 23.2 mm long and the included angle of the vee, measured at the apex between the two faces, is 120 degrees. The bar is 11.4 mm across, its centre is at F, and it seats down inside the vee until it touches the right-hand face at T and the left-hand face at U.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Round bar seated in a vee block\n\nOracle formalization for cad_t2_cad_vee_block_0081. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(31)\n\nZ, A, B, F, T, U = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: face ZA is the right-hand face and ZB the left-hand one\")\nscene.constraint.eq(Z.distance(A), 23.2, description=\"each vee face is 23.2 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 120, description=\"the included angle of the vee is 120 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nbar = scene.add.circle(F)\nscene.constraint.eq(2 * bar.radius, 11.4, description=\"the bar is 11.4 across\")\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.constraint.gt(F.y - Z.y, 0.05, description=\"the bar rests INSIDE the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [20.091789367798974, 11.600000000000001], "B": [-20.091789367798974, 11.600000000000001], "F": [0.0, 6.581793068761734], "T": [2.850000000000001, 1.6454482671904345], "U": [-2.850000000000001, 1.6454482671904345]}, "circles": {"F": 5.7}}, "implicit_facts": ["the bar radius is half the stated across-size", "'rests against both faces' means the bar circle is tangent to both faces", "by symmetry the bar centre lies on the bisector of the vee, i.e. the y-axis", "the centre height above the apex is the bar radius divided by the sine of HALF the included angle", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar sits inside the vee, above the apex, not on the far side of it"]} {"unique_id": "cad_t2_cad_vee_block_0082", "problem_id": "cad_t2_cad_vee_block_0082", "domain": "PyGeoX-CAD", "template": "cad_vee_block", "tier": 2, "nl_description": "A round bar is set down into a vee block and rests against both faces. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 23.9 mm long and the included angle of the vee, measured at the apex between the two faces, is 80 degrees. The bar is 9.2 mm across, its centre is at F, and it seats down inside the vee until it touches the right-hand face at T and the left-hand face at U.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Round bar seated in a vee block\n\nOracle formalization for cad_t2_cad_vee_block_0082. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(29)\n\nZ, A, B, F, T, U = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: face ZA is the right-hand face and ZB the left-hand one\")\nscene.constraint.eq(Z.distance(A), 23.9, description=\"each vee face is 23.9 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 80, description=\"the included angle of the vee is 80 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nbar = scene.add.circle(F)\nscene.constraint.eq(2 * bar.radius, 9.2, description=\"the bar is 9.2 across\")\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.constraint.gt(F.y - Z.y, 0.05, description=\"the bar rests INSIDE the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [15.362623871508287, 18.308462190543572], "B": [-15.362623871508287, 18.308462190543572], "F": [0.0, 7.156329603557897], "T": [3.5238044383472986, 4.1995065989998155], "U": [-3.5238044383472986, 4.1995065989998155]}, "circles": {"F": 4.6}}, "implicit_facts": ["the bar radius is half the stated across-size", "'rests against both faces' means the bar circle is tangent to both faces", "by symmetry the bar centre lies on the bisector of the vee, i.e. the y-axis", "the centre height above the apex is the bar radius divided by the sine of HALF the included angle", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar sits inside the vee, above the apex, not on the far side of it"]} {"unique_id": "cad_t2_cad_vee_block_0083", "problem_id": "cad_t2_cad_vee_block_0083", "domain": "PyGeoX-CAD", "template": "cad_vee_block", "tier": 2, "nl_description": "A round bar is set down into a vee block and rests against both faces. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 24.8 mm long and the included angle of the vee, measured at the apex between the two faces, is 120 degrees. The bar is 10.7 mm across, its centre is at F, and it seats down inside the vee until it touches the right-hand face at T and the left-hand face at U.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Round bar seated in a vee block\n\nOracle formalization for cad_t2_cad_vee_block_0083. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(33)\n\nZ, A, B, F, T, U = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: face ZA is the right-hand face and ZB the left-hand one\")\nscene.constraint.eq(Z.distance(A), 24.8, description=\"each vee face is 24.8 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 120, description=\"the included angle of the vee is 120 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nbar = scene.add.circle(F)\nscene.constraint.eq(2 * bar.radius, 10.7, description=\"the bar is 10.7 across\")\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.constraint.gt(F.y - Z.y, 0.05, description=\"the bar rests INSIDE the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [21.47743001385408, 12.400000000000004], "B": [-21.47743001385408, 12.400000000000004], "F": [0.0, 6.177647880328996], "T": [2.6750000000000007, 1.5444119700822498], "U": [-2.6750000000000007, 1.5444119700822498]}, "circles": {"F": 5.35}}, "implicit_facts": ["the bar radius is half the stated across-size", "'rests against both faces' means the bar circle is tangent to both faces", "by symmetry the bar centre lies on the bisector of the vee, i.e. the y-axis", "the centre height above the apex is the bar radius divided by the sine of HALF the included angle", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar sits inside the vee, above the apex, not on the far side of it"]} {"unique_id": "cad_t2_cad_vee_block_0084", "problem_id": "cad_t2_cad_vee_block_0084", "domain": "PyGeoX-CAD", "template": "cad_vee_block", "tier": 2, "nl_description": "A round bar is set down into a vee block and rests against both faces. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 22.4 mm long and the included angle of the vee, measured at the apex between the two faces, is 70 degrees. The bar is 15.3 mm across, its centre is at F, and it seats down inside the vee until it touches the right-hand face at T and the left-hand face at U.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Round bar seated in a vee block\n\nOracle formalization for cad_t2_cad_vee_block_0084. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(29)\n\nZ, A, B, F, T, U = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: face ZA is the right-hand face and ZB the left-hand one\")\nscene.constraint.eq(Z.distance(A), 22.4, description=\"each vee face is 22.4 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 70, description=\"the included angle of the vee is 70 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nbar = scene.add.circle(F)\nscene.constraint.eq(2 * bar.radius, 15.3, description=\"the bar is 15.3 across\")\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.constraint.gt(F.y - Z.y, 0.05, description=\"the bar rests INSIDE the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [12.84811217426343, 18.349005792073417], "B": [-12.84811217426343, 18.349005792073417], "F": [0.0, 13.337367986501402], "T": [6.2665131388107875, 8.9495082484159], "U": [-6.2665131388107875, 8.9495082484159]}, "circles": {"F": 7.65}}, "implicit_facts": ["the bar radius is half the stated across-size", "'rests against both faces' means the bar circle is tangent to both faces", "by symmetry the bar centre lies on the bisector of the vee, i.e. the y-axis", "the centre height above the apex is the bar radius divided by the sine of HALF the included angle", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar sits inside the vee, above the apex, not on the far side of it"]} {"unique_id": "cad_t3_cad_angled_slot_0061", "problem_id": "cad_t3_cad_angled_slot_0061", "domain": "PyGeoX-CAD", "template": "cad_angled_slot", "tier": 3, "nl_description": "A cover plate measures 42.6 mm by 30.7 mm. Set up coordinates with the bottom-left corner at the origin P, the 42.6 mm long edge running from P along the positive x-direction to Q, and the 30.7 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot 7.3 mm wide with rounded ends is cut across the plate at 25 degrees to the long edge, rotated anticlockwise from it. Its two end centres are C and D, 17.7 mm apart, and the slot is centred on the plate, so the midpoint between C and D is the centre of the plate. E is a construction point 5 mm from C in the datum direction, level with C, put there so the 25 degree angle can be measured as the angle DCE. One straight side of the slot touches the first end at T and the second at V; it is the side on the left when looking from C towards D.", "answer_schema": {"points": ["C", "D", "E", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Angled rounded-end slot\n\nOracle formalization for cad_t3_cad_angled_slot_0061. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(60)\n\nP, Q, R, S, C, D, E, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"E\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 42.6, description=\"plate is 42.6 long along PQ\")\nscene.constraint.eq(P.distance(S), 30.7, description=\"plate is 30.7 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(C.x + D.x, P.x + R.x, description=\"the slot is centred on the plate (x)\")\nscene.constraint.eq(C.y + D.y, P.y + R.y, description=\"the slot is centred on the plate (y)\")\nscene.constraint.eq(C.distance(D), 17.7, description=\"the two slot end centres are 17.7 apart\")\nscene.constraint.eq(E.y, C.y, description=\"construction point E marks the datum direction from C: it is level with C\")\nscene.constraint.eq(E.x - C.x, 5, description=\"construction point E is 5 mm from C along the datum direction\")\nscene.constraint.eq(scene.add.angle(D, C, E).value, 25, description=\"the slot axis CD is set at 25 degrees to the long edge of the plate\")\nscene.constraint.gt(cross(C, E, D), 0.05, description=\"the slot axis is rotated anticlockwise from the long edge\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 7.3, description=\"the slot is 7.3 wide and its first end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 7.3, description=\"the slot is 7.3 wide and its second end is a semicircle spanning that width\")\nside = scene.add.line(T, V)\nscene.relate.tangent_to_circle(side, arc_C, T)\nscene.relate.tangent_to_circle(side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the left-hand side of the slot, looking from C towards D\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the left-hand side of the slot, looking from C towards D\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [42.6, 0.0], "R": [42.6, 30.7], "S": [0.0, 30.7], "C": [13.279176084725648, 11.60982838359481], "D": [29.320823915274353, 19.09017161640519], "E": [18.27917608472565, 11.60982838359481], "T": [11.736619429372094, 14.917851806278582], "V": [27.7782672599208, 22.39819503908896]}, "circles": {"C": 3.65, "D": 3.65}}, "implicit_facts": ["each end arc radius is half the stated slot width", "'centred on the plate' means the midpoint of CD is the plate centre", "the end centres sit at +/- half the 17.7 mm spacing along the 25 degree direction", "the straight sides are parallel to CD, offset by the end radius, and tangent to both ends"], "branch_facts": ["the slot axis is rotated ANTICLOCKWISE from the long edge", "the side named is the left-hand one looking from C to D"]} {"unique_id": "cad_t3_cad_angled_slot_0062", "problem_id": "cad_t3_cad_angled_slot_0062", "domain": "PyGeoX-CAD", "template": "cad_angled_slot", "tier": 3, "nl_description": "A cover plate measures 39.4 mm by 33.6 mm. Set up coordinates with the bottom-left corner at the origin P, the 39.4 mm long edge running from P along the positive x-direction to Q, and the 33.6 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot 5.8 mm wide with rounded ends is cut across the plate at 40 degrees to the long edge, rotated anticlockwise from it. Its two end centres are C and D, 17.7 mm apart, and the slot is centred on the plate, so the midpoint between C and D is the centre of the plate. E is a construction point 5 mm from C in the datum direction, level with C, put there so the 40 degree angle can be measured as the angle DCE. One straight side of the slot touches the first end at T and the second at V; it is the side on the left when looking from C towards D.", "answer_schema": {"points": ["C", "D", "E", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Angled rounded-end slot\n\nOracle formalization for cad_t3_cad_angled_slot_0062. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(56)\n\nP, Q, R, S, C, D, E, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"E\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 39.4, description=\"plate is 39.4 long along PQ\")\nscene.constraint.eq(P.distance(S), 33.6, description=\"plate is 33.6 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(C.x + D.x, P.x + R.x, description=\"the slot is centred on the plate (x)\")\nscene.constraint.eq(C.y + D.y, P.y + R.y, description=\"the slot is centred on the plate (y)\")\nscene.constraint.eq(C.distance(D), 17.7, description=\"the two slot end centres are 17.7 apart\")\nscene.constraint.eq(E.y, C.y, description=\"construction point E marks the datum direction from C: it is level with C\")\nscene.constraint.eq(E.x - C.x, 5, description=\"construction point E is 5 mm from C along the datum direction\")\nscene.constraint.eq(scene.add.angle(D, C, E).value, 40, description=\"the slot axis CD is set at 40 degrees to the long edge of the plate\")\nscene.constraint.gt(cross(C, E, D), 0.05, description=\"the slot axis is rotated anticlockwise from the long edge\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 5.8, description=\"the slot is 5.8 wide and its first end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 5.8, description=\"the slot is 5.8 wide and its second end is a semicircle spanning that width\")\nside = scene.add.line(T, V)\nscene.relate.tangent_to_circle(side, arc_C, T)\nscene.relate.tangent_to_circle(side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the left-hand side of the slot, looking from C towards D\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the left-hand side of the slot, looking from C towards D\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [39.4, 0.0], "R": [39.4, 33.6], "S": [0.0, 33.6], "C": [12.920506678397043, 11.11132965427413], "D": [26.479493321602956, 22.488670345725872], "E": [17.920506678397043, 11.11132965427413], "T": [11.056422610306079, 13.332858539319165], "V": [24.61540925351199, 24.710199230770908]}, "circles": {"C": 2.9, "D": 2.9}}, "implicit_facts": ["each end arc radius is half the stated slot width", "'centred on the plate' means the midpoint of CD is the plate centre", "the end centres sit at +/- half the 17.7 mm spacing along the 40 degree direction", "the straight sides are parallel to CD, offset by the end radius, and tangent to both ends"], "branch_facts": ["the slot axis is rotated ANTICLOCKWISE from the long edge", "the side named is the left-hand one looking from C to D"]} {"unique_id": "cad_t3_cad_angled_slot_0063", "problem_id": "cad_t3_cad_angled_slot_0063", "domain": "PyGeoX-CAD", "template": "cad_angled_slot", "tier": 3, "nl_description": "A cover plate measures 37.8 mm by 31.5 mm. Set up coordinates with the bottom-left corner at the origin P, the 37.8 mm long edge running from P along the positive x-direction to Q, and the 31.5 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot 6.9 mm wide with rounded ends is cut across the plate at 20 degrees to the long edge, rotated anticlockwise from it. Its two end centres are C and D, 18.3 mm apart, and the slot is centred on the plate, so the midpoint between C and D is the centre of the plate. E is a construction point 5 mm from C in the datum direction, level with C, put there so the 20 degree angle can be measured as the angle DCE. One straight side of the slot touches the first end at T and the second at V; it is the side on the left when looking from C towards D.", "answer_schema": {"points": ["C", "D", "E", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Angled rounded-end slot\n\nOracle formalization for cad_t3_cad_angled_slot_0063. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(54)\n\nP, Q, R, S, C, D, E, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"E\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 37.8, description=\"plate is 37.8 long along PQ\")\nscene.constraint.eq(P.distance(S), 31.5, description=\"plate is 31.5 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(C.x + D.x, P.x + R.x, description=\"the slot is centred on the plate (x)\")\nscene.constraint.eq(C.y + D.y, P.y + R.y, description=\"the slot is centred on the plate (y)\")\nscene.constraint.eq(C.distance(D), 18.3, description=\"the two slot end centres are 18.3 apart\")\nscene.constraint.eq(E.y, C.y, description=\"construction point E marks the datum direction from C: it is level with C\")\nscene.constraint.eq(E.x - C.x, 5, description=\"construction point E is 5 mm from C along the datum direction\")\nscene.constraint.eq(scene.add.angle(D, C, E).value, 20, description=\"the slot axis CD is set at 20 degrees to the long edge of the plate\")\nscene.constraint.gt(cross(C, E, D), 0.05, description=\"the slot axis is rotated anticlockwise from the long edge\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 6.9, description=\"the slot is 6.9 wide and its first end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 6.9, description=\"the slot is 6.9 wide and its second end is a semicircle spanning that width\")\nside = scene.add.line(T, V)\nscene.relate.tangent_to_circle(side, arc_C, T)\nscene.relate.tangent_to_circle(side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the left-hand side of the slot, looking from C towards D\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the left-hand side of the slot, looking from C towards D\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [37.8, 0.0], "R": [37.8, 31.5], "S": [0.0, 31.5], "C": [10.301812519808935, 12.620515688570132], "D": [27.498187480191064, 18.87948431142987], "E": [15.301812519808935, 12.620515688570132], "T": [9.121843025335378, 15.862455230281515], "V": [26.318217985717506, 22.121423853141255]}, "circles": {"C": 3.45, "D": 3.45}}, "implicit_facts": ["each end arc radius is half the stated slot width", "'centred on the plate' means the midpoint of CD is the plate centre", "the end centres sit at +/- half the 18.3 mm spacing along the 20 degree direction", "the straight sides are parallel to CD, offset by the end radius, and tangent to both ends"], "branch_facts": ["the slot axis is rotated ANTICLOCKWISE from the long edge", "the side named is the left-hand one looking from C to D"]} {"unique_id": "cad_t3_cad_angled_slot_0064", "problem_id": "cad_t3_cad_angled_slot_0064", "domain": "PyGeoX-CAD", "template": "cad_angled_slot", "tier": 3, "nl_description": "A cover plate measures 35.5 mm by 28.8 mm. Set up coordinates with the bottom-left corner at the origin P, the 35.5 mm long edge running from P along the positive x-direction to Q, and the 28.8 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot 7.8 mm wide with rounded ends is cut across the plate at 30 degrees to the long edge, rotated anticlockwise from it. Its two end centres are C and D, 13 mm apart, and the slot is centred on the plate, so the midpoint between C and D is the centre of the plate. E is a construction point 5 mm from C in the datum direction, level with C, put there so the 30 degree angle can be measured as the angle DCE. One straight side of the slot touches the first end at T and the second at V; it is the side on the left when looking from C towards D.", "answer_schema": {"points": ["C", "D", "E", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Angled rounded-end slot\n\nOracle formalization for cad_t3_cad_angled_slot_0064. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(51)\n\nP, Q, R, S, C, D, E, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"E\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 35.5, description=\"plate is 35.5 long along PQ\")\nscene.constraint.eq(P.distance(S), 28.8, description=\"plate is 28.8 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(C.x + D.x, P.x + R.x, description=\"the slot is centred on the plate (x)\")\nscene.constraint.eq(C.y + D.y, P.y + R.y, description=\"the slot is centred on the plate (y)\")\nscene.constraint.eq(C.distance(D), 13, description=\"the two slot end centres are 13 apart\")\nscene.constraint.eq(E.y, C.y, description=\"construction point E marks the datum direction from C: it is level with C\")\nscene.constraint.eq(E.x - C.x, 5, description=\"construction point E is 5 mm from C along the datum direction\")\nscene.constraint.eq(scene.add.angle(D, C, E).value, 30, description=\"the slot axis CD is set at 30 degrees to the long edge of the plate\")\nscene.constraint.gt(cross(C, E, D), 0.05, description=\"the slot axis is rotated anticlockwise from the long edge\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 7.8, description=\"the slot is 7.8 wide and its first end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 7.8, description=\"the slot is 7.8 wide and its second end is a semicircle spanning that width\")\nside = scene.add.line(T, V)\nscene.relate.tangent_to_circle(side, arc_C, T)\nscene.relate.tangent_to_circle(side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the left-hand side of the slot, looking from C towards D\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the left-hand side of the slot, looking from C towards D\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [35.5, 0.0], "R": [35.5, 28.8], "S": [0.0, 28.8], "C": [12.120834875401147, 11.15], "D": [23.379165124598853, 17.65], "E": [17.120834875401147, 11.15], "T": [10.170834875401148, 14.527499074759312], "V": [21.429165124598853, 21.02749907475931]}, "circles": {"C": 3.9, "D": 3.9}}, "implicit_facts": ["each end arc radius is half the stated slot width", "'centred on the plate' means the midpoint of CD is the plate centre", "the end centres sit at +/- half the 13 mm spacing along the 30 degree direction", "the straight sides are parallel to CD, offset by the end radius, and tangent to both ends"], "branch_facts": ["the slot axis is rotated ANTICLOCKWISE from the long edge", "the side named is the left-hand one looking from C to D"]} {"unique_id": "cad_t3_cad_angled_slot_0065", "problem_id": "cad_t3_cad_angled_slot_0065", "domain": "PyGeoX-CAD", "template": "cad_angled_slot", "tier": 3, "nl_description": "A cover plate measures 44 mm by 26.7 mm. Set up coordinates with the bottom-left corner at the origin P, the 44 mm long edge running from P along the positive x-direction to Q, and the 26.7 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot 6.5 mm wide with rounded ends is cut across the plate at 20 degrees to the long edge, rotated anticlockwise from it. Its two end centres are C and D, 16.3 mm apart, and the slot is centred on the plate, so the midpoint between C and D is the centre of the plate. E is a construction point 5 mm from C in the datum direction, level with C, put there so the 20 degree angle can be measured as the angle DCE. One straight side of the slot touches the first end at T and the second at V; it is the side on the left when looking from C towards D.", "answer_schema": {"points": ["C", "D", "E", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Angled rounded-end slot\n\nOracle formalization for cad_t3_cad_angled_slot_0065. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(62)\n\nP, Q, R, S, C, D, E, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"E\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 44, description=\"plate is 44 long along PQ\")\nscene.constraint.eq(P.distance(S), 26.7, description=\"plate is 26.7 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(C.x + D.x, P.x + R.x, description=\"the slot is centred on the plate (x)\")\nscene.constraint.eq(C.y + D.y, P.y + R.y, description=\"the slot is centred on the plate (y)\")\nscene.constraint.eq(C.distance(D), 16.3, description=\"the two slot end centres are 16.3 apart\")\nscene.constraint.eq(E.y, C.y, description=\"construction point E marks the datum direction from C: it is level with C\")\nscene.constraint.eq(E.x - C.x, 5, description=\"construction point E is 5 mm from C along the datum direction\")\nscene.constraint.eq(scene.add.angle(D, C, E).value, 20, description=\"the slot axis CD is set at 20 degrees to the long edge of the plate\")\nscene.constraint.gt(cross(C, E, D), 0.05, description=\"the slot axis is rotated anticlockwise from the long edge\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 6.5, description=\"the slot is 6.5 wide and its first end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 6.5, description=\"the slot is 6.5 wide and its second end is a semicircle spanning that width\")\nside = scene.add.line(T, V)\nscene.relate.tangent_to_circle(side, arc_C, T)\nscene.relate.tangent_to_circle(side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the left-hand side of the slot, looking from C towards D\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the left-hand side of the slot, looking from C towards D\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [44.0, 0.0], "R": [44.0, 26.7], "S": [0.0, 26.7], "C": [14.341505140594846, 10.562535831895799], "D": [29.658494859405153, 16.1374641681042], "E": [19.341505140594847, 10.562535831895799], "T": [13.229939674786422, 13.616536849450002], "V": [28.54692939359673, 19.1914651856584]}, "circles": {"C": 3.25, "D": 3.25}}, "implicit_facts": ["each end arc radius is half the stated slot width", "'centred on the plate' means the midpoint of CD is the plate centre", "the end centres sit at +/- half the 16.3 mm spacing along the 20 degree direction", "the straight sides are parallel to CD, offset by the end radius, and tangent to both ends"], "branch_facts": ["the slot axis is rotated ANTICLOCKWISE from the long edge", "the side named is the left-hand one looking from C to D"]} {"unique_id": "cad_t3_cad_angled_slot_0066", "problem_id": "cad_t3_cad_angled_slot_0066", "domain": "PyGeoX-CAD", "template": "cad_angled_slot", "tier": 3, "nl_description": "A cover plate measures 45.5 mm by 29.8 mm. Set up coordinates with the bottom-left corner at the origin P, the 45.5 mm long edge running from P along the positive x-direction to Q, and the 29.8 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot 6.1 mm wide with rounded ends is cut across the plate at 50 degrees to the long edge, rotated anticlockwise from it. Its two end centres are C and D, 13.9 mm apart, and the slot is centred on the plate, so the midpoint between C and D is the centre of the plate. E is a construction point 5 mm from C in the datum direction, level with C, put there so the 50 degree angle can be measured as the angle DCE. One straight side of the slot touches the first end at T and the second at V; it is the side on the left when looking from C towards D.", "answer_schema": {"points": ["C", "D", "E", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Angled rounded-end slot\n\nOracle formalization for cad_t3_cad_angled_slot_0066. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(64)\n\nP, Q, R, S, C, D, E, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"E\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 45.5, description=\"plate is 45.5 long along PQ\")\nscene.constraint.eq(P.distance(S), 29.8, description=\"plate is 29.8 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(C.x + D.x, P.x + R.x, description=\"the slot is centred on the plate (x)\")\nscene.constraint.eq(C.y + D.y, P.y + R.y, description=\"the slot is centred on the plate (y)\")\nscene.constraint.eq(C.distance(D), 13.9, description=\"the two slot end centres are 13.9 apart\")\nscene.constraint.eq(E.y, C.y, description=\"construction point E marks the datum direction from C: it is level with C\")\nscene.constraint.eq(E.x - C.x, 5, description=\"construction point E is 5 mm from C along the datum direction\")\nscene.constraint.eq(scene.add.angle(D, C, E).value, 50, description=\"the slot axis CD is set at 50 degrees to the long edge of the plate\")\nscene.constraint.gt(cross(C, E, D), 0.05, description=\"the slot axis is rotated anticlockwise from the long edge\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 6.1, description=\"the slot is 6.1 wide and its first end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 6.1, description=\"the slot is 6.1 wide and its second end is a semicircle spanning that width\")\nside = scene.add.line(T, V)\nscene.relate.tangent_to_circle(side, arc_C, T)\nscene.relate.tangent_to_circle(side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the left-hand side of the slot, looking from C towards D\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the left-hand side of the slot, looking from C towards D\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [45.5, 0.0], "R": [45.5, 29.8], "S": [0.0, 29.8], "C": [18.28262611267855, 9.575991120323103], "D": [27.21737388732145, 20.224008879676898], "E": [23.28262611267855, 9.575991120323103], "T": [15.946190561165666, 11.536493329867048], "V": [24.88093833580857, 22.184511089220845]}, "circles": {"C": 3.05, "D": 3.05}}, "implicit_facts": ["each end arc radius is half the stated slot width", "'centred on the plate' means the midpoint of CD is the plate centre", "the end centres sit at +/- half the 13.9 mm spacing along the 50 degree direction", "the straight sides are parallel to CD, offset by the end radius, and tangent to both ends"], "branch_facts": ["the slot axis is rotated ANTICLOCKWISE from the long edge", "the side named is the left-hand one looking from C to D"]} {"unique_id": "cad_t3_cad_angled_slot_0067", "problem_id": "cad_t3_cad_angled_slot_0067", "domain": "PyGeoX-CAD", "template": "cad_angled_slot", "tier": 3, "nl_description": "A cover plate measures 36.4 mm by 30.8 mm. Set up coordinates with the bottom-left corner at the origin P, the 36.4 mm long edge running from P along the positive x-direction to Q, and the 30.8 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot 5.9 mm wide with rounded ends is cut across the plate at 30 degrees to the long edge, rotated anticlockwise from it. Its two end centres are C and D, 17.6 mm apart, and the slot is centred on the plate, so the midpoint between C and D is the centre of the plate. E is a construction point 5 mm from C in the datum direction, level with C, put there so the 30 degree angle can be measured as the angle DCE. One straight side of the slot touches the first end at T and the second at V; it is the side on the left when looking from C towards D.", "answer_schema": {"points": ["C", "D", "E", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Angled rounded-end slot\n\nOracle formalization for cad_t3_cad_angled_slot_0067. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(52)\n\nP, Q, R, S, C, D, E, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"E\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 36.4, description=\"plate is 36.4 long along PQ\")\nscene.constraint.eq(P.distance(S), 30.8, description=\"plate is 30.8 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(C.x + D.x, P.x + R.x, description=\"the slot is centred on the plate (x)\")\nscene.constraint.eq(C.y + D.y, P.y + R.y, description=\"the slot is centred on the plate (y)\")\nscene.constraint.eq(C.distance(D), 17.6, description=\"the two slot end centres are 17.6 apart\")\nscene.constraint.eq(E.y, C.y, description=\"construction point E marks the datum direction from C: it is level with C\")\nscene.constraint.eq(E.x - C.x, 5, description=\"construction point E is 5 mm from C along the datum direction\")\nscene.constraint.eq(scene.add.angle(D, C, E).value, 30, description=\"the slot axis CD is set at 30 degrees to the long edge of the plate\")\nscene.constraint.gt(cross(C, E, D), 0.05, description=\"the slot axis is rotated anticlockwise from the long edge\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 5.9, description=\"the slot is 5.9 wide and its first end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 5.9, description=\"the slot is 5.9 wide and its second end is a semicircle spanning that width\")\nside = scene.add.line(T, V)\nscene.relate.tangent_to_circle(side, arc_C, T)\nscene.relate.tangent_to_circle(side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the left-hand side of the slot, looking from C towards D\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the left-hand side of the slot, looking from C towards D\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [36.4, 0.0], "R": [36.4, 30.8], "S": [0.0, 30.8], "C": [10.578976446696938, 11.0], "D": [25.82102355330306, 19.8], "E": [15.578976446696938, 11.0], "T": [9.103976446696938, 13.554774941164094], "V": [24.34602355330306, 22.354774941164095]}, "circles": {"C": 2.95, "D": 2.95}}, "implicit_facts": ["each end arc radius is half the stated slot width", "'centred on the plate' means the midpoint of CD is the plate centre", "the end centres sit at +/- half the 17.6 mm spacing along the 30 degree direction", "the straight sides are parallel to CD, offset by the end radius, and tangent to both ends"], "branch_facts": ["the slot axis is rotated ANTICLOCKWISE from the long edge", "the side named is the left-hand one looking from C to D"]} {"unique_id": "cad_t3_cad_angled_slot_0068", "problem_id": "cad_t3_cad_angled_slot_0068", "domain": "PyGeoX-CAD", "template": "cad_angled_slot", "tier": 3, "nl_description": "A cover plate measures 36.9 mm by 30.3 mm. Set up coordinates with the bottom-left corner at the origin P, the 36.9 mm long edge running from P along the positive x-direction to Q, and the 30.3 mm edge running from P along the positive y-direction to S; R is the remaining corner. A slot 5.9 mm wide with rounded ends is cut across the plate at 20 degrees to the long edge, rotated anticlockwise from it. Its two end centres are C and D, 14.4 mm apart, and the slot is centred on the plate, so the midpoint between C and D is the centre of the plate. E is a construction point 5 mm from C in the datum direction, level with C, put there so the 20 degree angle can be measured as the angle DCE. One straight side of the slot touches the first end at T and the second at V; it is the side on the left when looking from C towards D.", "answer_schema": {"points": ["C", "D", "E", "P", "Q", "R", "S", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Angled rounded-end slot\n\nOracle formalization for cad_t3_cad_angled_slot_0068. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(53)\n\nP, Q, R, S, C, D, E, T, V = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"D\", \"E\", \"T\", \"V\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 36.9, description=\"plate is 36.9 long along PQ\")\nscene.constraint.eq(P.distance(S), 30.3, description=\"plate is 30.3 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.constraint.eq(C.x + D.x, P.x + R.x, description=\"the slot is centred on the plate (x)\")\nscene.constraint.eq(C.y + D.y, P.y + R.y, description=\"the slot is centred on the plate (y)\")\nscene.constraint.eq(C.distance(D), 14.4, description=\"the two slot end centres are 14.4 apart\")\nscene.constraint.eq(E.y, C.y, description=\"construction point E marks the datum direction from C: it is level with C\")\nscene.constraint.eq(E.x - C.x, 5, description=\"construction point E is 5 mm from C along the datum direction\")\nscene.constraint.eq(scene.add.angle(D, C, E).value, 20, description=\"the slot axis CD is set at 20 degrees to the long edge of the plate\")\nscene.constraint.gt(cross(C, E, D), 0.05, description=\"the slot axis is rotated anticlockwise from the long edge\")\narc_C = scene.add.circle(C)\narc_D = scene.add.circle(D)\nscene.constraint.eq(2 * arc_C.radius, 5.9, description=\"the slot is 5.9 wide and its first end is a semicircle spanning that width\")\nscene.constraint.eq(2 * arc_D.radius, 5.9, description=\"the slot is 5.9 wide and its second end is a semicircle spanning that width\")\nside = scene.add.line(T, V)\nscene.relate.tangent_to_circle(side, arc_C, T)\nscene.relate.tangent_to_circle(side, arc_D, V)\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"T is on the left-hand side of the slot, looking from C towards D\")\nscene.constraint.gt(cross(C, D, V), 0.05, description=\"V is on the left-hand side of the slot, looking from C towards D\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [36.9, 0.0], "R": [36.9, 30.3], "S": [0.0, 30.3], "C": [11.684213130341458, 12.687454968055185], "D": [25.21578686965854, 17.612545031944816], "E": [16.68421313034146, 12.687454968055185], "T": [10.675253707530736, 15.459548199373614], "V": [24.206827446847818, 20.384638263263245]}, "circles": {"C": 2.95, "D": 2.95}}, "implicit_facts": ["each end arc radius is half the stated slot width", "'centred on the plate' means the midpoint of CD is the plate centre", "the end centres sit at +/- half the 14.4 mm spacing along the 20 degree direction", "the straight sides are parallel to CD, offset by the end radius, and tangent to both ends"], "branch_facts": ["the slot axis is rotated ANTICLOCKWISE from the long edge", "the side named is the left-hand one looking from C to D"]} {"unique_id": "cad_t3_cad_flank_arc_0053", "problem_id": "cad_t3_cad_flank_arc_0053", "domain": "PyGeoX-CAD", "template": "cad_flank_arc", "tier": 3, "nl_description": "A lever is shaped from two circles joined by a curved flank. Set up coordinates with the hub centre at the origin C and the boss centre at D on the positive x-axis. The hub is 18.8 mm across, the boss is 11.2 mm across, and their centres are 35.4 mm apart. The flank is an arc of radius 37.9 mm centred at F. It touches the hub at T and the boss at V, and it bulges away from the line of centres, wrapping around neither circle - each of T and V lies between its own circle's centre and F. The flank shown is the one on the upper side of the line of centres.", "answer_schema": {"points": ["C", "D", "F", "T", "V"], "circles": ["C", "D", "F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Lever flank arc tangent to hub and boss\n\nOracle formalization for cad_t3_cad_flank_arc_0053. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(59)\n\nC, D, F, T, V = scene.add.points([\"C\", \"D\", \"F\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the hub centre (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the hub centre (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the boss lies in the +x direction from the hub\")\nscene.constraint.eq(C.distance(D), 35.4, description=\"hub and boss centres are 35.4 apart\")\nhub = scene.add.circle(C)\nboss = scene.add.circle(D)\nflank = scene.add.circle(F)\nscene.constraint.eq(2 * hub.radius, 18.8, description=\"the hub is 18.8 across\")\nscene.constraint.eq(2 * boss.radius, 11.2, description=\"the boss is 11.2 across\")\nscene.constraint.eq(flank.radius, 37.9, description=\"the flank arc has radius 37.9\")\nscene.relate.point_lies_on(T, hub)\nscene.relate.point_lies_on(T, flank)\nscene.relate.point_lies_on(T, scene.add.line_segment(C, F))\nscene.relate.point_lies_on(V, boss)\nscene.relate.point_lies_on(V, flank)\nscene.relate.point_lies_on(V, scene.add.line_segment(D, F))\nscene.constraint.gt(cross(C, D, F), 0.05, description=\"the flank shown is the one on the upper side of the line of centres\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [35.4, 0.0], "F": [22.57344632768361, 41.56596589628571], "T": [4.48605487273205, 8.26046679545636], "V": [33.74876550425352, 5.351020896993103]}, "circles": {"C": 9.4, "D": 5.6, "F": 37.9}}, "implicit_facts": ["hub and boss radii are half the stated across-sizes", "two circles touch externally exactly when the distance between centres equals the sum of the radii", "so F is at distance (flank radius + hub radius) from C and (flank radius + boss radius) from D", "each tangency point lies on the straight line joining the two centres involved"], "branch_facts": ["the flank named is the one above the line of centres, not its mirror image"]} {"unique_id": "cad_t3_cad_flank_arc_0054", "problem_id": "cad_t3_cad_flank_arc_0054", "domain": "PyGeoX-CAD", "template": "cad_flank_arc", "tier": 3, "nl_description": "A lever is shaped from two circles joined by a curved flank. Set up coordinates with the hub centre at the origin C and the boss centre at D on the positive x-axis. The hub is 14.9 mm across, the boss is 11.5 mm across, and their centres are 34.8 mm apart. The flank is an arc of radius 19.3 mm centred at F. It touches the hub at T and the boss at V, and it bulges away from the line of centres, wrapping around neither circle - each of T and V lies between its own circle's centre and F. The flank shown is the one on the upper side of the line of centres.", "answer_schema": {"points": ["C", "D", "F", "T", "V"], "circles": ["C", "D", "F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Lever flank arc tangent to hub and boss\n\nOracle formalization for cad_t3_cad_flank_arc_0054. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(50)\n\nC, D, F, T, V = scene.add.points([\"C\", \"D\", \"F\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the hub centre (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the hub centre (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the boss lies in the +x direction from the hub\")\nscene.constraint.eq(C.distance(D), 34.8, description=\"hub and boss centres are 34.8 apart\")\nhub = scene.add.circle(C)\nboss = scene.add.circle(D)\nflank = scene.add.circle(F)\nscene.constraint.eq(2 * hub.radius, 14.9, description=\"the hub is 14.9 across\")\nscene.constraint.eq(2 * boss.radius, 11.5, description=\"the boss is 11.5 across\")\nscene.constraint.eq(flank.radius, 19.3, description=\"the flank arc has radius 19.3\")\nscene.relate.point_lies_on(T, hub)\nscene.relate.point_lies_on(T, flank)\nscene.relate.point_lies_on(T, scene.add.line_segment(C, F))\nscene.relate.point_lies_on(V, boss)\nscene.relate.point_lies_on(V, flank)\nscene.relate.point_lies_on(V, scene.add.line_segment(D, F))\nscene.constraint.gt(cross(C, D, F), 0.05, description=\"the flank shown is the one on the upper side of the line of centres\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [34.8, 0.0], "F": [18.665229885057467, 19.16172469633038], "T": [5.198353743688902, 5.336629868697621], "V": [31.096410053456303, 4.398399880395196]}, "circles": {"C": 7.45, "D": 5.75, "F": 19.3}}, "implicit_facts": ["hub and boss radii are half the stated across-sizes", "two circles touch externally exactly when the distance between centres equals the sum of the radii", "so F is at distance (flank radius + hub radius) from C and (flank radius + boss radius) from D", "each tangency point lies on the straight line joining the two centres involved"], "branch_facts": ["the flank named is the one above the line of centres, not its mirror image"]} {"unique_id": "cad_t3_cad_flank_arc_0055", "problem_id": "cad_t3_cad_flank_arc_0055", "domain": "PyGeoX-CAD", "template": "cad_flank_arc", "tier": 3, "nl_description": "A lever is shaped from two circles joined by a curved flank. Set up coordinates with the hub centre at the origin C and the boss centre at D on the positive x-axis. The hub is 20.8 mm across, the boss is 10.4 mm across, and their centres are 36.6 mm apart. The flank is an arc of radius 37.1 mm centred at F. It touches the hub at T and the boss at V, and it bulges away from the line of centres, wrapping around neither circle - each of T and V lies between its own circle's centre and F. The flank shown is the one on the upper side of the line of centres.", "answer_schema": {"points": ["C", "D", "F", "T", "V"], "circles": ["C", "D", "F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Lever flank arc tangent to hub and boss\n\nOracle formalization for cad_t3_cad_flank_arc_0055. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(58)\n\nC, D, F, T, V = scene.add.points([\"C\", \"D\", \"F\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the hub centre (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the hub centre (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the boss lies in the +x direction from the hub\")\nscene.constraint.eq(C.distance(D), 36.6, description=\"hub and boss centres are 36.6 apart\")\nhub = scene.add.circle(C)\nboss = scene.add.circle(D)\nflank = scene.add.circle(F)\nscene.constraint.eq(2 * hub.radius, 20.8, description=\"the hub is 20.8 across\")\nscene.constraint.eq(2 * boss.radius, 10.4, description=\"the boss is 10.4 across\")\nscene.constraint.eq(flank.radius, 37.1, description=\"the flank arc has radius 37.1\")\nscene.relate.point_lies_on(T, hub)\nscene.relate.point_lies_on(T, flank)\nscene.relate.point_lies_on(T, scene.add.line_segment(C, F))\nscene.relate.point_lies_on(V, boss)\nscene.relate.point_lies_on(V, flank)\nscene.relate.point_lies_on(V, scene.add.line_segment(D, F))\nscene.constraint.gt(cross(C, D, F), 0.05, description=\"the flank shown is the one on the upper side of the line of centres\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [36.6, 0.0], "F": [24.67923497267759, 40.585531426400806], "T": [5.403453551912568, 8.886095301780388], "V": [35.13456316448992, 4.989237905845962]}, "circles": {"C": 10.4, "D": 5.2, "F": 37.1}}, "implicit_facts": ["hub and boss radii are half the stated across-sizes", "two circles touch externally exactly when the distance between centres equals the sum of the radii", "so F is at distance (flank radius + hub radius) from C and (flank radius + boss radius) from D", "each tangency point lies on the straight line joining the two centres involved"], "branch_facts": ["the flank named is the one above the line of centres, not its mirror image"]} {"unique_id": "cad_t3_cad_flank_arc_0056", "problem_id": "cad_t3_cad_flank_arc_0056", "domain": "PyGeoX-CAD", "template": "cad_flank_arc", "tier": 3, "nl_description": "A lever is shaped from two circles joined by a curved flank. Set up coordinates with the hub centre at the origin C and the boss centre at D on the positive x-axis. The hub is 21 mm across, the boss is 12 mm across, and their centres are 30.9 mm apart. The flank is an arc of radius 23.3 mm centred at F. It touches the hub at T and the boss at V, and it bulges away from the line of centres, wrapping around neither circle - each of T and V lies between its own circle's centre and F. The flank shown is the one on the upper side of the line of centres.", "answer_schema": {"points": ["C", "D", "F", "T", "V"], "circles": ["C", "D", "F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Lever flank arc tangent to hub and boss\n\nOracle formalization for cad_t3_cad_flank_arc_0056. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(45)\n\nC, D, F, T, V = scene.add.points([\"C\", \"D\", \"F\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the hub centre (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the hub centre (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the boss lies in the +x direction from the hub\")\nscene.constraint.eq(C.distance(D), 30.9, description=\"hub and boss centres are 30.9 apart\")\nhub = scene.add.circle(C)\nboss = scene.add.circle(D)\nflank = scene.add.circle(F)\nscene.constraint.eq(2 * hub.radius, 21, description=\"the hub is 21 across\")\nscene.constraint.eq(2 * boss.radius, 12, description=\"the boss is 12 across\")\nscene.constraint.eq(flank.radius, 23.3, description=\"the flank arc has radius 23.3\")\nscene.relate.point_lies_on(T, hub)\nscene.relate.point_lies_on(T, flank)\nscene.relate.point_lies_on(T, scene.add.line_segment(C, F))\nscene.relate.point_lies_on(V, boss)\nscene.relate.point_lies_on(V, flank)\nscene.relate.point_lies_on(V, scene.add.line_segment(D, F))\nscene.constraint.gt(cross(C, D, F), 0.05, description=\"the flank shown is the one on the upper side of the line of centres\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [30.9, 0.0], "F": [20.044660194174753, 27.214914986089262], "T": [6.226891480438903, 8.454337495678617], "V": [28.677063520991418, 5.5730201336701555]}, "circles": {"C": 10.5, "D": 6.0, "F": 23.3}}, "implicit_facts": ["hub and boss radii are half the stated across-sizes", "two circles touch externally exactly when the distance between centres equals the sum of the radii", "so F is at distance (flank radius + hub radius) from C and (flank radius + boss radius) from D", "each tangency point lies on the straight line joining the two centres involved"], "branch_facts": ["the flank named is the one above the line of centres, not its mirror image"]} {"unique_id": "cad_t3_cad_flank_arc_0057", "problem_id": "cad_t3_cad_flank_arc_0057", "domain": "PyGeoX-CAD", "template": "cad_flank_arc", "tier": 3, "nl_description": "A lever is shaped from two circles joined by a curved flank. Set up coordinates with the hub centre at the origin C and the boss centre at D on the positive x-axis. The hub is 15.6 mm across, the boss is 11 mm across, and their centres are 30.6 mm apart. The flank is an arc of radius 32.5 mm centred at F. It touches the hub at T and the boss at V, and it bulges away from the line of centres, wrapping around neither circle - each of T and V lies between its own circle's centre and F. The flank shown is the one on the upper side of the line of centres.", "answer_schema": {"points": ["C", "D", "F", "T", "V"], "circles": ["C", "D", "F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Lever flank arc tangent to hub and boss\n\nOracle formalization for cad_t3_cad_flank_arc_0057. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(52)\n\nC, D, F, T, V = scene.add.points([\"C\", \"D\", \"F\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the hub centre (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the hub centre (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the boss lies in the +x direction from the hub\")\nscene.constraint.eq(C.distance(D), 30.6, description=\"hub and boss centres are 30.6 apart\")\nhub = scene.add.circle(C)\nboss = scene.add.circle(D)\nflank = scene.add.circle(F)\nscene.constraint.eq(2 * hub.radius, 15.6, description=\"the hub is 15.6 across\")\nscene.constraint.eq(2 * boss.radius, 11, description=\"the boss is 11 across\")\nscene.constraint.eq(flank.radius, 32.5, description=\"the flank arc has radius 32.5\")\nscene.relate.point_lies_on(T, hub)\nscene.relate.point_lies_on(T, flank)\nscene.relate.point_lies_on(T, scene.add.line_segment(C, F))\nscene.relate.point_lies_on(V, boss)\nscene.relate.point_lies_on(V, flank)\nscene.relate.point_lies_on(V, scene.add.line_segment(D, F))\nscene.constraint.gt(cross(C, D, F), 0.05, description=\"the flank shown is the one on the upper side of the line of centres\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [30.6, 0.0], "F": [18.242647058823525, 35.934604885641875], "T": [3.530834914611005, 6.955084816575846], "V": [28.811435758513934, 5.201061233448166]}, "circles": {"C": 7.8, "D": 5.5, "F": 32.5}}, "implicit_facts": ["hub and boss radii are half the stated across-sizes", "two circles touch externally exactly when the distance between centres equals the sum of the radii", "so F is at distance (flank radius + hub radius) from C and (flank radius + boss radius) from D", "each tangency point lies on the straight line joining the two centres involved"], "branch_facts": ["the flank named is the one above the line of centres, not its mirror image"]} {"unique_id": "cad_t3_cad_flank_arc_0058", "problem_id": "cad_t3_cad_flank_arc_0058", "domain": "PyGeoX-CAD", "template": "cad_flank_arc", "tier": 3, "nl_description": "A lever is shaped from two circles joined by a curved flank. Set up coordinates with the hub centre at the origin C and the boss centre at D on the positive x-axis. The hub is 16.4 mm across, the boss is 9.3 mm across, and their centres are 29.4 mm apart. The flank is an arc of radius 21.9 mm centred at F. It touches the hub at T and the boss at V, and it bulges away from the line of centres, wrapping around neither circle - each of T and V lies between its own circle's centre and F. The flank shown is the one on the upper side of the line of centres.", "answer_schema": {"points": ["C", "D", "F", "T", "V"], "circles": ["C", "D", "F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Lever flank arc tangent to hub and boss\n\nOracle formalization for cad_t3_cad_flank_arc_0058. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(43)\n\nC, D, F, T, V = scene.add.points([\"C\", \"D\", \"F\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the hub centre (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the hub centre (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the boss lies in the +x direction from the hub\")\nscene.constraint.eq(C.distance(D), 29.4, description=\"hub and boss centres are 29.4 apart\")\nhub = scene.add.circle(C)\nboss = scene.add.circle(D)\nflank = scene.add.circle(F)\nscene.constraint.eq(2 * hub.radius, 16.4, description=\"the hub is 16.4 across\")\nscene.constraint.eq(2 * boss.radius, 9.3, description=\"the boss is 9.3 across\")\nscene.constraint.eq(flank.radius, 21.9, description=\"the flank arc has radius 21.9\")\nscene.relate.point_lies_on(T, hub)\nscene.relate.point_lies_on(T, flank)\nscene.relate.point_lies_on(T, scene.add.line_segment(C, F))\nscene.relate.point_lies_on(V, boss)\nscene.relate.point_lies_on(V, flank)\nscene.relate.point_lies_on(V, scene.add.line_segment(D, F))\nscene.constraint.gt(cross(C, D, F), 0.05, description=\"the flank shown is the one on the upper side of the line of centres\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [29.4, 0.0], "F": [18.120195578231293, 24.034735534360408], "T": [4.936398795398558, 6.547668816669613], "V": [27.424441033475535, 4.2094734551704684]}, "circles": {"C": 8.2, "D": 4.65, "F": 21.9}}, "implicit_facts": ["hub and boss radii are half the stated across-sizes", "two circles touch externally exactly when the distance between centres equals the sum of the radii", "so F is at distance (flank radius + hub radius) from C and (flank radius + boss radius) from D", "each tangency point lies on the straight line joining the two centres involved"], "branch_facts": ["the flank named is the one above the line of centres, not its mirror image"]} {"unique_id": "cad_t3_cad_flank_arc_0059", "problem_id": "cad_t3_cad_flank_arc_0059", "domain": "PyGeoX-CAD", "template": "cad_flank_arc", "tier": 3, "nl_description": "A lever is shaped from two circles joined by a curved flank. Set up coordinates with the hub centre at the origin C and the boss centre at D on the positive x-axis. The hub is 15.7 mm across, the boss is 9.2 mm across, and their centres are 24.8 mm apart. The flank is an arc of radius 21.3 mm centred at F. It touches the hub at T and the boss at V, and it bulges away from the line of centres, wrapping around neither circle - each of T and V lies between its own circle's centre and F. The flank shown is the one on the upper side of the line of centres.", "answer_schema": {"points": ["C", "D", "F", "T", "V"], "circles": ["C", "D", "F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Lever flank arc tangent to hub and boss\n\nOracle formalization for cad_t3_cad_flank_arc_0059. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(37)\n\nC, D, F, T, V = scene.add.points([\"C\", \"D\", \"F\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the hub centre (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the hub centre (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the boss lies in the +x direction from the hub\")\nscene.constraint.eq(C.distance(D), 24.8, description=\"hub and boss centres are 24.8 apart\")\nhub = scene.add.circle(C)\nboss = scene.add.circle(D)\nflank = scene.add.circle(F)\nscene.constraint.eq(2 * hub.radius, 15.7, description=\"the hub is 15.7 across\")\nscene.constraint.eq(2 * boss.radius, 9.2, description=\"the boss is 9.2 across\")\nscene.constraint.eq(flank.radius, 21.3, description=\"the flank arc has radius 21.3\")\nscene.relate.point_lies_on(T, hub)\nscene.relate.point_lies_on(T, flank)\nscene.relate.point_lies_on(T, scene.add.line_segment(C, F))\nscene.relate.point_lies_on(V, boss)\nscene.relate.point_lies_on(V, flank)\nscene.relate.point_lies_on(V, scene.add.line_segment(D, F))\nscene.constraint.gt(cross(C, D, F), 0.05, description=\"the flank shown is the one on the upper side of the line of centres\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [24.8, 0.0], "F": [16.00710685483871, 24.361753429048893], "T": [4.310661708764456, 6.5605408033630805], "V": [23.23832785838834, 4.326797906317564]}, "circles": {"C": 7.85, "D": 4.6, "F": 21.3}}, "implicit_facts": ["hub and boss radii are half the stated across-sizes", "two circles touch externally exactly when the distance between centres equals the sum of the radii", "so F is at distance (flank radius + hub radius) from C and (flank radius + boss radius) from D", "each tangency point lies on the straight line joining the two centres involved"], "branch_facts": ["the flank named is the one above the line of centres, not its mirror image"]} {"unique_id": "cad_t3_cad_flank_arc_0060", "problem_id": "cad_t3_cad_flank_arc_0060", "domain": "PyGeoX-CAD", "template": "cad_flank_arc", "tier": 3, "nl_description": "A lever is shaped from two circles joined by a curved flank. Set up coordinates with the hub centre at the origin C and the boss centre at D on the positive x-axis. The hub is 16.7 mm across, the boss is 11.1 mm across, and their centres are 28.3 mm apart. The flank is an arc of radius 23.3 mm centred at F. It touches the hub at T and the boss at V, and it bulges away from the line of centres, wrapping around neither circle - each of T and V lies between its own circle's centre and F. The flank shown is the one on the upper side of the line of centres.", "answer_schema": {"points": ["C", "D", "F", "T", "V"], "circles": ["C", "D", "F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Lever flank arc tangent to hub and boss\n\nOracle formalization for cad_t3_cad_flank_arc_0060. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(42)\n\nC, D, F, T, V = scene.add.points([\"C\", \"D\", \"F\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the hub centre (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the hub centre (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the boss lies in the +x direction from the hub\")\nscene.constraint.eq(C.distance(D), 28.3, description=\"hub and boss centres are 28.3 apart\")\nhub = scene.add.circle(C)\nboss = scene.add.circle(D)\nflank = scene.add.circle(F)\nscene.constraint.eq(2 * hub.radius, 16.7, description=\"the hub is 16.7 across\")\nscene.constraint.eq(2 * boss.radius, 11.1, description=\"the boss is 11.1 across\")\nscene.constraint.eq(flank.radius, 23.3, description=\"the flank arc has radius 23.3\")\nscene.relate.point_lies_on(T, hub)\nscene.relate.point_lies_on(T, flank)\nscene.relate.point_lies_on(T, scene.add.line_segment(C, F))\nscene.relate.point_lies_on(V, boss)\nscene.relate.point_lies_on(V, flank)\nscene.relate.point_lies_on(V, scene.add.line_segment(D, F))\nscene.constraint.gt(cross(C, D, F), 0.05, description=\"the flank shown is the one on the upper side of the line of centres\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [28.3, 0.0], "F": [17.142932862190808, 26.605306855633486], "T": [4.5227010868655055, 7.019093593824317], "V": [26.15366646049078, 5.118178615208521]}, "circles": {"C": 8.35, "D": 5.55, "F": 23.3}}, "implicit_facts": ["hub and boss radii are half the stated across-sizes", "two circles touch externally exactly when the distance between centres equals the sum of the radii", "so F is at distance (flank radius + hub radius) from C and (flank radius + boss radius) from D", "each tangency point lies on the straight line joining the two centres involved"], "branch_facts": ["the flank named is the one above the line of centres, not its mirror image"]} {"unique_id": "cadh_t4_cadh_arc_sagitta_0041", "problem_id": "cadh_t4_cadh_arc_sagitta_0041", "domain": "PyGeoX-CAD", "template": "cadh_arc_sagitta", "tier": 4, "nl_description": "A curved sheet-metal panel is set out from a chord and a bulge. Set up coordinates with the left-hand end of the chord at the origin A, the chord running from A along the positive x-direction to B. The chord from A to B measures 29 mm. Measured at the middle of the chord, the arc bulges 5.3 mm clear of it, reaching its crown at M. The arc is a true circular arc through A, M and B, centred at O, and it bulges upward - away from the side its centre is on. Work out where the centre is and how big the radius has to be.", "answer_schema": {"points": ["A", "B", "M", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Arc set out from chord and bulge\n\nOracle formalization for cadh_t4_cadh_arc_sagitta_0041. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(43)\n\nA, B, M, O = scene.add.points([\"A\", \"B\", \"M\", \"O\"])\n\nscene.constraint.eq(A.x, 0, description=\"datum: A is the origin, the left-hand end of the chord (x)\")\nscene.constraint.eq(A.y, 0, description=\"datum: A is the origin, the left-hand end of the chord (y)\")\nscene.constraint.eq(B.y, 0, description=\"datum: the chord AB runs along the x-axis\")\nscene.constraint.gt(B.x, 1.0, description=\"datum: B is to the right of A\")\nscene.constraint.eq(A.distance(B), 29, description=\"the chord from A to B measures 29\")\nscene.constraint.eq(2 * M.x, A.x + B.x, description=\"the crown M is measured at the middle of the chord\")\nscene.constraint.eq(M.y - A.y, 5.3, description=\"the arc bulges 5.3 clear of the chord at the crown\")\narc = scene.add.circle(O)\nscene.relate.point_lies_on(A, arc)\nscene.relate.point_lies_on(B, arc)\nscene.relate.point_lies_on(M, arc)\nscene.constraint.lt(O.y, -0.05, description=\"the arc bulges upward, so its centre lies below the chord\")\n", "possible_solution": {"points": {"A": [0.0, 0.0], "B": [29.0, 0.0], "M": [14.5, 5.3], "O": [14.5, -17.18490566037736]}, "circles": {"O": 22.48490566037736}}, "implicit_facts": ["the radius is NOT given - it follows from the chord and the bulge", "the crown lies at the middle of the chord, directly out from it", "A, M and B all lie on one circle, which pins the centre on the perpendicular bisector of the chord", "the radius works out as (chord squared over four, plus bulge squared) divided by twice the bulge"], "branch_facts": ["the arc bulges upward, so the centre is on the far side of the chord from the crown"]} {"unique_id": "cadh_t4_cadh_arc_sagitta_0042", "problem_id": "cadh_t4_cadh_arc_sagitta_0042", "domain": "PyGeoX-CAD", "template": "cadh_arc_sagitta", "tier": 4, "nl_description": "A curved sheet-metal panel is set out from a chord and a bulge. Set up coordinates with the left-hand end of the chord at the origin A, the chord running from A along the positive x-direction to B. The chord from A to B measures 26.3 mm. Measured at the middle of the chord, the arc bulges 5.6 mm clear of it, reaching its crown at M. The arc is a true circular arc through A, M and B, centred at O, and it bulges upward - away from the side its centre is on. Work out where the centre is and how big the radius has to be.", "answer_schema": {"points": ["A", "B", "M", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Arc set out from chord and bulge\n\nOracle formalization for cadh_t4_cadh_arc_sagitta_0042. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(39)\n\nA, B, M, O = scene.add.points([\"A\", \"B\", \"M\", \"O\"])\n\nscene.constraint.eq(A.x, 0, description=\"datum: A is the origin, the left-hand end of the chord (x)\")\nscene.constraint.eq(A.y, 0, description=\"datum: A is the origin, the left-hand end of the chord (y)\")\nscene.constraint.eq(B.y, 0, description=\"datum: the chord AB runs along the x-axis\")\nscene.constraint.gt(B.x, 1.0, description=\"datum: B is to the right of A\")\nscene.constraint.eq(A.distance(B), 26.3, description=\"the chord from A to B measures 26.3\")\nscene.constraint.eq(2 * M.x, A.x + B.x, description=\"the crown M is measured at the middle of the chord\")\nscene.constraint.eq(M.y - A.y, 5.6, description=\"the arc bulges 5.6 clear of the chord at the crown\")\narc = scene.add.circle(O)\nscene.relate.point_lies_on(A, arc)\nscene.relate.point_lies_on(B, arc)\nscene.relate.point_lies_on(M, arc)\nscene.constraint.lt(O.y, -0.05, description=\"the arc bulges upward, so its centre lies below the chord\")\n", "possible_solution": {"points": {"A": [0.0, 0.0], "B": [26.3, 0.0], "M": [13.15, 5.6], "O": [13.15, -12.639508928571429]}, "circles": {"O": 18.23950892857143}}, "implicit_facts": ["the radius is NOT given - it follows from the chord and the bulge", "the crown lies at the middle of the chord, directly out from it", "A, M and B all lie on one circle, which pins the centre on the perpendicular bisector of the chord", "the radius works out as (chord squared over four, plus bulge squared) divided by twice the bulge"], "branch_facts": ["the arc bulges upward, so the centre is on the far side of the chord from the crown"]} {"unique_id": "cadh_t4_cadh_arc_sagitta_0043", "problem_id": "cadh_t4_cadh_arc_sagitta_0043", "domain": "PyGeoX-CAD", "template": "cadh_arc_sagitta", "tier": 4, "nl_description": "A curved sheet-metal panel is set out from a chord and a bulge. Set up coordinates with the left-hand end of the chord at the origin A, the chord running from A along the positive x-direction to B. The chord from A to B measures 28.4 mm. Measured at the middle of the chord, the arc bulges 5.3 mm clear of it, reaching its crown at M. The arc is a true circular arc through A, M and B, centred at O, and it bulges upward - away from the side its centre is on. Work out where the centre is and how big the radius has to be.", "answer_schema": {"points": ["A", "B", "M", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Arc set out from chord and bulge\n\nOracle formalization for cadh_t4_cadh_arc_sagitta_0043. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(42)\n\nA, B, M, O = scene.add.points([\"A\", \"B\", \"M\", \"O\"])\n\nscene.constraint.eq(A.x, 0, description=\"datum: A is the origin, the left-hand end of the chord (x)\")\nscene.constraint.eq(A.y, 0, description=\"datum: A is the origin, the left-hand end of the chord (y)\")\nscene.constraint.eq(B.y, 0, description=\"datum: the chord AB runs along the x-axis\")\nscene.constraint.gt(B.x, 1.0, description=\"datum: B is to the right of A\")\nscene.constraint.eq(A.distance(B), 28.4, description=\"the chord from A to B measures 28.4\")\nscene.constraint.eq(2 * M.x, A.x + B.x, description=\"the crown M is measured at the middle of the chord\")\nscene.constraint.eq(M.y - A.y, 5.3, description=\"the arc bulges 5.3 clear of the chord at the crown\")\narc = scene.add.circle(O)\nscene.relate.point_lies_on(A, arc)\nscene.relate.point_lies_on(B, arc)\nscene.relate.point_lies_on(M, arc)\nscene.constraint.lt(O.y, -0.05, description=\"the arc bulges upward, so its centre lies below the chord\")\n", "possible_solution": {"points": {"A": [0.0, 0.0], "B": [28.4, 0.0], "M": [14.2, 5.3], "O": [14.2, -16.37264150943396]}, "circles": {"O": 21.672641509433962}}, "implicit_facts": ["the radius is NOT given - it follows from the chord and the bulge", "the crown lies at the middle of the chord, directly out from it", "A, M and B all lie on one circle, which pins the centre on the perpendicular bisector of the chord", "the radius works out as (chord squared over four, plus bulge squared) divided by twice the bulge"], "branch_facts": ["the arc bulges upward, so the centre is on the far side of the chord from the crown"]} {"unique_id": "cadh_t4_cadh_arc_sagitta_0044", "problem_id": "cadh_t4_cadh_arc_sagitta_0044", "domain": "PyGeoX-CAD", "template": "cadh_arc_sagitta", "tier": 4, "nl_description": "A curved sheet-metal panel is set out from a chord and a bulge. Set up coordinates with the left-hand end of the chord at the origin A, the chord running from A along the positive x-direction to B. The chord from A to B measures 25.4 mm. Measured at the middle of the chord, the arc bulges 9.1 mm clear of it, reaching its crown at M. The arc is a true circular arc through A, M and B, centred at O, and it bulges upward - away from the side its centre is on. Work out where the centre is and how big the radius has to be.", "answer_schema": {"points": ["A", "B", "M", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Arc set out from chord and bulge\n\nOracle formalization for cadh_t4_cadh_arc_sagitta_0044. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(38)\n\nA, B, M, O = scene.add.points([\"A\", \"B\", \"M\", \"O\"])\n\nscene.constraint.eq(A.x, 0, description=\"datum: A is the origin, the left-hand end of the chord (x)\")\nscene.constraint.eq(A.y, 0, description=\"datum: A is the origin, the left-hand end of the chord (y)\")\nscene.constraint.eq(B.y, 0, description=\"datum: the chord AB runs along the x-axis\")\nscene.constraint.gt(B.x, 1.0, description=\"datum: B is to the right of A\")\nscene.constraint.eq(A.distance(B), 25.4, description=\"the chord from A to B measures 25.4\")\nscene.constraint.eq(2 * M.x, A.x + B.x, description=\"the crown M is measured at the middle of the chord\")\nscene.constraint.eq(M.y - A.y, 9.1, description=\"the arc bulges 9.1 clear of the chord at the crown\")\narc = scene.add.circle(O)\nscene.relate.point_lies_on(A, arc)\nscene.relate.point_lies_on(B, arc)\nscene.relate.point_lies_on(M, arc)\nscene.constraint.lt(O.y, -0.05, description=\"the arc bulges upward, so its centre lies below the chord\")\n", "possible_solution": {"points": {"A": [0.0, 0.0], "B": [25.4, 0.0], "M": [12.7, 9.1], "O": [12.7, -4.312087912087911]}, "circles": {"O": 13.41208791208791}}, "implicit_facts": ["the radius is NOT given - it follows from the chord and the bulge", "the crown lies at the middle of the chord, directly out from it", "A, M and B all lie on one circle, which pins the centre on the perpendicular bisector of the chord", "the radius works out as (chord squared over four, plus bulge squared) divided by twice the bulge"], "branch_facts": ["the arc bulges upward, so the centre is on the far side of the chord from the crown"]} {"unique_id": "cadh_t4_cadh_arc_sagitta_0045", "problem_id": "cadh_t4_cadh_arc_sagitta_0045", "domain": "PyGeoX-CAD", "template": "cadh_arc_sagitta", "tier": 4, "nl_description": "A curved sheet-metal panel is set out from a chord and a bulge. Set up coordinates with the left-hand end of the chord at the origin A, the chord running from A along the positive x-direction to B. The chord from A to B measures 28.2 mm. Measured at the middle of the chord, the arc bulges 5.2 mm clear of it, reaching its crown at M. The arc is a true circular arc through A, M and B, centred at O, and it bulges upward - away from the side its centre is on. Work out where the centre is and how big the radius has to be.", "answer_schema": {"points": ["A", "B", "M", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Arc set out from chord and bulge\n\nOracle formalization for cadh_t4_cadh_arc_sagitta_0045. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(42)\n\nA, B, M, O = scene.add.points([\"A\", \"B\", \"M\", \"O\"])\n\nscene.constraint.eq(A.x, 0, description=\"datum: A is the origin, the left-hand end of the chord (x)\")\nscene.constraint.eq(A.y, 0, description=\"datum: A is the origin, the left-hand end of the chord (y)\")\nscene.constraint.eq(B.y, 0, description=\"datum: the chord AB runs along the x-axis\")\nscene.constraint.gt(B.x, 1.0, description=\"datum: B is to the right of A\")\nscene.constraint.eq(A.distance(B), 28.2, description=\"the chord from A to B measures 28.2\")\nscene.constraint.eq(2 * M.x, A.x + B.x, description=\"the crown M is measured at the middle of the chord\")\nscene.constraint.eq(M.y - A.y, 5.2, description=\"the arc bulges 5.2 clear of the chord at the crown\")\narc = scene.add.circle(O)\nscene.relate.point_lies_on(A, arc)\nscene.relate.point_lies_on(B, arc)\nscene.relate.point_lies_on(M, arc)\nscene.constraint.lt(O.y, -0.05, description=\"the arc bulges upward, so its centre lies below the chord\")\n", "possible_solution": {"points": {"A": [0.0, 0.0], "B": [28.2, 0.0], "M": [14.1, 5.2], "O": [14.1, -16.516346153846154]}, "circles": {"O": 21.716346153846153}}, "implicit_facts": ["the radius is NOT given - it follows from the chord and the bulge", "the crown lies at the middle of the chord, directly out from it", "A, M and B all lie on one circle, which pins the centre on the perpendicular bisector of the chord", "the radius works out as (chord squared over four, plus bulge squared) divided by twice the bulge"], "branch_facts": ["the arc bulges upward, so the centre is on the far side of the chord from the crown"]} {"unique_id": "cadh_t4_cadh_hole_exact_clearance_0031", "problem_id": "cadh_t4_cadh_hole_exact_clearance_0031", "domain": "PyGeoX-CAD", "template": "cadh_hole_exact_clearance", "tier": 4, "nl_description": "A rectangular plate measures 37.3 mm by 25.5 mm. Set up coordinates with the bottom-left corner at the origin P, the 37.3 mm edge running from P along the positive x-direction to Q, and the 25.5 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut taking 12.8 mm off along each of the two edges meeting there: the cut runs from G1 on edge QR to G2 on edge SR, both lying on the plate between the corners named. A hole 6.3 mm across is to be drilled with its centre N on the line 14.6 mm up from edge PQ. The hole is pushed as far towards the chamfered corner as it will go, stopping at the point where exactly 2.5 mm of material is left between the hole and the chamfer face.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Hole pushed up to its exact clearance limit\n\nOracle formalization for cadh_t4_cadh_hole_exact_clearance_0031. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(53)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 37.3, description=\"plate is 37.3 long along PQ\")\nscene.constraint.eq(P.distance(S), 25.5, description=\"plate is 25.5 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 12.8, description=\"the chamfer takes 12.8 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 12.8, description=\"the chamfer takes 12.8 off along edge SR\")\nscene.constraint.eq(N.y, 14.6, description=\"the hole centre sits on the line 14.6 up from edge PQ\")\nscene.constraint.eq(cross(G1, G2, N) / G1.distance(G2), 5.65, description=\"the hole is pushed towards the chamfer until exactly 2.5 of material is left between the hole and the chamfer face\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [37.3, 0.0], "R": [37.3, 25.5], "S": [0.0, 25.5], "G1": [37.3, 12.7], "G2": [24.499999999999996, 25.5], "N": [27.40969337259201, 14.6]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'exactly X of material left' is an EQUATION, not a limit to stay under - the clearance is used up completely", "the material left is measured from the hole EDGE, so the centre is (clearance + radius) from the chamfer face", "that distance is the perpendicular distance from N to the line through G1 and G2"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cadh_t4_cadh_hole_exact_clearance_0032", "problem_id": "cadh_t4_cadh_hole_exact_clearance_0032", "domain": "PyGeoX-CAD", "template": "cadh_hole_exact_clearance", "tier": 4, "nl_description": "A rectangular plate measures 34.9 mm by 28.2 mm. Set up coordinates with the bottom-left corner at the origin P, the 34.9 mm edge running from P along the positive x-direction to Q, and the 28.2 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut taking 13.8 mm off along each of the two edges meeting there: the cut runs from G1 on edge QR to G2 on edge SR, both lying on the plate between the corners named. A hole 7 mm across is to be drilled with its centre N on the line 14.9 mm up from edge PQ. The hole is pushed as far towards the chamfered corner as it will go, stopping at the point where exactly 2.1 mm of material is left between the hole and the chamfer face.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Hole pushed up to its exact clearance limit\n\nOracle formalization for cadh_t4_cadh_hole_exact_clearance_0032. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(50)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 34.9, description=\"plate is 34.9 long along PQ\")\nscene.constraint.eq(P.distance(S), 28.2, description=\"plate is 28.2 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 13.8, description=\"the chamfer takes 13.8 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 13.8, description=\"the chamfer takes 13.8 off along edge SR\")\nscene.constraint.eq(N.y, 14.9, description=\"the hole centre sits on the line 14.9 up from edge PQ\")\nscene.constraint.eq(cross(G1, G2, N) / G1.distance(G2), 5.6, description=\"the hole is pushed towards the chamfer until exactly 2.1 of material is left between the hole and the chamfer face\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [34.9, 0.0], "R": [34.9, 28.2], "S": [0.0, 28.2], "G1": [34.9, 14.399999999999999], "G2": [21.099999999999998, 28.2], "N": [26.480404050710668, 14.9]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'exactly X of material left' is an EQUATION, not a limit to stay under - the clearance is used up completely", "the material left is measured from the hole EDGE, so the centre is (clearance + radius) from the chamfer face", "that distance is the perpendicular distance from N to the line through G1 and G2"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cadh_t4_cadh_hole_exact_clearance_0033", "problem_id": "cadh_t4_cadh_hole_exact_clearance_0033", "domain": "PyGeoX-CAD", "template": "cadh_hole_exact_clearance", "tier": 4, "nl_description": "A rectangular plate measures 44.8 mm by 25.1 mm. Set up coordinates with the bottom-left corner at the origin P, the 44.8 mm edge running from P along the positive x-direction to Q, and the 25.1 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut taking 14.4 mm off along each of the two edges meeting there: the cut runs from G1 on edge QR to G2 on edge SR, both lying on the plate between the corners named. A hole 7.3 mm across is to be drilled with its centre N on the line 15 mm up from edge PQ. The hole is pushed as far towards the chamfered corner as it will go, stopping at the point where exactly 2.1 mm of material is left between the hole and the chamfer face.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Hole pushed up to its exact clearance limit\n\nOracle formalization for cadh_t4_cadh_hole_exact_clearance_0033. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(63)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 44.8, description=\"plate is 44.8 long along PQ\")\nscene.constraint.eq(P.distance(S), 25.1, description=\"plate is 25.1 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 14.4, description=\"the chamfer takes 14.4 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 14.4, description=\"the chamfer takes 14.4 off along edge SR\")\nscene.constraint.eq(N.y, 15, description=\"the hole centre sits on the line 15 up from edge PQ\")\nscene.constraint.eq(cross(G1, G2, N) / G1.distance(G2), 5.75, description=\"the hole is pushed towards the chamfer until exactly 2.1 of material is left between the hole and the chamfer face\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [44.8, 0.0], "R": [44.8, 25.1], "S": [0.0, 25.1], "G1": [44.8, 10.700000000000001], "G2": [30.4, 25.1], "N": [32.36827201635471, 15.0]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'exactly X of material left' is an EQUATION, not a limit to stay under - the clearance is used up completely", "the material left is measured from the hole EDGE, so the centre is (clearance + radius) from the chamfer face", "that distance is the perpendicular distance from N to the line through G1 and G2"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cadh_t4_cadh_hole_exact_clearance_0034", "problem_id": "cadh_t4_cadh_hole_exact_clearance_0034", "domain": "PyGeoX-CAD", "template": "cadh_hole_exact_clearance", "tier": 4, "nl_description": "A rectangular plate measures 40.2 mm by 24.4 mm. Set up coordinates with the bottom-left corner at the origin P, the 40.2 mm edge running from P along the positive x-direction to Q, and the 24.4 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut taking 14.5 mm off along each of the two edges meeting there: the cut runs from G1 on edge QR to G2 on edge SR, both lying on the plate between the corners named. A hole 5.9 mm across is to be drilled with its centre N on the line 14.6 mm up from edge PQ. The hole is pushed as far towards the chamfered corner as it will go, stopping at the point where exactly 3.4 mm of material is left between the hole and the chamfer face.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Hole pushed up to its exact clearance limit\n\nOracle formalization for cadh_t4_cadh_hole_exact_clearance_0034. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(57)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 40.2, description=\"plate is 40.2 long along PQ\")\nscene.constraint.eq(P.distance(S), 24.4, description=\"plate is 24.4 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 14.5, description=\"the chamfer takes 14.5 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 14.5, description=\"the chamfer takes 14.5 off along edge SR\")\nscene.constraint.eq(N.y, 14.6, description=\"the hole centre sits on the line 14.6 up from edge PQ\")\nscene.constraint.eq(cross(G1, G2, N) / G1.distance(G2), 6.35, description=\"the hole is pushed towards the chamfer until exactly 3.4 of material is left between the hole and the chamfer face\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [40.2, 0.0], "R": [40.2, 24.4], "S": [0.0, 24.4], "G1": [40.2, 9.899999999999999], "G2": [25.700000000000003, 24.4], "N": [26.51974387893084, 14.6]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'exactly X of material left' is an EQUATION, not a limit to stay under - the clearance is used up completely", "the material left is measured from the hole EDGE, so the centre is (clearance + radius) from the chamfer face", "that distance is the perpendicular distance from N to the line through G1 and G2"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cadh_t4_cadh_hole_exact_clearance_0035", "problem_id": "cadh_t4_cadh_hole_exact_clearance_0035", "domain": "PyGeoX-CAD", "template": "cadh_hole_exact_clearance", "tier": 4, "nl_description": "A rectangular plate measures 36 mm by 27.1 mm. Set up coordinates with the bottom-left corner at the origin P, the 36 mm edge running from P along the positive x-direction to Q, and the 27.1 mm edge running from P along the positive y-direction to S; R is the far corner, diagonally opposite P. The corner at R is chamfered off by a straight cut taking 14.7 mm off along each of the two edges meeting there: the cut runs from G1 on edge QR to G2 on edge SR, both lying on the plate between the corners named. A hole 6.3 mm across is to be drilled with its centre N on the line 12 mm up from edge PQ. The hole is pushed as far towards the chamfered corner as it will go, stopping at the point where exactly 3.5 mm of material is left between the hole and the chamfer face.", "answer_schema": {"points": ["G1", "G2", "N", "P", "Q", "R", "S"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Hole pushed up to its exact clearance limit\n\nOracle formalization for cadh_t4_cadh_hole_exact_clearance_0035. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(52)\n\nP, Q, R, S, G1, G2, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"G1\", \"G2\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 36, description=\"plate is 36 long along PQ\")\nscene.constraint.eq(P.distance(S), 27.1, description=\"plate is 27.1 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nscene.relate.point_lies_on(G1, scene.add.line_segment(Q, R))\nscene.relate.point_lies_on(G2, scene.add.line_segment(S, R))\nscene.constraint.eq(G1.distance(R), 14.7, description=\"the chamfer takes 14.7 off along edge QR\")\nscene.constraint.eq(G2.distance(R), 14.7, description=\"the chamfer takes 14.7 off along edge SR\")\nscene.constraint.eq(N.y, 12, description=\"the hole centre sits on the line 12 up from edge PQ\")\nscene.constraint.eq(cross(G1, G2, N) / G1.distance(G2), 6.65, description=\"the hole is pushed towards the chamfer until exactly 3.5 of material is left between the hole and the chamfer face\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [36.0, 0.0], "R": [36.0, 27.1], "S": [0.0, 27.1], "G1": [36.0, 12.400000000000002], "G2": [21.3, 27.1], "N": [26.995479810218924, 12.0]}, "circles": {}}, "implicit_facts": ["the hole radius is half the stated across-size", "'exactly X of material left' is an EQUATION, not a limit to stay under - the clearance is used up completely", "the material left is measured from the hole EDGE, so the centre is (clearance + radius) from the chamfer face", "that distance is the perpendicular distance from N to the line through G1 and G2"], "branch_facts": ["G1 and G2 lie on the plate between the named corners, not on the extensions of those edges"]} {"unique_id": "cadh_t4_cadh_incircle_vee_0001", "problem_id": "cadh_t4_cadh_incircle_vee_0001", "domain": "PyGeoX-CAD", "template": "cadh_incircle_vee", "tier": 4, "nl_description": "A round bar is dropped into a vee block and a flat cover plate is then clamped down across the top of the vee, trapping the bar so that it touches all three faces at once. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 22.3 mm long and the included angle of the vee, measured at the apex between the two faces, is 55 degrees. The cover plate lies along the straight line from A to B. The bar's centre is F; it touches the right-hand face at T, the left-hand face at U, and the underside of the cover at W. The bar is the one that fits inside the vee, between the apex and the cover. Work out how big the bar has to be.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "W", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bar trapped in a vee under a cover plate\n\nOracle formalization for cadh_t4_cadh_incircle_vee_0001. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(31)\n\nZ, A, B, F, T, U, W = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\", \"W\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 22.3, description=\"each vee face is 22.3 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 55, description=\"the included angle of the vee is 55 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\ncover = scene.add.line_segment(A, B)\nbar = scene.add.circle(F)\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.relate.tangent_to_circle(cover, bar, W)\nscene.constraint.gt(F.y, 0.05, description=\"the bar sits inside the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [10.296994075141257, 19.780341579874346], "B": [-10.296994075141257, 19.780341579874346], "F": [0.0, 13.531972187815494], "T": [5.542371340455421, 10.646796286052089], "U": [-5.542371340455421, 10.646796286052089], "W": [0.0, 19.780341579874346]}, "circles": {"F": 6.248369392058852}}, "implicit_facts": ["the bar size is NOT given - it is fixed by having to touch all three faces at once", "touching all three sides of the triangle ZAB makes the bar the inscribed circle of that triangle", "so its radius is the triangle's area divided by its semi-perimeter", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar is the one inside the vee, not the larger circle that touches the same three lines from beyond the cover"]} {"unique_id": "cadh_t4_cadh_incircle_vee_0002", "problem_id": "cadh_t4_cadh_incircle_vee_0002", "domain": "PyGeoX-CAD", "template": "cadh_incircle_vee", "tier": 4, "nl_description": "A round bar is dropped into a vee block and a flat cover plate is then clamped down across the top of the vee, trapping the bar so that it touches all three faces at once. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 24.4 mm long and the included angle of the vee, measured at the apex between the two faces, is 70 degrees. The cover plate lies along the straight line from A to B. The bar's centre is F; it touches the right-hand face at T, the left-hand face at U, and the underside of the cover at W. The bar is the one that fits inside the vee, between the apex and the cover. Work out how big the bar has to be.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "W", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bar trapped in a vee under a cover plate\n\nOracle formalization for cadh_t4_cadh_incircle_vee_0002. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(31)\n\nZ, A, B, F, T, U, W = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\", \"W\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 24.4, description=\"each vee face is 24.4 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 70, description=\"the included angle of the vee is 70 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\ncover = scene.add.line_segment(A, B)\nbar = scene.add.circle(F)\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.relate.tangent_to_circle(cover, bar, W)\nscene.constraint.gt(F.y, 0.05, description=\"the bar sits inside the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [13.995265046965523, 19.9873098806514], "B": [-13.995265046965523, 19.9873098806514], "F": [0.0, 12.701836033462609], "T": [5.967910795538683, 8.523059907063319], "U": [-5.967910795538683, 8.523059907063319], "W": [0.0, 19.9873098806514]}, "circles": {"F": 7.285473847188789}}, "implicit_facts": ["the bar size is NOT given - it is fixed by having to touch all three faces at once", "touching all three sides of the triangle ZAB makes the bar the inscribed circle of that triangle", "so its radius is the triangle's area divided by its semi-perimeter", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar is the one inside the vee, not the larger circle that touches the same three lines from beyond the cover"]} {"unique_id": "cadh_t4_cadh_incircle_vee_0003", "problem_id": "cadh_t4_cadh_incircle_vee_0003", "domain": "PyGeoX-CAD", "template": "cadh_incircle_vee", "tier": 4, "nl_description": "A round bar is dropped into a vee block and a flat cover plate is then clamped down across the top of the vee, trapping the bar so that it touches all three faces at once. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 25 mm long and the included angle of the vee, measured at the apex between the two faces, is 90 degrees. The cover plate lies along the straight line from A to B. The bar's centre is F; it touches the right-hand face at T, the left-hand face at U, and the underside of the cover at W. The bar is the one that fits inside the vee, between the apex and the cover. Work out how big the bar has to be.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "W", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bar trapped in a vee under a cover plate\n\nOracle formalization for cadh_t4_cadh_incircle_vee_0003. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(28)\n\nZ, A, B, F, T, U, W = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\", \"W\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 25, description=\"each vee face is 25 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 90, description=\"the included angle of the vee is 90 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\ncover = scene.add.line_segment(A, B)\nbar = scene.add.circle(F)\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.relate.tangent_to_circle(cover, bar, W)\nscene.constraint.gt(F.y, 0.05, description=\"the bar sits inside the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [17.67766952966369, 17.67766952966369], "B": [-17.67766952966369, 17.67766952966369], "F": [0.0, 10.355339059327378], "T": [5.17766952966369, 5.17766952966369], "U": [-5.17766952966369, 5.17766952966369], "W": [0.0, 17.67766952966369]}, "circles": {"F": 7.322330470336313}}, "implicit_facts": ["the bar size is NOT given - it is fixed by having to touch all three faces at once", "touching all three sides of the triangle ZAB makes the bar the inscribed circle of that triangle", "so its radius is the triangle's area divided by its semi-perimeter", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar is the one inside the vee, not the larger circle that touches the same three lines from beyond the cover"]} {"unique_id": "cadh_t4_cadh_incircle_vee_0004", "problem_id": "cadh_t4_cadh_incircle_vee_0004", "domain": "PyGeoX-CAD", "template": "cadh_incircle_vee", "tier": 4, "nl_description": "A round bar is dropped into a vee block and a flat cover plate is then clamped down across the top of the vee, trapping the bar so that it touches all three faces at once. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 24.8 mm long and the included angle of the vee, measured at the apex between the two faces, is 60 degrees. The cover plate lies along the straight line from A to B. The bar's centre is F; it touches the right-hand face at T, the left-hand face at U, and the underside of the cover at W. The bar is the one that fits inside the vee, between the apex and the cover. Work out how big the bar has to be.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "W", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bar trapped in a vee under a cover plate\n\nOracle formalization for cadh_t4_cadh_incircle_vee_0004. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(33)\n\nZ, A, B, F, T, U, W = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\", \"W\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 24.8, description=\"each vee face is 24.8 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 60, description=\"the included angle of the vee is 60 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\ncover = scene.add.line_segment(A, B)\nbar = scene.add.circle(F)\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.relate.tangent_to_circle(cover, bar, W)\nscene.constraint.gt(F.y, 0.05, description=\"the bar sits inside the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [12.399999999999999, 21.47743001385408], "B": [-12.399999999999999, 21.47743001385408], "F": [0.0, 14.31828667590272], "T": [6.199999999999999, 10.73871500692704], "U": [-6.199999999999999, 10.73871500692704], "W": [0.0, 21.47743001385408]}, "circles": {"F": 7.159143337951359}}, "implicit_facts": ["the bar size is NOT given - it is fixed by having to touch all three faces at once", "touching all three sides of the triangle ZAB makes the bar the inscribed circle of that triangle", "so its radius is the triangle's area divided by its semi-perimeter", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar is the one inside the vee, not the larger circle that touches the same three lines from beyond the cover"]} {"unique_id": "cadh_t4_cadh_incircle_vee_0005", "problem_id": "cadh_t4_cadh_incircle_vee_0005", "domain": "PyGeoX-CAD", "template": "cadh_incircle_vee", "tier": 4, "nl_description": "A round bar is dropped into a vee block and a flat cover plate is then clamped down across the top of the vee, trapping the bar so that it touches all three faces at once. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 22.8 mm long and the included angle of the vee, measured at the apex between the two faces, is 55 degrees. The cover plate lies along the straight line from A to B. The bar's centre is F; it touches the right-hand face at T, the left-hand face at U, and the underside of the cover at W. The bar is the one that fits inside the vee, between the apex and the cover. Work out how big the bar has to be.", "answer_schema": {"points": ["A", "B", "F", "T", "U", "W", "Z"], "circles": ["F"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bar trapped in a vee under a cover plate\n\nOracle formalization for cadh_t4_cadh_incircle_vee_0005. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(31)\n\nZ, A, B, F, T, U, W = scene.add.points([\"Z\", \"A\", \"B\", \"F\", \"T\", \"U\", \"W\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 22.8, description=\"each vee face is 22.8 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 55, description=\"the included angle of the vee is 55 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\ncover = scene.add.line_segment(A, B)\nbar = scene.add.circle(F)\nscene.relate.tangent_to_circle(face_r, bar, T)\nscene.relate.tangent_to_circle(face_l, bar, U)\nscene.relate.tangent_to_circle(cover, bar, W)\nscene.constraint.gt(F.y, 0.05, description=\"the bar sits inside the vee, above the apex\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [10.527868381758774, 20.223846996463454], "B": [-10.527868381758774, 20.223846996463454], "F": [0.0, 13.83537963597279], "T": [5.666639756160699, 10.88551369156895], "U": [-5.666639756160699, 10.88551369156895], "W": [0.0, 20.223846996463454]}, "circles": {"F": 6.388467360490664}}, "implicit_facts": ["the bar size is NOT given - it is fixed by having to touch all three faces at once", "touching all three sides of the triangle ZAB makes the bar the inscribed circle of that triangle", "so its radius is the triangle's area divided by its semi-perimeter", "each contact point is the foot of the perpendicular from F onto that face"], "branch_facts": ["the bar is the one inside the vee, not the larger circle that touches the same three lines from beyond the cover"]} {"unique_id": "cadh_t4_cadh_offset_corner_0061", "problem_id": "cadh_t4_cadh_offset_corner_0061", "domain": "PyGeoX-CAD", "template": "cadh_offset_corner", "tier": 4, "nl_description": "A stiffening rib is folded to a vee and then machined to a constant wall thickness, so the inner profile runs parallel to the outer one all the way round. Set up coordinates with the outside corner of the rib at the origin Z, opening upward and symmetric about the positive y-direction; the right-hand outer face runs from Z to A and the left-hand outer face from Z to B. Each outer face is 35.9 mm long and the two meet at 110 degrees. The wall is 7 mm thick, measured square to each face. K is the point where the two inner faces meet; Ta is the point on the right-hand outer face nearest to K, and Tb the point on the left-hand outer face nearest to K. Work out how far K sits from the outside corner.", "answer_schema": {"points": ["A", "B", "K", "Ta", "Tb", "Z"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Inner corner of a constant-thickness offset at a non-right angle\n\nOracle formalization for cadh_t4_cadh_offset_corner_0061. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(43)\n\nZ, A, B, K, Ta, Tb = scene.add.points([\"Z\", \"A\", \"B\", \"K\", \"Ta\", \"Tb\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the outside corner of the rib (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the outside corner of the rib (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand outer face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 35.9, description=\"each outer face is 35.9 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 110, description=\"the two outer faces meet at 110 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nscene.relate.point_lies_on(Ta, face_r)\nscene.relate.point_lies_on(Tb, face_l)\nscene.relate.perpendicular(scene.add.line_segment(K, Ta), face_r)\nscene.relate.perpendicular(scene.add.line_segment(K, Tb), face_l)\nscene.constraint.eq(K.distance(Ta), 7, description=\"the rib is 7 thick, measured square to the right-hand face\")\nscene.constraint.eq(K.distance(Tb), 7, description=\"the rib is 7 thick, measured square to the left-hand face\")\nscene.constraint.gt(K.y, 0.05, description=\"the inner corner K lies inside the rib, on the same side as the material\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [29.407558389974806, 20.591394065002554], "B": [-29.407558389974806, 20.591394065002554], "K": [0.0, 8.545422121330192], "Ta": [4.015035054457322, 2.8113578113072495], "Tb": [-4.015035054457322, 2.8113578113072495]}, "circles": {}}, "implicit_facts": ["the inner corner is NOT the wall thickness from the outer corner - that only holds at a right angle", "the nearest point on a face is the foot of the perpendicular, so K is the stated thickness square to BOTH faces", "being the same distance from both faces puts K on the bisector of the corner", "the distance from the outer corner along the bisector is the thickness divided by the sine of HALF the included angle"], "branch_facts": ["K lies inside the rib on the material side, not on the mirror-image side of the apex"]} {"unique_id": "cadh_t4_cadh_offset_corner_0062", "problem_id": "cadh_t4_cadh_offset_corner_0062", "domain": "PyGeoX-CAD", "template": "cadh_offset_corner", "tier": 4, "nl_description": "A stiffening rib is folded to a vee and then machined to a constant wall thickness, so the inner profile runs parallel to the outer one all the way round. Set up coordinates with the outside corner of the rib at the origin Z, opening upward and symmetric about the positive y-direction; the right-hand outer face runs from Z to A and the left-hand outer face from Z to B. Each outer face is 27.5 mm long and the two meet at 60 degrees. The wall is 3.8 mm thick, measured square to each face. K is the point where the two inner faces meet; Ta is the point on the right-hand outer face nearest to K, and Tb the point on the left-hand outer face nearest to K. Work out how far K sits from the outside corner.", "answer_schema": {"points": ["A", "B", "K", "Ta", "Tb", "Z"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Inner corner of a constant-thickness offset at a non-right angle\n\nOracle formalization for cadh_t4_cadh_offset_corner_0062. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(36)\n\nZ, A, B, K, Ta, Tb = scene.add.points([\"Z\", \"A\", \"B\", \"K\", \"Ta\", \"Tb\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the outside corner of the rib (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the outside corner of the rib (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand outer face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 27.5, description=\"each outer face is 27.5 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 60, description=\"the two outer faces meet at 60 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nscene.relate.point_lies_on(Ta, face_r)\nscene.relate.point_lies_on(Tb, face_l)\nscene.relate.perpendicular(scene.add.line_segment(K, Ta), face_r)\nscene.relate.perpendicular(scene.add.line_segment(K, Tb), face_l)\nscene.constraint.eq(K.distance(Ta), 3.8, description=\"the rib is 3.8 thick, measured square to the right-hand face\")\nscene.constraint.eq(K.distance(Tb), 3.8, description=\"the rib is 3.8 thick, measured square to the left-hand face\")\nscene.constraint.gt(K.y, 0.05, description=\"the inner corner K lies inside the rib, on the same side as the material\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [13.749999999999998, 23.815698604072065], "B": [-13.749999999999998, 23.815698604072065], "K": [0.0, 7.6000000000000005], "Ta": [3.2908965343808667, 5.700000000000001], "Tb": [-3.2908965343808667, 5.700000000000001]}, "circles": {}}, "implicit_facts": ["the inner corner is NOT the wall thickness from the outer corner - that only holds at a right angle", "the nearest point on a face is the foot of the perpendicular, so K is the stated thickness square to BOTH faces", "being the same distance from both faces puts K on the bisector of the corner", "the distance from the outer corner along the bisector is the thickness divided by the sine of HALF the included angle"], "branch_facts": ["K lies inside the rib on the material side, not on the mirror-image side of the apex"]} {"unique_id": "cadh_t4_cadh_offset_corner_0063", "problem_id": "cadh_t4_cadh_offset_corner_0063", "domain": "PyGeoX-CAD", "template": "cadh_offset_corner", "tier": 4, "nl_description": "A stiffening rib is folded to a vee and then machined to a constant wall thickness, so the inner profile runs parallel to the outer one all the way round. Set up coordinates with the outside corner of the rib at the origin Z, opening upward and symmetric about the positive y-direction; the right-hand outer face runs from Z to A and the left-hand outer face from Z to B. Each outer face is 34.6 mm long and the two meet at 50 degrees. The wall is 3.7 mm thick, measured square to each face. K is the point where the two inner faces meet; Ta is the point on the right-hand outer face nearest to K, and Tb the point on the left-hand outer face nearest to K. Work out how far K sits from the outside corner.", "answer_schema": {"points": ["A", "B", "K", "Ta", "Tb", "Z"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Inner corner of a constant-thickness offset at a non-right angle\n\nOracle formalization for cadh_t4_cadh_offset_corner_0063. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(46)\n\nZ, A, B, K, Ta, Tb = scene.add.points([\"Z\", \"A\", \"B\", \"K\", \"Ta\", \"Tb\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the outside corner of the rib (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the outside corner of the rib (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand outer face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 34.6, description=\"each outer face is 34.6 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 50, description=\"the two outer faces meet at 50 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nscene.relate.point_lies_on(Ta, face_r)\nscene.relate.point_lies_on(Tb, face_l)\nscene.relate.perpendicular(scene.add.line_segment(K, Ta), face_r)\nscene.relate.perpendicular(scene.add.line_segment(K, Tb), face_l)\nscene.constraint.eq(K.distance(Ta), 3.7, description=\"the rib is 3.7 thick, measured square to the right-hand face\")\nscene.constraint.eq(K.distance(Tb), 3.7, description=\"the rib is 3.7 thick, measured square to the left-hand face\")\nscene.constraint.gt(K.y, 0.05, description=\"the inner corner K lies inside the rib, on the same side as the material\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [14.622591856228201, 31.35824943146809], "B": [-14.622591856228201, 31.35824943146809], "K": [0.0, 8.754945857664245], "Ta": [3.353338812035605, 7.191258289223657], "Tb": [-3.353338812035605, 7.191258289223657]}, "circles": {}}, "implicit_facts": ["the inner corner is NOT the wall thickness from the outer corner - that only holds at a right angle", "the nearest point on a face is the foot of the perpendicular, so K is the stated thickness square to BOTH faces", "being the same distance from both faces puts K on the bisector of the corner", "the distance from the outer corner along the bisector is the thickness divided by the sine of HALF the included angle"], "branch_facts": ["K lies inside the rib on the material side, not on the mirror-image side of the apex"]} {"unique_id": "cadh_t4_cadh_offset_corner_0064", "problem_id": "cadh_t4_cadh_offset_corner_0064", "domain": "PyGeoX-CAD", "template": "cadh_offset_corner", "tier": 4, "nl_description": "A stiffening rib is folded to a vee and then machined to a constant wall thickness, so the inner profile runs parallel to the outer one all the way round. Set up coordinates with the outside corner of the rib at the origin Z, opening upward and symmetric about the positive y-direction; the right-hand outer face runs from Z to A and the left-hand outer face from Z to B. Each outer face is 37.9 mm long and the two meet at 50 degrees. The wall is 3.3 mm thick, measured square to each face. K is the point where the two inner faces meet; Ta is the point on the right-hand outer face nearest to K, and Tb the point on the left-hand outer face nearest to K. Work out how far K sits from the outside corner.", "answer_schema": {"points": ["A", "B", "K", "Ta", "Tb", "Z"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Inner corner of a constant-thickness offset at a non-right angle\n\nOracle formalization for cadh_t4_cadh_offset_corner_0064. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(50)\n\nZ, A, B, K, Ta, Tb = scene.add.points([\"Z\", \"A\", \"B\", \"K\", \"Ta\", \"Tb\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the outside corner of the rib (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the outside corner of the rib (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand outer face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 37.9, description=\"each outer face is 37.9 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 50, description=\"the two outer faces meet at 50 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nscene.relate.point_lies_on(Ta, face_r)\nscene.relate.point_lies_on(Tb, face_l)\nscene.relate.perpendicular(scene.add.line_segment(K, Ta), face_r)\nscene.relate.perpendicular(scene.add.line_segment(K, Tb), face_l)\nscene.constraint.eq(K.distance(Ta), 3.3, description=\"the rib is 3.3 thick, measured square to the right-hand face\")\nscene.constraint.eq(K.distance(Tb), 3.3, description=\"the rib is 3.3 thick, measured square to the left-hand face\")\nscene.constraint.gt(K.y, 0.05, description=\"the inner corner K lies inside the rib, on the same side as the material\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [16.01723211997251, 34.349065128689034], "B": [-16.01723211997251, 34.349065128689034], "K": [0.0, 7.808465224403244], "Ta": [2.9908156972209445, 6.413824960658936], "Tb": [-2.9908156972209445, 6.413824960658936]}, "circles": {}}, "implicit_facts": ["the inner corner is NOT the wall thickness from the outer corner - that only holds at a right angle", "the nearest point on a face is the foot of the perpendicular, so K is the stated thickness square to BOTH faces", "being the same distance from both faces puts K on the bisector of the corner", "the distance from the outer corner along the bisector is the thickness divided by the sine of HALF the included angle"], "branch_facts": ["K lies inside the rib on the material side, not on the mirror-image side of the apex"]} {"unique_id": "cadh_t4_cadh_offset_corner_0065", "problem_id": "cadh_t4_cadh_offset_corner_0065", "domain": "PyGeoX-CAD", "template": "cadh_offset_corner", "tier": 4, "nl_description": "A stiffening rib is folded to a vee and then machined to a constant wall thickness, so the inner profile runs parallel to the outer one all the way round. Set up coordinates with the outside corner of the rib at the origin Z, opening upward and symmetric about the positive y-direction; the right-hand outer face runs from Z to A and the left-hand outer face from Z to B. Each outer face is 34.7 mm long and the two meet at 70 degrees. The wall is 5.7 mm thick, measured square to each face. K is the point where the two inner faces meet; Ta is the point on the right-hand outer face nearest to K, and Tb the point on the left-hand outer face nearest to K. Work out how far K sits from the outside corner.", "answer_schema": {"points": ["A", "B", "K", "Ta", "Tb", "Z"], "circles": [], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Inner corner of a constant-thickness offset at a non-right angle\n\nOracle formalization for cadh_t4_cadh_offset_corner_0065. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(42)\n\nZ, A, B, K, Ta, Tb = scene.add.points([\"Z\", \"A\", \"B\", \"K\", \"Ta\", \"Tb\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the outside corner of the rib (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the outside corner of the rib (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the rib is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand outer face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 34.7, description=\"each outer face is 34.7 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 70, description=\"the two outer faces meet at 70 degrees\")\nface_r = scene.add.line_segment(Z, A)\nface_l = scene.add.line_segment(Z, B)\nscene.relate.point_lies_on(Ta, face_r)\nscene.relate.point_lies_on(Tb, face_l)\nscene.relate.perpendicular(scene.add.line_segment(K, Ta), face_r)\nscene.relate.perpendicular(scene.add.line_segment(K, Tb), face_l)\nscene.constraint.eq(K.distance(Ta), 5.7, description=\"the rib is 5.7 thick, measured square to the right-hand face\")\nscene.constraint.eq(K.distance(Tb), 5.7, description=\"the rib is 5.7 thick, measured square to the left-hand face\")\nscene.constraint.gt(K.y, 0.05, description=\"the inner corner K lies inside the rib, on the same side as the material\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [19.9031023413813, 28.424575936828017], "B": [-19.9031023413813, 28.424575936828017], "K": [0.0, 9.93764673504026], "Ta": [4.669166652447254, 6.668261047839298], "Tb": [-4.669166652447254, 6.668261047839298]}, "circles": {}}, "implicit_facts": ["the inner corner is NOT the wall thickness from the outer corner - that only holds at a right angle", "the nearest point on a face is the foot of the perpendicular, so K is the stated thickness square to BOTH faces", "being the same distance from both faces puts K on the bisector of the corner", "the distance from the outer corner along the bisector is the thickness divided by the sine of HALF the included angle"], "branch_facts": ["K lies inside the rib on the material side, not on the mirror-image side of the apex"]} {"unique_id": "cadh_t4_cadh_ogee_blend_0021", "problem_id": "cadh_t4_cadh_ogee_blend_0021", "domain": "PyGeoX-CAD", "template": "cadh_ogee_blend", "tier": 4, "nl_description": "A step in a pressed panel is blended with an S-curve made of two arcs. Set up coordinates with the left-hand end of the lower face at the origin P; the lower face runs from P along the positive x-direction to Q and is 37.1 mm long. The upper face runs parallel to it 18.4 mm above, from S (directly above P) to R (directly above Q). The first arc, radius 9.5 mm and centred at C1, runs smoothly out of the lower face, leaving it at T1 which is 10.6 mm along from P. The second arc, radius 7.3 mm and centred at C2, runs smoothly into the upper face at T2. The two arcs meet each other at J, and the curve is smooth there too - there is no kink where one arc becomes the other. The S-curve runs left to right, climbing from the lower face to the upper one.", "answer_schema": {"points": ["C1", "C2", "J", "P", "Q", "R", "S", "T1", "T2"], "circles": ["C1", "C2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Ogee (S-curve) blend between two parallel faces\n\nOracle formalization for cadh_t4_cadh_ogee_blend_0021. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(53)\n\nP, Q, S, R, C1, C2, T1, T2, J = scene.add.points([\"P\", \"Q\", \"S\", \"R\", \"C1\", \"C2\", \"T1\", \"T2\", \"J\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the left-hand end of the lower face (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the left-hand end of the lower face (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: the lower face PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the part extends in the +x direction\")\nscene.constraint.eq(P.distance(Q), 37.1, description=\"the lower face is 37.1 long\")\nscene.constraint.eq(S.y - P.y, 18.4, description=\"the upper face is 18.4 above the lower one\")\nscene.constraint.eq(R.y - Q.y, 18.4, description=\"the upper face is parallel to the lower one\")\nscene.constraint.eq(S.x, P.x, description=\"the upper face starts directly above P\")\nscene.constraint.eq(R.x, Q.x, description=\"the upper face ends directly above Q\")\nlower = scene.add.line_segment(P, Q)\nupper = scene.add.line_segment(S, R)\narc1 = scene.add.circle(C1)\narc2 = scene.add.circle(C2)\nscene.constraint.eq(arc1.radius, 9.5, description=\"the first blend arc has radius 9.5\")\nscene.constraint.eq(arc2.radius, 7.3, description=\"the second blend arc has radius 7.3\")\nscene.relate.tangent_to_circle(lower, arc1, T1)\nscene.relate.tangent_to_circle(upper, arc2, T2)\nscene.relate.point_lies_on(J, arc1)\nscene.relate.point_lies_on(J, arc2)\nscene.relate.point_lies_on(J, scene.add.line_segment(C1, C2))\nscene.constraint.eq(T1.x, 10.6, description=\"the S-curve leaves the lower face 10.6 along from P\")\nscene.constraint.gt(C1.y, 0.05, description=\"the first arc curves up away from the lower face\")\nscene.constraint.gt(T2.x - T1.x, 0.05, description=\"the S-curve runs left to right, from the lower face up to the upper one\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [37.1, 0.0], "S": [0.0, 18.4], "R": [37.1, 18.4], "C1": [10.6, 9.5], "C2": [27.323635968293495, 11.099999999999998], "T1": [10.6, 0.0], "T2": [27.323635968293495, 18.4], "J": [20.0568179582612, 10.404761904761903]}, "circles": {"C1": 9.5, "C2": 7.3}}, "implicit_facts": ["'runs smoothly out of' a straight face means the arc is tangent to it, so its centre sits one radius away, square to the face", "'no kink where one arc becomes the other' means the two arcs are tangent to EACH OTHER at J", "two arcs tangent to each other with the curve bending opposite ways have centre distance equal to the SUM of the radii, with J on the line joining the centres", "that fixes how far along the second arc must sit - it is not a free choice"], "branch_facts": ["the first arc curves upward away from the lower face, and the curve runs left to right"]} {"unique_id": "cadh_t4_cadh_ogee_blend_0022", "problem_id": "cadh_t4_cadh_ogee_blend_0022", "domain": "PyGeoX-CAD", "template": "cadh_ogee_blend", "tier": 4, "nl_description": "A step in a pressed panel is blended with an S-curve made of two arcs. Set up coordinates with the left-hand end of the lower face at the origin P; the lower face runs from P along the positive x-direction to Q and is 37 mm long. The upper face runs parallel to it 19.7 mm above, from S (directly above P) to R (directly above Q). The first arc, radius 12.7 mm and centred at C1, runs smoothly out of the lower face, leaving it at T1 which is 8 mm along from P. The second arc, radius 7.1 mm and centred at C2, runs smoothly into the upper face at T2. The two arcs meet each other at J, and the curve is smooth there too - there is no kink where one arc becomes the other. The S-curve runs left to right, climbing from the lower face to the upper one.", "answer_schema": {"points": ["C1", "C2", "J", "P", "Q", "R", "S", "T1", "T2"], "circles": ["C1", "C2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Ogee (S-curve) blend between two parallel faces\n\nOracle formalization for cadh_t4_cadh_ogee_blend_0022. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(53)\n\nP, Q, S, R, C1, C2, T1, T2, J = scene.add.points([\"P\", \"Q\", \"S\", \"R\", \"C1\", \"C2\", \"T1\", \"T2\", \"J\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the left-hand end of the lower face (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the left-hand end of the lower face (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: the lower face PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the part extends in the +x direction\")\nscene.constraint.eq(P.distance(Q), 37, description=\"the lower face is 37 long\")\nscene.constraint.eq(S.y - P.y, 19.7, description=\"the upper face is 19.7 above the lower one\")\nscene.constraint.eq(R.y - Q.y, 19.7, description=\"the upper face is parallel to the lower one\")\nscene.constraint.eq(S.x, P.x, description=\"the upper face starts directly above P\")\nscene.constraint.eq(R.x, Q.x, description=\"the upper face ends directly above Q\")\nlower = scene.add.line_segment(P, Q)\nupper = scene.add.line_segment(S, R)\narc1 = scene.add.circle(C1)\narc2 = scene.add.circle(C2)\nscene.constraint.eq(arc1.radius, 12.7, description=\"the first blend arc has radius 12.7\")\nscene.constraint.eq(arc2.radius, 7.1, description=\"the second blend arc has radius 7.1\")\nscene.relate.tangent_to_circle(lower, arc1, T1)\nscene.relate.tangent_to_circle(upper, arc2, T2)\nscene.relate.point_lies_on(J, arc1)\nscene.relate.point_lies_on(J, arc2)\nscene.relate.point_lies_on(J, scene.add.line_segment(C1, C2))\nscene.constraint.eq(T1.x, 8, description=\"the S-curve leaves the lower face 8 along from P\")\nscene.constraint.gt(C1.y, 0.05, description=\"the first arc curves up away from the lower face\")\nscene.constraint.gt(T2.x - T1.x, 0.05, description=\"the S-curve runs left to right, from the lower face up to the upper one\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [37.0, 0.0], "S": [0.0, 19.7], "R": [37.0, 19.7], "C1": [8.0, 12.7], "C2": [27.799747473137124, 12.6], "T1": [8.0, 0.0], "T2": [27.799747473137124, 19.7], "J": [20.699838025699066, 12.635858585858585]}, "circles": {"C1": 12.7, "C2": 7.1}}, "implicit_facts": ["'runs smoothly out of' a straight face means the arc is tangent to it, so its centre sits one radius away, square to the face", "'no kink where one arc becomes the other' means the two arcs are tangent to EACH OTHER at J", "two arcs tangent to each other with the curve bending opposite ways have centre distance equal to the SUM of the radii, with J on the line joining the centres", "that fixes how far along the second arc must sit - it is not a free choice"], "branch_facts": ["the first arc curves upward away from the lower face, and the curve runs left to right"]} {"unique_id": "cadh_t4_cadh_ogee_blend_0023", "problem_id": "cadh_t4_cadh_ogee_blend_0023", "domain": "PyGeoX-CAD", "template": "cadh_ogee_blend", "tier": 4, "nl_description": "A step in a pressed panel is blended with an S-curve made of two arcs. Set up coordinates with the left-hand end of the lower face at the origin P; the lower face runs from P along the positive x-direction to Q and is 41.3 mm long. The upper face runs parallel to it 17.7 mm above, from S (directly above P) to R (directly above Q). The first arc, radius 12.9 mm and centred at C1, runs smoothly out of the lower face, leaving it at T1 which is 11.9 mm along from P. The second arc, radius 13.6 mm and centred at C2, runs smoothly into the upper face at T2. The two arcs meet each other at J, and the curve is smooth there too - there is no kink where one arc becomes the other. The S-curve runs left to right, climbing from the lower face to the upper one.", "answer_schema": {"points": ["C1", "C2", "J", "P", "Q", "R", "S", "T1", "T2"], "circles": ["C1", "C2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Ogee (S-curve) blend between two parallel faces\n\nOracle formalization for cadh_t4_cadh_ogee_blend_0023. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(59)\n\nP, Q, S, R, C1, C2, T1, T2, J = scene.add.points([\"P\", \"Q\", \"S\", \"R\", \"C1\", \"C2\", \"T1\", \"T2\", \"J\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the left-hand end of the lower face (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the left-hand end of the lower face (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: the lower face PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the part extends in the +x direction\")\nscene.constraint.eq(P.distance(Q), 41.3, description=\"the lower face is 41.3 long\")\nscene.constraint.eq(S.y - P.y, 17.7, description=\"the upper face is 17.7 above the lower one\")\nscene.constraint.eq(R.y - Q.y, 17.7, description=\"the upper face is parallel to the lower one\")\nscene.constraint.eq(S.x, P.x, description=\"the upper face starts directly above P\")\nscene.constraint.eq(R.x, Q.x, description=\"the upper face ends directly above Q\")\nlower = scene.add.line_segment(P, Q)\nupper = scene.add.line_segment(S, R)\narc1 = scene.add.circle(C1)\narc2 = scene.add.circle(C2)\nscene.constraint.eq(arc1.radius, 12.9, description=\"the first blend arc has radius 12.9\")\nscene.constraint.eq(arc2.radius, 13.6, description=\"the second blend arc has radius 13.6\")\nscene.relate.tangent_to_circle(lower, arc1, T1)\nscene.relate.tangent_to_circle(upper, arc2, T2)\nscene.relate.point_lies_on(J, arc1)\nscene.relate.point_lies_on(J, arc2)\nscene.relate.point_lies_on(J, scene.add.line_segment(C1, C2))\nscene.constraint.eq(T1.x, 11.9, description=\"the S-curve leaves the lower face 11.9 along from P\")\nscene.constraint.gt(C1.y, 0.05, description=\"the first arc curves up away from the lower face\")\nscene.constraint.gt(T2.x - T1.x, 0.05, description=\"the S-curve runs left to right, from the lower face up to the upper one\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [41.3, 0.0], "S": [0.0, 17.7], "R": [41.3, 17.7], "C1": [11.9, 12.9], "C2": [36.89619971115609, 4.1], "T1": [11.9, 0.0], "T2": [36.89619971115609, 17.7], "J": [24.06796136882693, 8.616226415094339]}, "circles": {"C1": 12.9, "C2": 13.6}}, "implicit_facts": ["'runs smoothly out of' a straight face means the arc is tangent to it, so its centre sits one radius away, square to the face", "'no kink where one arc becomes the other' means the two arcs are tangent to EACH OTHER at J", "two arcs tangent to each other with the curve bending opposite ways have centre distance equal to the SUM of the radii, with J on the line joining the centres", "that fixes how far along the second arc must sit - it is not a free choice"], "branch_facts": ["the first arc curves upward away from the lower face, and the curve runs left to right"]} {"unique_id": "cadh_t4_cadh_ogee_blend_0024", "problem_id": "cadh_t4_cadh_ogee_blend_0024", "domain": "PyGeoX-CAD", "template": "cadh_ogee_blend", "tier": 4, "nl_description": "A step in a pressed panel is blended with an S-curve made of two arcs. Set up coordinates with the left-hand end of the lower face at the origin P; the lower face runs from P along the positive x-direction to Q and is 35.3 mm long. The upper face runs parallel to it 12.9 mm above, from S (directly above P) to R (directly above Q). The first arc, radius 8.2 mm and centred at C1, runs smoothly out of the lower face, leaving it at T1 which is 11.2 mm along from P. The second arc, radius 7.1 mm and centred at C2, runs smoothly into the upper face at T2. The two arcs meet each other at J, and the curve is smooth there too - there is no kink where one arc becomes the other. The S-curve runs left to right, climbing from the lower face to the upper one.", "answer_schema": {"points": ["C1", "C2", "J", "P", "Q", "R", "S", "T1", "T2"], "circles": ["C1", "C2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Ogee (S-curve) blend between two parallel faces\n\nOracle formalization for cadh_t4_cadh_ogee_blend_0024. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(51)\n\nP, Q, S, R, C1, C2, T1, T2, J = scene.add.points([\"P\", \"Q\", \"S\", \"R\", \"C1\", \"C2\", \"T1\", \"T2\", \"J\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the left-hand end of the lower face (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the left-hand end of the lower face (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: the lower face PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the part extends in the +x direction\")\nscene.constraint.eq(P.distance(Q), 35.3, description=\"the lower face is 35.3 long\")\nscene.constraint.eq(S.y - P.y, 12.9, description=\"the upper face is 12.9 above the lower one\")\nscene.constraint.eq(R.y - Q.y, 12.9, description=\"the upper face is parallel to the lower one\")\nscene.constraint.eq(S.x, P.x, description=\"the upper face starts directly above P\")\nscene.constraint.eq(R.x, Q.x, description=\"the upper face ends directly above Q\")\nlower = scene.add.line_segment(P, Q)\nupper = scene.add.line_segment(S, R)\narc1 = scene.add.circle(C1)\narc2 = scene.add.circle(C2)\nscene.constraint.eq(arc1.radius, 8.2, description=\"the first blend arc has radius 8.2\")\nscene.constraint.eq(arc2.radius, 7.1, description=\"the second blend arc has radius 7.1\")\nscene.relate.tangent_to_circle(lower, arc1, T1)\nscene.relate.tangent_to_circle(upper, arc2, T2)\nscene.relate.point_lies_on(J, arc1)\nscene.relate.point_lies_on(J, arc2)\nscene.relate.point_lies_on(J, scene.add.line_segment(C1, C2))\nscene.constraint.eq(T1.x, 11.2, description=\"the S-curve leaves the lower face 11.2 along from P\")\nscene.constraint.gt(C1.y, 0.05, description=\"the first arc curves up away from the lower face\")\nscene.constraint.gt(T2.x - T1.x, 0.05, description=\"the S-curve runs left to right, from the lower face up to the upper one\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [35.3, 0.0], "S": [0.0, 12.9], "R": [35.3, 12.9], "C1": [11.2, 8.2], "C2": [26.310592311355634, 5.800000000000001], "T1": [11.2, 0.0], "T2": [26.310592311355634, 12.9], "J": [19.298487382556615, 6.913725490196079]}, "circles": {"C1": 8.2, "C2": 7.1}}, "implicit_facts": ["'runs smoothly out of' a straight face means the arc is tangent to it, so its centre sits one radius away, square to the face", "'no kink where one arc becomes the other' means the two arcs are tangent to EACH OTHER at J", "two arcs tangent to each other with the curve bending opposite ways have centre distance equal to the SUM of the radii, with J on the line joining the centres", "that fixes how far along the second arc must sit - it is not a free choice"], "branch_facts": ["the first arc curves upward away from the lower face, and the curve runs left to right"]} {"unique_id": "cadh_t4_cadh_ogee_blend_0025", "problem_id": "cadh_t4_cadh_ogee_blend_0025", "domain": "PyGeoX-CAD", "template": "cadh_ogee_blend", "tier": 4, "nl_description": "A step in a pressed panel is blended with an S-curve made of two arcs. Set up coordinates with the left-hand end of the lower face at the origin P; the lower face runs from P along the positive x-direction to Q and is 39.6 mm long. The upper face runs parallel to it 18.3 mm above, from S (directly above P) to R (directly above Q). The first arc, radius 13.1 mm and centred at C1, runs smoothly out of the lower face, leaving it at T1 which is 8.7 mm along from P. The second arc, radius 6.1 mm and centred at C2, runs smoothly into the upper face at T2. The two arcs meet each other at J, and the curve is smooth there too - there is no kink where one arc becomes the other. The S-curve runs left to right, climbing from the lower face to the upper one.", "answer_schema": {"points": ["C1", "C2", "J", "P", "Q", "R", "S", "T1", "T2"], "circles": ["C1", "C2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Ogee (S-curve) blend between two parallel faces\n\nOracle formalization for cadh_t4_cadh_ogee_blend_0025. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(56)\n\nP, Q, S, R, C1, C2, T1, T2, J = scene.add.points([\"P\", \"Q\", \"S\", \"R\", \"C1\", \"C2\", \"T1\", \"T2\", \"J\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the left-hand end of the lower face (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the left-hand end of the lower face (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: the lower face PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the part extends in the +x direction\")\nscene.constraint.eq(P.distance(Q), 39.6, description=\"the lower face is 39.6 long\")\nscene.constraint.eq(S.y - P.y, 18.3, description=\"the upper face is 18.3 above the lower one\")\nscene.constraint.eq(R.y - Q.y, 18.3, description=\"the upper face is parallel to the lower one\")\nscene.constraint.eq(S.x, P.x, description=\"the upper face starts directly above P\")\nscene.constraint.eq(R.x, Q.x, description=\"the upper face ends directly above Q\")\nlower = scene.add.line_segment(P, Q)\nupper = scene.add.line_segment(S, R)\narc1 = scene.add.circle(C1)\narc2 = scene.add.circle(C2)\nscene.constraint.eq(arc1.radius, 13.1, description=\"the first blend arc has radius 13.1\")\nscene.constraint.eq(arc2.radius, 6.1, description=\"the second blend arc has radius 6.1\")\nscene.relate.tangent_to_circle(lower, arc1, T1)\nscene.relate.tangent_to_circle(upper, arc2, T2)\nscene.relate.point_lies_on(J, arc1)\nscene.relate.point_lies_on(J, arc2)\nscene.relate.point_lies_on(J, scene.add.line_segment(C1, C2))\nscene.constraint.eq(T1.x, 8.7, description=\"the S-curve leaves the lower face 8.7 along from P\")\nscene.constraint.gt(C1.y, 0.05, description=\"the first arc curves up away from the lower face\")\nscene.constraint.gt(T2.x - T1.x, 0.05, description=\"the S-curve runs left to right, from the lower face up to the upper one\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [39.6, 0.0], "S": [0.0, 18.3], "R": [39.6, 18.3], "C1": [8.7, 13.1], "C2": [27.87889465010953, 12.200000000000001], "T1": [8.7, 0.0], "T2": [27.87889465010953, 18.3], "J": [21.785599995647647, 12.4859375]}, "circles": {"C1": 13.1, "C2": 6.1}}, "implicit_facts": ["'runs smoothly out of' a straight face means the arc is tangent to it, so its centre sits one radius away, square to the face", "'no kink where one arc becomes the other' means the two arcs are tangent to EACH OTHER at J", "two arcs tangent to each other with the curve bending opposite ways have centre distance equal to the SUM of the radii, with J on the line joining the centres", "that fixes how far along the second arc must sit - it is not a free choice"], "branch_facts": ["the first arc curves upward away from the lower face, and the curve runs left to right"]} {"unique_id": "cadh_t4_cadh_pitch_from_chord_0071", "problem_id": "cadh_t4_cadh_pitch_from_chord_0071", "domain": "PyGeoX-CAD", "template": "cadh_pitch_from_chord", "tier": 4, "nl_description": "A flange is drilled with 6 bolt holes spaced evenly around a bolt circle, but the drawing gives the hole spacing rather than the bolt circle itself. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; the hole centres are numbered going anticlockwise. Measured in a straight line from one hole centre across to the next one round, the spacing is 11.5 mm. Give the centres of the first 4 holes, and the radius of the bolt circle they sit on.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle recovered from the hole spacing\n\nOracle formalization for cadh_t4_cadh_pitch_from_chord_0071. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(20)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(H1.distance(H2), 11.5, description=\"the straight distance from one hole centre to the next is 11.5\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 60, description=\"the 6 holes are evenly spaced, so H1 and H2 are adjacent\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 60, description=\"the 6 holes are evenly spaced, so H2 and H3 are adjacent\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 60, description=\"the 6 holes are evenly spaced, so H3 and H4 are adjacent\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [11.500000000000002, 0.0], "H2": [5.750000000000002, 9.959292143521045], "H3": [-5.749999999999998, 9.959292143521047], "H4": [-11.500000000000002, 1.4083438190194564e-15]}, "circles": {"O": 11.500000000000002}}, "implicit_facts": ["the bolt circle size is NOT given - it must be recovered from the hole-to-hole spacing", "evenly spaced means adjacent hole centres subtend 60 degrees at the centre", "the straight distance between adjacent holes is a CHORD of the bolt circle, not an arc", "so the radius is the spacing divided by twice the sine of half that angle"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cadh_t4_cadh_pitch_from_chord_0072", "problem_id": "cadh_t4_cadh_pitch_from_chord_0072", "domain": "PyGeoX-CAD", "template": "cadh_pitch_from_chord", "tier": 4, "nl_description": "A flange is drilled with 6 bolt holes spaced evenly around a bolt circle, but the drawing gives the hole spacing rather than the bolt circle itself. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; the hole centres are numbered going anticlockwise. Measured in a straight line from one hole centre across to the next one round, the spacing is 12.3 mm. Give the centres of the first 4 holes, and the radius of the bolt circle they sit on.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle recovered from the hole spacing\n\nOracle formalization for cadh_t4_cadh_pitch_from_chord_0072. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(21)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(H1.distance(H2), 12.3, description=\"the straight distance from one hole centre to the next is 12.3\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 60, description=\"the 6 holes are evenly spaced, so H1 and H2 are adjacent\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 60, description=\"the 6 holes are evenly spaced, so H2 and H3 are adjacent\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 60, description=\"the 6 holes are evenly spaced, so H3 and H4 are adjacent\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [12.300000000000002, 0.0], "H2": [6.150000000000003, 10.652112466548596], "H3": [-6.149999999999999, 10.652112466548598], "H4": [-12.300000000000002, 1.5063155629512448e-15]}, "circles": {"O": 12.300000000000002}}, "implicit_facts": ["the bolt circle size is NOT given - it must be recovered from the hole-to-hole spacing", "evenly spaced means adjacent hole centres subtend 60 degrees at the centre", "the straight distance between adjacent holes is a CHORD of the bolt circle, not an arc", "so the radius is the spacing divided by twice the sine of half that angle"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cadh_t4_cadh_pitch_from_chord_0073", "problem_id": "cadh_t4_cadh_pitch_from_chord_0073", "domain": "PyGeoX-CAD", "template": "cadh_pitch_from_chord", "tier": 4, "nl_description": "A flange is drilled with 6 bolt holes spaced evenly around a bolt circle, but the drawing gives the hole spacing rather than the bolt circle itself. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; the hole centres are numbered going anticlockwise. Measured in a straight line from one hole centre across to the next one round, the spacing is 9.6 mm. Give the centres of the first 4 holes, and the radius of the bolt circle they sit on.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle recovered from the hole spacing\n\nOracle formalization for cadh_t4_cadh_pitch_from_chord_0073. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(17)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(H1.distance(H2), 9.6, description=\"the straight distance from one hole centre to the next is 9.6\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 60, description=\"the 6 holes are evenly spaced, so H1 and H2 are adjacent\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 60, description=\"the 6 holes are evenly spaced, so H2 and H3 are adjacent\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 60, description=\"the 6 holes are evenly spaced, so H3 and H4 are adjacent\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [9.600000000000001, 0.0], "H2": [4.800000000000002, 8.313843876330612], "H3": [-4.799999999999999, 8.313843876330612], "H4": [-9.600000000000001, 1.1756609271814592e-15]}, "circles": {"O": 9.600000000000001}}, "implicit_facts": ["the bolt circle size is NOT given - it must be recovered from the hole-to-hole spacing", "evenly spaced means adjacent hole centres subtend 60 degrees at the centre", "the straight distance between adjacent holes is a CHORD of the bolt circle, not an arc", "so the radius is the spacing divided by twice the sine of half that angle"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cadh_t4_cadh_pitch_from_chord_0074", "problem_id": "cadh_t4_cadh_pitch_from_chord_0074", "domain": "PyGeoX-CAD", "template": "cadh_pitch_from_chord", "tier": 4, "nl_description": "A flange is drilled with 5 bolt holes spaced evenly around a bolt circle, but the drawing gives the hole spacing rather than the bolt circle itself. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; the hole centres are numbered going anticlockwise. Measured in a straight line from one hole centre across to the next one round, the spacing is 15.3 mm. Give the centres of the first 4 holes, and the radius of the bolt circle they sit on.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle recovered from the hole spacing\n\nOracle formalization for cadh_t4_cadh_pitch_from_chord_0074. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(22)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(H1.distance(H2), 15.3, description=\"the straight distance from one hole centre to the next is 15.3\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 72, description=\"the 5 holes are evenly spaced, so H1 and H2 are adjacent\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 72, description=\"the 5 holes are evenly spaced, so H2 and H3 are adjacent\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 72, description=\"the 5 holes are evenly spaced, so H3 and H4 are adjacent\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [13.014957367786211, 0.0], "H2": [4.0218430077113725, 12.377960013936695], "H3": [-10.529321691604476, 7.650000000000001], "H4": [-10.529321691604478, -7.649999999999999]}, "circles": {"O": 13.014957367786211}}, "implicit_facts": ["the bolt circle size is NOT given - it must be recovered from the hole-to-hole spacing", "evenly spaced means adjacent hole centres subtend 72 degrees at the centre", "the straight distance between adjacent holes is a CHORD of the bolt circle, not an arc", "so the radius is the spacing divided by twice the sine of half that angle"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cadh_t4_cadh_pitch_from_chord_0075", "problem_id": "cadh_t4_cadh_pitch_from_chord_0075", "domain": "PyGeoX-CAD", "template": "cadh_pitch_from_chord", "tier": 4, "nl_description": "A flange is drilled with 6 bolt holes spaced evenly around a bolt circle, but the drawing gives the hole spacing rather than the bolt circle itself. Set up coordinates with the flange centre at the origin O and the first hole centre H1 on the positive x-axis; the hole centres are numbered going anticlockwise. Measured in a straight line from one hole centre across to the next one round, the spacing is 10.6 mm. Give the centres of the first 4 holes, and the radius of the bolt circle they sit on.", "answer_schema": {"points": ["H1", "H2", "H3", "H4", "O"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Bolt circle recovered from the hole spacing\n\nOracle formalization for cadh_t4_cadh_pitch_from_chord_0075. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(19)\n\nO, H1, H2, H3, H4 = scene.add.points([\"O\", \"H1\", \"H2\", \"H3\", \"H4\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the flange centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the flange centre (y)\")\nscene.constraint.eq(H1.y, 0, description=\"datum: the first hole centre H1 lies on the positive x-axis\")\nscene.constraint.gt(H1.x, 1.0, description=\"datum: H1 is on the POSITIVE x side of the centre\")\nbolt_circle = scene.add.circle(O)\nscene.constraint.eq(H1.distance(H2), 10.6, description=\"the straight distance from one hole centre to the next is 10.6\")\nscene.relate.point_lies_on(H1, bolt_circle)\nscene.relate.point_lies_on(H2, bolt_circle)\nscene.relate.point_lies_on(H3, bolt_circle)\nscene.relate.point_lies_on(H4, bolt_circle)\nscene.constraint.eq(scene.add.angle(H1, O, H2).value, 60, description=\"the 6 holes are evenly spaced, so H1 and H2 are adjacent\")\nscene.constraint.gt(cross(O, H1, H2), 0.05, description=\"the holes are numbered anticlockwise, so H2 follows H1\")\nscene.constraint.eq(scene.add.angle(H2, O, H3).value, 60, description=\"the 6 holes are evenly spaced, so H2 and H3 are adjacent\")\nscene.constraint.gt(cross(O, H2, H3), 0.05, description=\"the holes are numbered anticlockwise, so H3 follows H2\")\nscene.constraint.eq(scene.add.angle(H3, O, H4).value, 60, description=\"the 6 holes are evenly spaced, so H3 and H4 are adjacent\")\nscene.constraint.gt(cross(O, H3, H4), 0.05, description=\"the holes are numbered anticlockwise, so H4 follows H3\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "H1": [10.600000000000001, 0.0], "H2": [5.300000000000002, 9.179869280115051], "H3": [-5.299999999999998, 9.179869280115051], "H4": [-10.600000000000001, 1.2981256070961946e-15]}, "circles": {"O": 10.600000000000001}}, "implicit_facts": ["the bolt circle size is NOT given - it must be recovered from the hole-to-hole spacing", "evenly spaced means adjacent hole centres subtend 60 degrees at the centre", "the straight distance between adjacent holes is a CHORD of the bolt circle, not an arc", "so the radius is the spacing divided by twice the sine of half that angle"], "branch_facts": ["the holes are numbered anticlockwise, which fixes the sense of rotation"]} {"unique_id": "cadh_t4_cadh_slot_touch_hole_0081", "problem_id": "cadh_t4_cadh_slot_touch_hole_0081", "domain": "PyGeoX-CAD", "template": "cadh_slot_touch_hole", "tier": 4, "nl_description": "A cover plate measures 45.1 mm by 30.9 mm. Set up coordinates with the bottom-left corner at the origin P, the 45.1 mm edge running from P along the positive x-direction to Q, and the 30.9 mm edge running from P along the positive y-direction to S; R is the remaining corner. A hole 8.9 mm across is already drilled with its centre N at 35 mm from edge PS and 13.2 mm from edge PQ. A slot 8.7 mm wide with a rounded end is machined along the line 11.6 mm up from edge PQ, approaching the hole from the left. The slot is extended towards the hole until its rounded end just touches the hole - the two exactly meet, with no material left between them and no overlap. C is the centre of the slot's rounded end.", "answer_schema": {"points": ["C", "N", "P", "Q", "R", "S"], "circles": ["C"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Slot extended until it just touches a fixed hole\n\nOracle formalization for cadh_t4_cadh_slot_touch_hole_0081. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(64)\n\nP, Q, R, S, C, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 45.1, description=\"plate is 45.1 long along PQ\")\nscene.constraint.eq(P.distance(S), 30.9, description=\"plate is 30.9 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nslot_end = scene.add.circle(C)\nscene.constraint.eq(2 * slot_end.radius, 8.7, description=\"the slot is 8.7 wide, so its rounded end spans that width\")\nscene.constraint.eq(C.y, 11.6, description=\"the slot runs along the line 11.6 up from edge PQ\")\nscene.constraint.eq(N.x, 35, description=\"the fixed hole centre is 35 from edge PS\")\nscene.constraint.eq(N.y, 13.2, description=\"the fixed hole centre is 13.2 from edge PQ\")\nscene.constraint.eq(C.distance(N), slot_end.radius + 4.45, description=\"the slot is extended until its rounded end just touches the hole, with no material left between them\")\nscene.constraint.lt(C.x - N.x, -0.05, description=\"the slot approaches the hole from the left, so its end centre is to the left of the hole\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [45.1, 0.0], "R": [45.1, 30.9], "S": [0.0, 30.9], "C": [26.346676938886425, 11.6], "N": [35.0, 13.2]}, "circles": {"C": 4.35}}, "implicit_facts": ["the slot end radius is half the stated slot width, and the hole radius is half its across-size", "'just touches, no material left and no overlap' means the two circles are externally tangent", "externally tangent circles have their centre distance equal to the SUM of the two radii", "the slot centreline height and that centre distance together fix how far along the slot end sits"], "branch_facts": ["the slot comes in from the left, so its end centre is on the left of the hole, not past it"]} {"unique_id": "cadh_t4_cadh_slot_touch_hole_0082", "problem_id": "cadh_t4_cadh_slot_touch_hole_0082", "domain": "PyGeoX-CAD", "template": "cadh_slot_touch_hole", "tier": 4, "nl_description": "A cover plate measures 44.2 mm by 30.9 mm. Set up coordinates with the bottom-left corner at the origin P, the 44.2 mm edge running from P along the positive x-direction to Q, and the 30.9 mm edge running from P along the positive y-direction to S; R is the remaining corner. A hole 6.2 mm across is already drilled with its centre N at 31.6 mm from edge PS and 10.5 mm from edge PQ. A slot 8.7 mm wide with a rounded end is machined along the line 14.1 mm up from edge PQ, approaching the hole from the left. The slot is extended towards the hole until its rounded end just touches the hole - the two exactly meet, with no material left between them and no overlap. C is the centre of the slot's rounded end.", "answer_schema": {"points": ["C", "N", "P", "Q", "R", "S"], "circles": ["C"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Slot extended until it just touches a fixed hole\n\nOracle formalization for cadh_t4_cadh_slot_touch_hole_0082. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(62)\n\nP, Q, R, S, C, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 44.2, description=\"plate is 44.2 long along PQ\")\nscene.constraint.eq(P.distance(S), 30.9, description=\"plate is 30.9 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nslot_end = scene.add.circle(C)\nscene.constraint.eq(2 * slot_end.radius, 8.7, description=\"the slot is 8.7 wide, so its rounded end spans that width\")\nscene.constraint.eq(C.y, 14.1, description=\"the slot runs along the line 14.1 up from edge PQ\")\nscene.constraint.eq(N.x, 31.6, description=\"the fixed hole centre is 31.6 from edge PS\")\nscene.constraint.eq(N.y, 10.5, description=\"the fixed hole centre is 10.5 from edge PQ\")\nscene.constraint.eq(C.distance(N), slot_end.radius + 3.1, description=\"the slot is extended until its rounded end just touches the hole, with no material left between them\")\nscene.constraint.lt(C.x - N.x, -0.05, description=\"the slot approaches the hole from the left, so its end centre is to the left of the hole\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [44.2, 0.0], "R": [44.2, 30.9], "S": [0.0, 30.9], "C": [25.07753880808786, 14.1], "N": [31.6, 10.5]}, "circles": {"C": 4.35}}, "implicit_facts": ["the slot end radius is half the stated slot width, and the hole radius is half its across-size", "'just touches, no material left and no overlap' means the two circles are externally tangent", "externally tangent circles have their centre distance equal to the SUM of the two radii", "the slot centreline height and that centre distance together fix how far along the slot end sits"], "branch_facts": ["the slot comes in from the left, so its end centre is on the left of the hole, not past it"]} {"unique_id": "cadh_t4_cadh_slot_touch_hole_0083", "problem_id": "cadh_t4_cadh_slot_touch_hole_0083", "domain": "PyGeoX-CAD", "template": "cadh_slot_touch_hole", "tier": 4, "nl_description": "A cover plate measures 47.3 mm by 31 mm. Set up coordinates with the bottom-left corner at the origin P, the 47.3 mm edge running from P along the positive x-direction to Q, and the 31 mm edge running from P along the positive y-direction to S; R is the remaining corner. A hole 7.8 mm across is already drilled with its centre N at 35.3 mm from edge PS and 17.7 mm from edge PQ. A slot 9.7 mm wide with a rounded end is machined along the line 12.5 mm up from edge PQ, approaching the hole from the left. The slot is extended towards the hole until its rounded end just touches the hole - the two exactly meet, with no material left between them and no overlap. C is the centre of the slot's rounded end.", "answer_schema": {"points": ["C", "N", "P", "Q", "R", "S"], "circles": ["C"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Slot extended until it just touches a fixed hole\n\nOracle formalization for cadh_t4_cadh_slot_touch_hole_0083. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(66)\n\nP, Q, R, S, C, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 47.3, description=\"plate is 47.3 long along PQ\")\nscene.constraint.eq(P.distance(S), 31, description=\"plate is 31 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nslot_end = scene.add.circle(C)\nscene.constraint.eq(2 * slot_end.radius, 9.7, description=\"the slot is 9.7 wide, so its rounded end spans that width\")\nscene.constraint.eq(C.y, 12.5, description=\"the slot runs along the line 12.5 up from edge PQ\")\nscene.constraint.eq(N.x, 35.3, description=\"the fixed hole centre is 35.3 from edge PS\")\nscene.constraint.eq(N.y, 17.7, description=\"the fixed hole centre is 17.7 from edge PQ\")\nscene.constraint.eq(C.distance(N), slot_end.radius + 3.9, description=\"the slot is extended until its rounded end just touches the hole, with no material left between them\")\nscene.constraint.lt(C.x - N.x, -0.05, description=\"the slot approaches the hole from the left, so its end centre is to the left of the hole\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [47.3, 0.0], "R": [47.3, 31.0], "S": [0.0, 31.0], "C": [28.262777536556058, 12.5], "N": [35.3, 17.7]}, "circles": {"C": 4.85}}, "implicit_facts": ["the slot end radius is half the stated slot width, and the hole radius is half its across-size", "'just touches, no material left and no overlap' means the two circles are externally tangent", "externally tangent circles have their centre distance equal to the SUM of the two radii", "the slot centreline height and that centre distance together fix how far along the slot end sits"], "branch_facts": ["the slot comes in from the left, so its end centre is on the left of the hole, not past it"]} {"unique_id": "cadh_t4_cadh_slot_touch_hole_0084", "problem_id": "cadh_t4_cadh_slot_touch_hole_0084", "domain": "PyGeoX-CAD", "template": "cadh_slot_touch_hole", "tier": 4, "nl_description": "A cover plate measures 38.9 mm by 24.3 mm. Set up coordinates with the bottom-left corner at the origin P, the 38.9 mm edge running from P along the positive x-direction to Q, and the 24.3 mm edge running from P along the positive y-direction to S; R is the remaining corner. A hole 7.9 mm across is already drilled with its centre N at 31.1 mm from edge PS and 13.2 mm from edge PQ. A slot 7.3 mm wide with a rounded end is machined along the line 11.7 mm up from edge PQ, approaching the hole from the left. The slot is extended towards the hole until its rounded end just touches the hole - the two exactly meet, with no material left between them and no overlap. C is the centre of the slot's rounded end.", "answer_schema": {"points": ["C", "N", "P", "Q", "R", "S"], "circles": ["C"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Slot extended until it just touches a fixed hole\n\nOracle formalization for cadh_t4_cadh_slot_touch_hole_0084. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(56)\n\nP, Q, R, S, C, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 38.9, description=\"plate is 38.9 long along PQ\")\nscene.constraint.eq(P.distance(S), 24.3, description=\"plate is 24.3 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nslot_end = scene.add.circle(C)\nscene.constraint.eq(2 * slot_end.radius, 7.3, description=\"the slot is 7.3 wide, so its rounded end spans that width\")\nscene.constraint.eq(C.y, 11.7, description=\"the slot runs along the line 11.7 up from edge PQ\")\nscene.constraint.eq(N.x, 31.1, description=\"the fixed hole centre is 31.1 from edge PS\")\nscene.constraint.eq(N.y, 13.2, description=\"the fixed hole centre is 13.2 from edge PQ\")\nscene.constraint.eq(C.distance(N), slot_end.radius + 3.95, description=\"the slot is extended until its rounded end just touches the hole, with no material left between them\")\nscene.constraint.lt(C.x - N.x, -0.05, description=\"the slot approaches the hole from the left, so its end centre is to the left of the hole\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [38.9, 0.0], "R": [38.9, 24.3], "S": [0.0, 24.3], "C": [23.64949666129865, 11.7], "N": [31.1, 13.2]}, "circles": {"C": 3.65}}, "implicit_facts": ["the slot end radius is half the stated slot width, and the hole radius is half its across-size", "'just touches, no material left and no overlap' means the two circles are externally tangent", "externally tangent circles have their centre distance equal to the SUM of the two radii", "the slot centreline height and that centre distance together fix how far along the slot end sits"], "branch_facts": ["the slot comes in from the left, so its end centre is on the left of the hole, not past it"]} {"unique_id": "cadh_t4_cadh_slot_touch_hole_0085", "problem_id": "cadh_t4_cadh_slot_touch_hole_0085", "domain": "PyGeoX-CAD", "template": "cadh_slot_touch_hole", "tier": 4, "nl_description": "A cover plate measures 43.5 mm by 27.6 mm. Set up coordinates with the bottom-left corner at the origin P, the 43.5 mm edge running from P along the positive x-direction to Q, and the 27.6 mm edge running from P along the positive y-direction to S; R is the remaining corner. A hole 8.7 mm across is already drilled with its centre N at 34.7 mm from edge PS and 14.2 mm from edge PQ. A slot 7.9 mm wide with a rounded end is machined along the line 15.1 mm up from edge PQ, approaching the hole from the left. The slot is extended towards the hole until its rounded end just touches the hole - the two exactly meet, with no material left between them and no overlap. C is the centre of the slot's rounded end.", "answer_schema": {"points": ["C", "N", "P", "Q", "R", "S"], "circles": ["C"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Slot extended until it just touches a fixed hole\n\nOracle formalization for cadh_t4_cadh_slot_touch_hole_0085. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(62)\n\nP, Q, R, S, C, N = scene.add.points([\"P\", \"Q\", \"R\", \"S\", \"C\", \"N\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the datum corner of the plate (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the datum corner of the plate (y)\")\nscene.constraint.eq(Q.y, 0, description=\"datum: edge PQ runs along the x-axis\")\nscene.constraint.gt(Q.x, 1.0, description=\"datum: the plate lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 43.5, description=\"plate is 43.5 long along PQ\")\nscene.constraint.eq(P.distance(S), 27.6, description=\"plate is 27.6 wide along PS\")\nscene.relate.perpendicular(scene.add.line_segment(P, Q), scene.add.line_segment(P, S))\nscene.constraint.gt(S.y, 1.0, description=\"datum: the plate lies in the +y direction from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"plate corner R closes the rectangle (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"plate corner R closes the rectangle (y)\")\nslot_end = scene.add.circle(C)\nscene.constraint.eq(2 * slot_end.radius, 7.9, description=\"the slot is 7.9 wide, so its rounded end spans that width\")\nscene.constraint.eq(C.y, 15.1, description=\"the slot runs along the line 15.1 up from edge PQ\")\nscene.constraint.eq(N.x, 34.7, description=\"the fixed hole centre is 34.7 from edge PS\")\nscene.constraint.eq(N.y, 14.2, description=\"the fixed hole centre is 14.2 from edge PQ\")\nscene.constraint.eq(C.distance(N), slot_end.radius + 4.35, description=\"the slot is extended until its rounded end just touches the hole, with no material left between them\")\nscene.constraint.lt(C.x - N.x, -0.05, description=\"the slot approaches the hole from the left, so its end centre is to the left of the hole\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [43.5, 0.0], "R": [43.5, 27.6], "S": [0.0, 27.6], "C": [26.448939462105493, 15.1], "N": [34.7, 14.2]}, "circles": {"C": 3.95}}, "implicit_facts": ["the slot end radius is half the stated slot width, and the hole radius is half its across-size", "'just touches, no material left and no overlap' means the two circles are externally tangent", "externally tangent circles have their centre distance equal to the SUM of the two radii", "the slot centreline height and that centre distance together fix how far along the slot end sits"], "branch_facts": ["the slot comes in from the left, so its end centre is on the left of the hole, not past it"]} {"unique_id": "cadh_t4_cadh_tangent_from_corner_0091", "problem_id": "cadh_t4_cadh_tangent_from_corner_0091", "domain": "PyGeoX-CAD", "template": "cadh_tangent_from_corner", "tier": 4, "nl_description": "A bore is to be positioned relative to a corner of a casting by scribing tangent lines. Set up coordinates with the centre of the bore at the origin O, and the corner E on the positive x-axis from it. The bore is 14.6 mm across. Two straight lines are scribed from the corner E so that each one just grazes the bore, touching it at a single point - T1 above the line OE and T2 below it. Measured from the corner along either scribed line to the point where it meets the bore, the distance is 17.8 mm. Work out how far the corner has to be from the bore centre.", "answer_schema": {"points": ["E", "O", "T1", "T2"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Corner standoff from a scribed tangent length\n\nOracle formalization for cadh_t4_cadh_tangent_from_corner_0091. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(30)\n\nO, E, T1, T2 = scene.add.points([\"O\", \"E\", \"T1\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the centre of the bore (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the centre of the bore (y)\")\nscene.constraint.eq(E.y, 0, description=\"datum: the corner E lies on the positive x-axis from O\")\nscene.constraint.gt(E.x, 1.0, description=\"datum: E is on the POSITIVE x side of the bore centre\")\nbore = scene.add.circle(O)\nscene.constraint.eq(2 * bore.radius, 14.6, description=\"the bore is 14.6 across\")\nscene.relate.point_lies_on(T1, bore)\nscene.relate.point_lies_on(T2, bore)\nscene.relate.perpendicular(scene.add.line_segment(O, T1), scene.add.line_segment(E, T1))\nscene.relate.perpendicular(scene.add.line_segment(O, T2), scene.add.line_segment(E, T2))\nscene.constraint.eq(E.distance(T1), 17.8, description=\"the straight scribed line from the corner to where it meets the bore measures 17.8\")\nscene.constraint.gt(T1.y, 0.05, description=\"T1 is the upper of the two touch points\")\nscene.constraint.lt(T2.y, -0.05, description=\"T2 is the lower of the two touch points\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [19.23876295399473, 0.0], "T1": [2.7699286137799666, 6.754072510312795], "T2": [2.7699286137799666, -6.754072510312795]}, "circles": {"O": 7.3}}, "implicit_facts": ["the standoff distance is NOT given - it must be recovered from the scribed length", "a line that 'just grazes' the bore is tangent to it, so the radius to the touch point is square to that line", "that makes O, T1 and E a right-angled triangle with the right angle at T1", "so the centre-to-corner distance is the square root of (scribed length squared plus radius squared)"], "branch_facts": ["T1 is the upper touch point and T2 the lower one"]} {"unique_id": "cadh_t4_cadh_tangent_from_corner_0092", "problem_id": "cadh_t4_cadh_tangent_from_corner_0092", "domain": "PyGeoX-CAD", "template": "cadh_tangent_from_corner", "tier": 4, "nl_description": "A bore is to be positioned relative to a corner of a casting by scribing tangent lines. Set up coordinates with the centre of the bore at the origin O, and the corner E on the positive x-axis from it. The bore is 16.1 mm across. Two straight lines are scribed from the corner E so that each one just grazes the bore, touching it at a single point - T1 above the line OE and T2 below it. Measured from the corner along either scribed line to the point where it meets the bore, the distance is 15.3 mm. Work out how far the corner has to be from the bore centre.", "answer_schema": {"points": ["E", "O", "T1", "T2"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Corner standoff from a scribed tangent length\n\nOracle formalization for cadh_t4_cadh_tangent_from_corner_0092. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(27)\n\nO, E, T1, T2 = scene.add.points([\"O\", \"E\", \"T1\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the centre of the bore (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the centre of the bore (y)\")\nscene.constraint.eq(E.y, 0, description=\"datum: the corner E lies on the positive x-axis from O\")\nscene.constraint.gt(E.x, 1.0, description=\"datum: E is on the POSITIVE x side of the bore centre\")\nbore = scene.add.circle(O)\nscene.constraint.eq(2 * bore.radius, 16.1, description=\"the bore is 16.1 across\")\nscene.relate.point_lies_on(T1, bore)\nscene.relate.point_lies_on(T2, bore)\nscene.relate.perpendicular(scene.add.line_segment(O, T1), scene.add.line_segment(E, T1))\nscene.relate.perpendicular(scene.add.line_segment(O, T2), scene.add.line_segment(E, T2))\nscene.constraint.eq(E.distance(T1), 15.3, description=\"the straight scribed line from the corner to where it meets the bore measures 15.3\")\nscene.constraint.gt(T1.y, 0.05, description=\"T1 is the upper of the two touch points\")\nscene.constraint.lt(T2.y, -0.05, description=\"T2 is the lower of the two touch points\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [17.288507743585043, 0.0], "T1": [3.7482992147801295, 7.124096644240495], "T2": [3.7482992147801295, -7.124096644240495]}, "circles": {"O": 8.05}}, "implicit_facts": ["the standoff distance is NOT given - it must be recovered from the scribed length", "a line that 'just grazes' the bore is tangent to it, so the radius to the touch point is square to that line", "that makes O, T1 and E a right-angled triangle with the right angle at T1", "so the centre-to-corner distance is the square root of (scribed length squared plus radius squared)"], "branch_facts": ["T1 is the upper touch point and T2 the lower one"]} {"unique_id": "cadh_t4_cadh_tangent_from_corner_0093", "problem_id": "cadh_t4_cadh_tangent_from_corner_0093", "domain": "PyGeoX-CAD", "template": "cadh_tangent_from_corner", "tier": 4, "nl_description": "A bore is to be positioned relative to a corner of a casting by scribing tangent lines. Set up coordinates with the centre of the bore at the origin O, and the corner E on the positive x-axis from it. The bore is 20.1 mm across. Two straight lines are scribed from the corner E so that each one just grazes the bore, touching it at a single point - T1 above the line OE and T2 below it. Measured from the corner along either scribed line to the point where it meets the bore, the distance is 25 mm. Work out how far the corner has to be from the bore centre.", "answer_schema": {"points": ["E", "O", "T1", "T2"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Corner standoff from a scribed tangent length\n\nOracle formalization for cadh_t4_cadh_tangent_from_corner_0093. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(40)\n\nO, E, T1, T2 = scene.add.points([\"O\", \"E\", \"T1\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the centre of the bore (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the centre of the bore (y)\")\nscene.constraint.eq(E.y, 0, description=\"datum: the corner E lies on the positive x-axis from O\")\nscene.constraint.gt(E.x, 1.0, description=\"datum: E is on the POSITIVE x side of the bore centre\")\nbore = scene.add.circle(O)\nscene.constraint.eq(2 * bore.radius, 20.1, description=\"the bore is 20.1 across\")\nscene.relate.point_lies_on(T1, bore)\nscene.relate.point_lies_on(T2, bore)\nscene.relate.perpendicular(scene.add.line_segment(O, T1), scene.add.line_segment(E, T1))\nscene.relate.perpendicular(scene.add.line_segment(O, T2), scene.add.line_segment(E, T2))\nscene.constraint.eq(E.distance(T1), 25, description=\"the straight scribed line from the corner to where it meets the bore measures 25\")\nscene.constraint.gt(T1.y, 0.05, description=\"T1 is the upper of the two touch points\")\nscene.constraint.lt(T2.y, -0.05, description=\"T2 is the lower of the two touch points\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [26.944433562426212, 0.0], "T1": [3.7485479056738145, 9.324746034014463], "T2": [3.7485479056738145, -9.324746034014463]}, "circles": {"O": 10.05}}, "implicit_facts": ["the standoff distance is NOT given - it must be recovered from the scribed length", "a line that 'just grazes' the bore is tangent to it, so the radius to the touch point is square to that line", "that makes O, T1 and E a right-angled triangle with the right angle at T1", "so the centre-to-corner distance is the square root of (scribed length squared plus radius squared)"], "branch_facts": ["T1 is the upper touch point and T2 the lower one"]} {"unique_id": "cadh_t4_cadh_tangent_from_corner_0094", "problem_id": "cadh_t4_cadh_tangent_from_corner_0094", "domain": "PyGeoX-CAD", "template": "cadh_tangent_from_corner", "tier": 4, "nl_description": "A bore is to be positioned relative to a corner of a casting by scribing tangent lines. Set up coordinates with the centre of the bore at the origin O, and the corner E on the positive x-axis from it. The bore is 21.1 mm across. Two straight lines are scribed from the corner E so that each one just grazes the bore, touching it at a single point - T1 above the line OE and T2 below it. Measured from the corner along either scribed line to the point where it meets the bore, the distance is 15.5 mm. Work out how far the corner has to be from the bore centre.", "answer_schema": {"points": ["E", "O", "T1", "T2"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Corner standoff from a scribed tangent length\n\nOracle formalization for cadh_t4_cadh_tangent_from_corner_0094. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(29)\n\nO, E, T1, T2 = scene.add.points([\"O\", \"E\", \"T1\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the centre of the bore (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the centre of the bore (y)\")\nscene.constraint.eq(E.y, 0, description=\"datum: the corner E lies on the positive x-axis from O\")\nscene.constraint.gt(E.x, 1.0, description=\"datum: E is on the POSITIVE x side of the bore centre\")\nbore = scene.add.circle(O)\nscene.constraint.eq(2 * bore.radius, 21.1, description=\"the bore is 21.1 across\")\nscene.relate.point_lies_on(T1, bore)\nscene.relate.point_lies_on(T2, bore)\nscene.relate.perpendicular(scene.add.line_segment(O, T1), scene.add.line_segment(E, T1))\nscene.relate.perpendicular(scene.add.line_segment(O, T2), scene.add.line_segment(E, T2))\nscene.constraint.eq(E.distance(T1), 15.5, description=\"the straight scribed line from the corner to where it meets the bore measures 15.5\")\nscene.constraint.gt(T1.y, 0.05, description=\"T1 is the upper of the two touch points\")\nscene.constraint.lt(T2.y, -0.05, description=\"T2 is the lower of the two touch points\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [18.74973333143701, 0.0], "T1": [5.936217760141851, 8.721457372720254], "T2": [5.936217760141851, -8.721457372720254]}, "circles": {"O": 10.55}}, "implicit_facts": ["the standoff distance is NOT given - it must be recovered from the scribed length", "a line that 'just grazes' the bore is tangent to it, so the radius to the touch point is square to that line", "that makes O, T1 and E a right-angled triangle with the right angle at T1", "so the centre-to-corner distance is the square root of (scribed length squared plus radius squared)"], "branch_facts": ["T1 is the upper touch point and T2 the lower one"]} {"unique_id": "cadh_t4_cadh_tangent_from_corner_0095", "problem_id": "cadh_t4_cadh_tangent_from_corner_0095", "domain": "PyGeoX-CAD", "template": "cadh_tangent_from_corner", "tier": 4, "nl_description": "A bore is to be positioned relative to a corner of a casting by scribing tangent lines. Set up coordinates with the centre of the bore at the origin O, and the corner E on the positive x-axis from it. The bore is 14.1 mm across. Two straight lines are scribed from the corner E so that each one just grazes the bore, touching it at a single point - T1 above the line OE and T2 below it. Measured from the corner along either scribed line to the point where it meets the bore, the distance is 16.3 mm. Work out how far the corner has to be from the bore centre.", "answer_schema": {"points": ["E", "O", "T1", "T2"], "circles": ["O"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Corner standoff from a scribed tangent length\n\nOracle formalization for cadh_t4_cadh_tangent_from_corner_0095. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(28)\n\nO, E, T1, T2 = scene.add.points([\"O\", \"E\", \"T1\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the centre of the bore (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the centre of the bore (y)\")\nscene.constraint.eq(E.y, 0, description=\"datum: the corner E lies on the positive x-axis from O\")\nscene.constraint.gt(E.x, 1.0, description=\"datum: E is on the POSITIVE x side of the bore centre\")\nbore = scene.add.circle(O)\nscene.constraint.eq(2 * bore.radius, 14.1, description=\"the bore is 14.1 across\")\nscene.relate.point_lies_on(T1, bore)\nscene.relate.point_lies_on(T2, bore)\nscene.relate.perpendicular(scene.add.line_segment(O, T1), scene.add.line_segment(E, T1))\nscene.relate.perpendicular(scene.add.line_segment(O, T2), scene.add.line_segment(E, T2))\nscene.constraint.eq(E.distance(T1), 16.3, description=\"the straight scribed line from the corner to where it meets the bore measures 16.3\")\nscene.constraint.gt(T1.y, 0.05, description=\"T1 is the upper of the two touch points\")\nscene.constraint.lt(T2.y, -0.05, description=\"T2 is the lower of the two touch points\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [17.759293341797132, 0.0], "T1": [2.7986755465671256, 6.470696653765127], "T2": [2.7986755465671256, -6.470696653765127]}, "circles": {"O": 7.05}}, "implicit_facts": ["the standoff distance is NOT given - it must be recovered from the scribed length", "a line that 'just grazes' the bore is tangent to it, so the radius to the touch point is square to that line", "that makes O, T1 and E a right-angled triangle with the right angle at T1", "so the centre-to-corner distance is the square root of (scribed length squared plus radius squared)"], "branch_facts": ["T1 is the upper touch point and T2 the lower one"]} {"unique_id": "cadh_t4_cadh_tangent_length_inverse_0051", "problem_id": "cadh_t4_cadh_tangent_length_inverse_0051", "domain": "PyGeoX-CAD", "template": "cadh_tangent_length_inverse", "tier": 4, "nl_description": "Two pulleys are to be set on a bracket and the centre distance has to be worked out from the belt. Set up coordinates with the centre of the large pulley at the origin C and the centre of the small pulley at D on the positive x-axis. The large pulley is 22.6 mm across and the small one is 9.3 mm across. A straight belt run passes over the tops of both, touching the large one at T and the small one at V, and it does not cross over between them. The straight run of belt between those two touch points measures 35 mm. Work out how far apart the two centres must be.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Centre distance from a stated belt run\n\nOracle formalization for cadh_t4_cadh_tangent_length_inverse_0051. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(52)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small pulley lies in the +x direction\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 22.6, description=\"the large pulley is 22.6 across\")\nscene.constraint.eq(2 * pulley_D.radius, 9.3, description=\"the small pulley is 9.3 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.eq(T.distance(V), 35, description=\"the straight run of belt between the two touch points measures 35\")\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run shown passes over the top of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [35.62614910427452, 0.0], "T": [2.109265297802953, 11.10139630422607], "V": [36.494121107352726, 4.568273700411613]}, "circles": {"C": 11.3, "D": 4.65}}, "implicit_facts": ["the centre distance is NOT given - it has to be recovered from the belt length", "each pulley radius is half the stated across-size", "for a belt that does not cross over, the straight run, the centre distance and the difference of the radii form a right-angled triangle", "so the centre distance is the square root of (belt run squared plus the radius difference squared)"], "branch_facts": ["'does not cross over' selects the external tangent, not the crossing one; 'over the tops' picks the upper one"]} {"unique_id": "cadh_t4_cadh_tangent_length_inverse_0052", "problem_id": "cadh_t4_cadh_tangent_length_inverse_0052", "domain": "PyGeoX-CAD", "template": "cadh_tangent_length_inverse", "tier": 4, "nl_description": "Two pulleys are to be set on a bracket and the centre distance has to be worked out from the belt. Set up coordinates with the centre of the large pulley at the origin C and the centre of the small pulley at D on the positive x-axis. The large pulley is 19.7 mm across and the small one is 7.3 mm across. A straight belt run passes over the tops of both, touching the large one at T and the small one at V, and it does not cross over between them. The straight run of belt between those two touch points measures 29.6 mm. Work out how far apart the two centres must be.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Centre distance from a stated belt run\n\nOracle formalization for cadh_t4_cadh_tangent_length_inverse_0052. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(45)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small pulley lies in the +x direction\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 19.7, description=\"the large pulley is 19.7 across\")\nscene.constraint.eq(2 * pulley_D.radius, 7.3, description=\"the small pulley is 7.3 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.eq(T.distance(V), 29.6, description=\"the straight run of belt between the two touch points measures 29.6\")\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run shown passes over the top of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [30.242354405700624, 0.0], "T": [2.019353360546837, 9.640783785836511], "V": [30.990642706816963, 3.5724731795231746]}, "circles": {"C": 9.85, "D": 3.65}}, "implicit_facts": ["the centre distance is NOT given - it has to be recovered from the belt length", "each pulley radius is half the stated across-size", "for a belt that does not cross over, the straight run, the centre distance and the difference of the radii form a right-angled triangle", "so the centre distance is the square root of (belt run squared plus the radius difference squared)"], "branch_facts": ["'does not cross over' selects the external tangent, not the crossing one; 'over the tops' picks the upper one"]} {"unique_id": "cadh_t4_cadh_tangent_length_inverse_0053", "problem_id": "cadh_t4_cadh_tangent_length_inverse_0053", "domain": "PyGeoX-CAD", "template": "cadh_tangent_length_inverse", "tier": 4, "nl_description": "Two pulleys are to be set on a bracket and the centre distance has to be worked out from the belt. Set up coordinates with the centre of the large pulley at the origin C and the centre of the small pulley at D on the positive x-axis. The large pulley is 25.5 mm across and the small one is 9.5 mm across. A straight belt run passes over the tops of both, touching the large one at T and the small one at V, and it does not cross over between them. The straight run of belt between those two touch points measures 31.4 mm. Work out how far apart the two centres must be.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Centre distance from a stated belt run\n\nOracle formalization for cadh_t4_cadh_tangent_length_inverse_0053. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(49)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small pulley lies in the +x direction\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 25.5, description=\"the large pulley is 25.5 across\")\nscene.constraint.eq(2 * pulley_D.radius, 9.5, description=\"the small pulley is 9.5 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.eq(T.distance(V), 31.4, description=\"the straight run of belt between the two touch points measures 31.4\")\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run shown passes over the top of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [32.403086272761115, 0.0], "T": [3.14784829881294, 12.355304572840788], "V": [33.57581407035809, 4.602956605568137]}, "circles": {"C": 12.75, "D": 4.75}}, "implicit_facts": ["the centre distance is NOT given - it has to be recovered from the belt length", "each pulley radius is half the stated across-size", "for a belt that does not cross over, the straight run, the centre distance and the difference of the radii form a right-angled triangle", "so the centre distance is the square root of (belt run squared plus the radius difference squared)"], "branch_facts": ["'does not cross over' selects the external tangent, not the crossing one; 'over the tops' picks the upper one"]} {"unique_id": "cadh_t4_cadh_tangent_length_inverse_0054", "problem_id": "cadh_t4_cadh_tangent_length_inverse_0054", "domain": "PyGeoX-CAD", "template": "cadh_tangent_length_inverse", "tier": 4, "nl_description": "Two pulleys are to be set on a bracket and the centre distance has to be worked out from the belt. Set up coordinates with the centre of the large pulley at the origin C and the centre of the small pulley at D on the positive x-axis. The large pulley is 20.4 mm across and the small one is 10.7 mm across. A straight belt run passes over the tops of both, touching the large one at T and the small one at V, and it does not cross over between them. The straight run of belt between those two touch points measures 35.1 mm. Work out how far apart the two centres must be.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Centre distance from a stated belt run\n\nOracle formalization for cadh_t4_cadh_tangent_length_inverse_0054. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(52)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small pulley lies in the +x direction\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 20.4, description=\"the large pulley is 20.4 across\")\nscene.constraint.eq(2 * pulley_D.radius, 10.7, description=\"the small pulley is 10.7 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.eq(T.distance(V), 35.1, description=\"the straight run of belt between the two touch points measures 35.1\")\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run shown passes over the top of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [35.433494041654995, 0.0], "T": [1.3961366593383064, 10.103999328407127], "V": [36.16578140709225, 5.299646706566483]}, "circles": {"C": 10.2, "D": 5.35}}, "implicit_facts": ["the centre distance is NOT given - it has to be recovered from the belt length", "each pulley radius is half the stated across-size", "for a belt that does not cross over, the straight run, the centre distance and the difference of the radii form a right-angled triangle", "so the centre distance is the square root of (belt run squared plus the radius difference squared)"], "branch_facts": ["'does not cross over' selects the external tangent, not the crossing one; 'over the tops' picks the upper one"]} {"unique_id": "cadh_t4_cadh_tangent_length_inverse_0055", "problem_id": "cadh_t4_cadh_tangent_length_inverse_0055", "domain": "PyGeoX-CAD", "template": "cadh_tangent_length_inverse", "tier": 4, "nl_description": "Two pulleys are to be set on a bracket and the centre distance has to be worked out from the belt. Set up coordinates with the centre of the large pulley at the origin C and the centre of the small pulley at D on the positive x-axis. The large pulley is 18.7 mm across and the small one is 8.3 mm across. A straight belt run passes over the tops of both, touching the large one at T and the small one at V, and it does not cross over between them. The straight run of belt between those two touch points measures 29.5 mm. Work out how far apart the two centres must be.", "answer_schema": {"points": ["C", "D", "T", "V"], "circles": ["C", "D"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Centre distance from a stated belt run\n\nOracle formalization for cadh_t4_cadh_tangent_length_inverse_0055. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(45)\n\nC, D, T, V = scene.add.points([\"C\", \"D\", \"T\", \"V\"])\n\nscene.constraint.eq(C.x, 0, description=\"datum: C is the origin, the centre of the large pulley (x)\")\nscene.constraint.eq(C.y, 0, description=\"datum: C is the origin, the centre of the large pulley (y)\")\nscene.constraint.eq(D.y, 0, description=\"datum: the line of centres CD runs along the x-axis\")\nscene.constraint.gt(D.x, 1.0, description=\"datum: the small pulley lies in the +x direction\")\npulley_C = scene.add.circle(C)\npulley_D = scene.add.circle(D)\nscene.constraint.eq(2 * pulley_C.radius, 18.7, description=\"the large pulley is 18.7 across\")\nscene.constraint.eq(2 * pulley_D.radius, 8.3, description=\"the small pulley is 8.3 across\")\nbelt = scene.add.line(T, V)\nscene.relate.tangent_to_circle(belt, pulley_C, T)\nscene.relate.tangent_to_circle(belt, pulley_D, V)\nscene.constraint.eq(T.distance(V), 29.5, description=\"the straight run of belt between the two touch points measures 29.5\")\nscene.constraint.gt(cross(C, D, T), 0.05, description=\"the belt run shown passes over the top of both pulleys\")\nscene.constraint.gt(dot(C, T, D, V), 0.05, description=\"the belt does not cross over between the pulleys\")\n", "possible_solution": {"points": {"C": [0.0, 0.0], "D": [29.954799281584243, 0.0], "T": [1.6231121945754727, 9.208040334610855], "V": [30.675218063133784, 4.086991164559898]}, "circles": {"C": 9.35, "D": 4.15}}, "implicit_facts": ["the centre distance is NOT given - it has to be recovered from the belt length", "each pulley radius is half the stated across-size", "for a belt that does not cross over, the straight run, the centre distance and the difference of the radii form a right-angled triangle", "so the centre distance is the square root of (belt run squared plus the radius difference squared)"], "branch_facts": ["'does not cross over' selects the external tangent, not the crossing one; 'over the tops' picks the upper one"]} {"unique_id": "cadh_t4_cadh_two_bars_vee_0011", "problem_id": "cadh_t4_cadh_two_bars_vee_0011", "domain": "PyGeoX-CAD", "template": "cadh_two_bars_vee", "tier": 4, "nl_description": "Two round bars of different sizes are nested in a vee block, one resting on top of the other, both sitting centrally in the vee. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 38 mm long and the included angle of the vee is 55 degrees. The lower bar is 11.5 mm across; its centre is F1 and it touches the right-hand face at T1. The upper bar rests on top of the lower one, touching it, and is itself pushed out until it also touches the vee faces; its centre is F2 and it touches the right-hand face at T2. Work out how big the upper bar has to be.", "answer_schema": {"points": ["A", "B", "F1", "F2", "T1", "T2", "Z"], "circles": ["F1", "F2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Two bars nested in a vee block\n\nOracle formalization for cadh_t4_cadh_two_bars_vee_0011. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(49)\n\nZ, A, B, F1, F2, T1, T2 = scene.add.points([\"Z\", \"A\", \"B\", \"F1\", \"F2\", \"T1\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 38, description=\"each vee face is 38 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 55, description=\"the included angle of the vee is 55 degrees\")\nface_r = scene.add.line_segment(Z, A)\nbar_low = scene.add.circle(F1)\nbar_up = scene.add.circle(F2)\nscene.constraint.eq(2 * bar_low.radius, 11.5, description=\"the lower bar is 11.5 across\")\nscene.relate.tangent_to_circle(face_r, bar_low, T1)\nscene.relate.tangent_to_circle(face_r, bar_up, T2)\nscene.constraint.eq(F1.x, Z.x, description=\"the lower bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F2.x, Z.x, description=\"the upper bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F1.distance(F2), bar_low.radius + bar_up.radius, description=\"the two bars touch each other\")\nscene.constraint.gt(F1.y, 0.05, description=\"the lower bar sits inside the vee, above the apex\")\nscene.constraint.gt(F2.y - F1.y, 0.05, description=\"the upper bar rests on top of the lower one\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [17.546447302931288, 33.706411660772424], "B": [-17.546447302931288, 33.706411660772424], "F1": [0.0, 12.452663278651151], "F2": [0.0, 33.81814469267603], "T1": [5.100312290774775, 9.797608752549706], "T2": [13.851101179533243, 26.607717804752596]}, "circles": {"F1": 5.75, "F2": 15.61548141402488}}, "implicit_facts": ["the upper bar's size is NOT given - it follows from touching the faces AND the lower bar", "a bar sitting centrally and touching both faces has its centre on the vee axis at height radius/sin(half the included angle)", "two bars touching each other have their centre distance equal to the SUM of their radii", "combining those gives the upper radius as the lower radius times (1+sin a)/(1-sin a)"], "branch_facts": ["the upper bar rests ON TOP of the lower one, not below it"]} {"unique_id": "cadh_t4_cadh_two_bars_vee_0012", "problem_id": "cadh_t4_cadh_two_bars_vee_0012", "domain": "PyGeoX-CAD", "template": "cadh_two_bars_vee", "tier": 4, "nl_description": "Two round bars of different sizes are nested in a vee block, one resting on top of the other, both sitting centrally in the vee. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 39.1 mm long and the included angle of the vee is 50 degrees. The lower bar is 11.8 mm across; its centre is F1 and it touches the right-hand face at T1. The upper bar rests on top of the lower one, touching it, and is itself pushed out until it also touches the vee faces; its centre is F2 and it touches the right-hand face at T2. Work out how big the upper bar has to be.", "answer_schema": {"points": ["A", "B", "F1", "F2", "T1", "T2", "Z"], "circles": ["F1", "F2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Two bars nested in a vee block\n\nOracle formalization for cadh_t4_cadh_two_bars_vee_0012. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(51)\n\nZ, A, B, F1, F2, T1, T2 = scene.add.points([\"Z\", \"A\", \"B\", \"F1\", \"F2\", \"T1\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 39.1, description=\"each vee face is 39.1 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 50, description=\"the included angle of the vee is 50 degrees\")\nface_r = scene.add.line_segment(Z, A)\nbar_low = scene.add.circle(F1)\nbar_up = scene.add.circle(F2)\nscene.constraint.eq(2 * bar_low.radius, 11.8, description=\"the lower bar is 11.8 across\")\nscene.relate.tangent_to_circle(face_r, bar_low, T1)\nscene.relate.tangent_to_circle(face_r, bar_up, T2)\nscene.constraint.eq(F1.x, Z.x, description=\"the lower bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F2.x, Z.x, description=\"the upper bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F1.distance(F2), bar_low.radius + bar_up.radius, description=\"the two bars touch each other\")\nscene.constraint.gt(F1.y, 0.05, description=\"the lower bar sits inside the vee, above the apex\")\nscene.constraint.gt(F2.y - F1.y, 0.05, description=\"the upper bar rests on top of the lower one\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [16.52437403406135, 35.43663447313301], "B": [-16.52437403406135, 35.43663447313301], "F1": [0.0, 13.960589340599743], "F2": [0.0, 34.3976749255627], "T1": [5.347215943516235, 11.467141596329615], "T2": [13.175073866470155, 28.254037084869875]}, "circles": {"F1": 5.9, "F2": 14.53708558496295}}, "implicit_facts": ["the upper bar's size is NOT given - it follows from touching the faces AND the lower bar", "a bar sitting centrally and touching both faces has its centre on the vee axis at height radius/sin(half the included angle)", "two bars touching each other have their centre distance equal to the SUM of their radii", "combining those gives the upper radius as the lower radius times (1+sin a)/(1-sin a)"], "branch_facts": ["the upper bar rests ON TOP of the lower one, not below it"]} {"unique_id": "cadh_t4_cadh_two_bars_vee_0013", "problem_id": "cadh_t4_cadh_two_bars_vee_0013", "domain": "PyGeoX-CAD", "template": "cadh_two_bars_vee", "tier": 4, "nl_description": "Two round bars of different sizes are nested in a vee block, one resting on top of the other, both sitting centrally in the vee. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 29 mm long and the included angle of the vee is 55 degrees. The lower bar is 7.3 mm across; its centre is F1 and it touches the right-hand face at T1. The upper bar rests on top of the lower one, touching it, and is itself pushed out until it also touches the vee faces; its centre is F2 and it touches the right-hand face at T2. Work out how big the upper bar has to be.", "answer_schema": {"points": ["A", "B", "F1", "F2", "T1", "T2", "Z"], "circles": ["F1", "F2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Two bars nested in a vee block\n\nOracle formalization for cadh_t4_cadh_two_bars_vee_0013. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(38)\n\nZ, A, B, F1, F2, T1, T2 = scene.add.points([\"Z\", \"A\", \"B\", \"F1\", \"F2\", \"T1\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 29, description=\"each vee face is 29 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 55, description=\"the included angle of the vee is 55 degrees\")\nface_r = scene.add.line_segment(Z, A)\nbar_low = scene.add.circle(F1)\nbar_up = scene.add.circle(F2)\nscene.constraint.eq(2 * bar_low.radius, 7.3, description=\"the lower bar is 7.3 across\")\nscene.relate.tangent_to_circle(face_r, bar_low, T1)\nscene.relate.tangent_to_circle(face_r, bar_up, T2)\nscene.constraint.eq(F1.x, Z.x, description=\"the lower bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F2.x, Z.x, description=\"the upper bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F1.distance(F2), bar_low.radius + bar_up.radius, description=\"the two bars touch each other\")\nscene.constraint.gt(F1.y, 0.05, description=\"the lower bar sits inside the vee, above the apex\")\nscene.constraint.gt(F2.y - F1.y, 0.05, description=\"the upper bar rests on top of the lower one\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [13.390709783815984, 25.72331416216843], "B": [-13.390709783815984, 25.72331416216843], "F1": [0.0, 7.904734081230731], "F2": [0.0, 21.467170109263915], "T1": [3.237589541100509, 6.219351642922857], "T2": [8.792438140051537, 16.890116519538605]}, "circles": {"F1": 3.65, "F2": 9.912436028033184}}, "implicit_facts": ["the upper bar's size is NOT given - it follows from touching the faces AND the lower bar", "a bar sitting centrally and touching both faces has its centre on the vee axis at height radius/sin(half the included angle)", "two bars touching each other have their centre distance equal to the SUM of their radii", "combining those gives the upper radius as the lower radius times (1+sin a)/(1-sin a)"], "branch_facts": ["the upper bar rests ON TOP of the lower one, not below it"]} {"unique_id": "cadh_t4_cadh_two_bars_vee_0014", "problem_id": "cadh_t4_cadh_two_bars_vee_0014", "domain": "PyGeoX-CAD", "template": "cadh_two_bars_vee", "tier": 4, "nl_description": "Two round bars of different sizes are nested in a vee block, one resting on top of the other, both sitting centrally in the vee. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 36.1 mm long and the included angle of the vee is 70 degrees. The lower bar is 8.4 mm across; its centre is F1 and it touches the right-hand face at T1. The upper bar rests on top of the lower one, touching it, and is itself pushed out until it also touches the vee faces; its centre is F2 and it touches the right-hand face at T2. Work out how big the upper bar has to be.", "answer_schema": {"points": ["A", "B", "F1", "F2", "T1", "T2", "Z"], "circles": ["F1", "F2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Two bars nested in a vee block\n\nOracle formalization for cadh_t4_cadh_two_bars_vee_0014. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(43)\n\nZ, A, B, F1, F2, T1, T2 = scene.add.points([\"Z\", \"A\", \"B\", \"F1\", \"F2\", \"T1\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 36.1, description=\"each vee face is 36.1 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 70, description=\"the included angle of the vee is 70 degrees\")\nface_r = scene.add.line_segment(Z, A)\nbar_low = scene.add.circle(F1)\nbar_up = scene.add.circle(F2)\nscene.constraint.eq(2 * bar_low.radius, 8.4, description=\"the lower bar is 8.4 across\")\nscene.relate.tangent_to_circle(face_r, bar_low, T1)\nscene.relate.tangent_to_circle(face_r, bar_up, T2)\nscene.constraint.eq(F1.x, Z.x, description=\"the lower bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F2.x, Z.x, description=\"the upper bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F1.distance(F2), bar_low.radius + bar_up.radius, description=\"the two bars touch each other\")\nscene.constraint.gt(F1.y, 0.05, description=\"the lower bar sits inside the vee, above the apex\")\nscene.constraint.gt(F2.y - F1.y, 0.05, description=\"the upper bar rests on top of the lower one\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [20.70610935227276, 29.571388798832604], "B": [-20.70610935227276, 29.571388798832604], "F1": [0.0, 7.322476541608612], "F2": [0.0, 27.021200336607805], "T1": [3.4404385860137654, 4.913455508934218], "T2": [12.695811280544028, 18.13149757428301]}, "circles": {"F1": 4.2, "F2": 15.49872379499919}}, "implicit_facts": ["the upper bar's size is NOT given - it follows from touching the faces AND the lower bar", "a bar sitting centrally and touching both faces has its centre on the vee axis at height radius/sin(half the included angle)", "two bars touching each other have their centre distance equal to the SUM of their radii", "combining those gives the upper radius as the lower radius times (1+sin a)/(1-sin a)"], "branch_facts": ["the upper bar rests ON TOP of the lower one, not below it"]} {"unique_id": "cadh_t4_cadh_two_bars_vee_0015", "problem_id": "cadh_t4_cadh_two_bars_vee_0015", "domain": "PyGeoX-CAD", "template": "cadh_two_bars_vee", "tier": 4, "nl_description": "Two round bars of different sizes are nested in a vee block, one resting on top of the other, both sitting centrally in the vee. Set up coordinates with the apex of the vee at the origin Z, the vee opening upward and symmetric about the positive y-direction; the right-hand face runs from Z to A and the left-hand face runs from Z to B. Each face is 36.9 mm long and the included angle of the vee is 60 degrees. The lower bar is 7.3 mm across; its centre is F1 and it touches the right-hand face at T1. The upper bar rests on top of the lower one, touching it, and is itself pushed out until it also touches the vee faces; its centre is F2 and it touches the right-hand face at T2. Work out how big the upper bar has to be.", "answer_schema": {"points": ["A", "B", "F1", "F2", "T1", "T2", "Z"], "circles": ["F1", "F2"], "units": "millimetres", "note": "Report every point as [x, y] in the datum frame described in the text, and every circle as {centre_label: radius}. Give at least 4 decimal places: a stated fact counts as satisfied only when its residual is below 5e-3 mm."}, "full_code": "\"\"\"Two bars nested in a vee block\n\nOracle formalization for cadh_t4_cadh_two_bars_vee_0015. Every constraint below corresponds to one\nstatement in the problem text; the datum constraints fix the coordinate frame\nthe text describes, and the inequality constraints encode the discrete choices\nthe text makes explicit (which side, which branch).\n\"\"\"\nfrom pygeox import GeoScene\n\n\ndef cross(A, B, P):\n \"\"\"z-component of (B-A) x (P-A). Positive iff P lies to the LEFT of the\n directed line A->B. Used to state 'above', 'on the far side', 'anticlockwise'\n without any appeal to a picture.\"\"\"\n return (B.x - A.x) * (P.y - A.y) - (B.y - A.y) * (P.x - A.x)\n\n\ndef dot(A, B, C, D):\n \"\"\"(B-A) . (D-C). Positive iff the two directions point the same way.\"\"\"\n return (B.x - A.x) * (D.x - C.x) + (B.y - A.y) * (D.y - C.y)\n\n\nscene = GeoScene(47)\n\nZ, A, B, F1, F2, T1, T2 = scene.add.points([\"Z\", \"A\", \"B\", \"F1\", \"F2\", \"T1\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the apex of the vee (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the apex of the vee (y)\")\nscene.constraint.eq(A.x + B.x, 0, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.eq(A.y, B.y, description=\"datum: the vee is symmetric about the y-axis\")\nscene.constraint.gt(A.x, 1.0, description=\"datum: A is the right-hand face end, B the left-hand one\")\nscene.constraint.eq(Z.distance(A), 36.9, description=\"each vee face is 36.9 long\")\nscene.constraint.eq(scene.add.angle(A, Z, B).value, 60, description=\"the included angle of the vee is 60 degrees\")\nface_r = scene.add.line_segment(Z, A)\nbar_low = scene.add.circle(F1)\nbar_up = scene.add.circle(F2)\nscene.constraint.eq(2 * bar_low.radius, 7.3, description=\"the lower bar is 7.3 across\")\nscene.relate.tangent_to_circle(face_r, bar_low, T1)\nscene.relate.tangent_to_circle(face_r, bar_up, T2)\nscene.constraint.eq(F1.x, Z.x, description=\"the lower bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F2.x, Z.x, description=\"the upper bar sits centrally in the vee, on its axis of symmetry\")\nscene.constraint.eq(F1.distance(F2), bar_low.radius + bar_up.radius, description=\"the two bars touch each other\")\nscene.constraint.gt(F1.y, 0.05, description=\"the lower bar sits inside the vee, above the apex\")\nscene.constraint.gt(F2.y - F1.y, 0.05, description=\"the upper bar rests on top of the lower one\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "A": [18.449999999999996, 31.956337399645786], "B": [-18.449999999999996, 31.956337399645786], "F1": [0.0, 7.300000000000001], "F2": [0.0, 21.900000000000002], "T1": [3.160992723813201, 5.475000000000001], "T2": [9.482978171439603, 16.425000000000004]}, "circles": {"F1": 3.65, "F2": 10.95}}, "implicit_facts": ["the upper bar's size is NOT given - it follows from touching the faces AND the lower bar", "a bar sitting centrally and touching both faces has its centre on the vee axis at height radius/sin(half the included angle)", "two bars touching each other have their centre distance equal to the SUM of their radii", "combining those gives the upper radius as the lower radius times (1+sin a)/(1-sin a)"], "branch_facts": ["the upper bar rests ON TOP of the lower one, not below it"]}