{"unique_id": "mech_t1_mech_pantograph_0025", "problem_id": "mech_t1_mech_pantograph_0025", "domain": "PyGeoX-Mech", "template": "mech_pantograph", "tier": 1, "nl_description": "Four bars are pinned together so that they always form a parallelogram, with pins at P, Q, R and S taken in order round the loop: bar PQ stays parallel and equal to bar SR, and bar QR stays parallel and equal to bar PS. Pin P is fixed to the bench. Bar PQ is 18.8 mm long and bar QR is 11 mm long. Set up coordinates with P at the origin and with the mechanism opened out so that pin R sits 17 mm from P along the positive x-direction. In this position pin Q lies above the line PR and pin S below it.", "answer_schema": {"points": ["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": "\"\"\"Pantograph parallelogram set to a diagonal\n\nOracle formalization for mech_t1_mech_pantograph_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(27)\n\nP, Q, R, S = scene.add.points([\"P\", \"Q\", \"R\", \"S\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (y)\")\nscene.constraint.eq(R.y, P.y, description=\"datum: the diagonal PR is laid along the x-axis\")\nscene.constraint.gt(R.x, 1.0, description=\"datum: R lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 18.8, description=\"bar PQ is 18.8 long\")\nscene.constraint.eq(Q.distance(R), 11, description=\"bar QR is 11 long\")\nscene.constraint.eq(P.distance(R), 17, description=\"the mechanism is set so that R sits 17 from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"the four bars always form a parallelogram (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"the four bars always form a parallelogram (y)\")\nscene.constraint.gt(cross(P, R, Q), 0.05, description=\"pin Q lies above the diagonal PR and pin S below it\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [15.336470588235295, 10.873484717246527], "R": [17.0, 0.0], "S": [1.6635294117647046, -10.873484717246527]}, "circles": {}}, "implicit_facts": ["the parallelogram condition is the vector relation R = Q + S - P", "equivalently the two diagonals PR and QS bisect each other", "the opposite bars are equal, so |SR| = |PQ| and |PS| = |QR| without being stated again", "the diagonal is limited to between |PQ - QR| and PQ + QR, and the stated value lies inside that range"], "branch_facts": ["Q above the diagonal and S below it fixes which way the parallelogram folds"]} {"unique_id": "mech_t1_mech_pantograph_0026", "problem_id": "mech_t1_mech_pantograph_0026", "domain": "PyGeoX-Mech", "template": "mech_pantograph", "tier": 1, "nl_description": "Four bars are pinned together so that they always form a parallelogram, with pins at P, Q, R and S taken in order round the loop: bar PQ stays parallel and equal to bar SR, and bar QR stays parallel and equal to bar PS. Pin P is fixed to the bench. Bar PQ is 16.8 mm long and bar QR is 8.6 mm long. Set up coordinates with P at the origin and with the mechanism opened out so that pin R sits 21.9 mm from P along the positive x-direction. In this position pin Q lies above the line PR and pin S below it.", "answer_schema": {"points": ["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": "\"\"\"Pantograph parallelogram set to a diagonal\n\nOracle formalization for mech_t1_mech_pantograph_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(33)\n\nP, Q, R, S = scene.add.points([\"P\", \"Q\", \"R\", \"S\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (y)\")\nscene.constraint.eq(R.y, P.y, description=\"datum: the diagonal PR is laid along the x-axis\")\nscene.constraint.gt(R.x, 1.0, description=\"datum: R lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 16.8, description=\"bar PQ is 16.8 long\")\nscene.constraint.eq(Q.distance(R), 8.6, description=\"bar QR is 8.6 long\")\nscene.constraint.eq(P.distance(R), 21.9, description=\"the mechanism is set so that R sits 21.9 from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"the four bars always form a parallelogram (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"the four bars always form a parallelogram (y)\")\nscene.constraint.gt(cross(P, R, Q), 0.05, description=\"pin Q lies above the diagonal PR and pin S below it\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [15.705251141552512, 5.965323677786773], "R": [21.9, 0.0], "S": [6.194748858447486, -5.965323677786773]}, "circles": {}}, "implicit_facts": ["the parallelogram condition is the vector relation R = Q + S - P", "equivalently the two diagonals PR and QS bisect each other", "the opposite bars are equal, so |SR| = |PQ| and |PS| = |QR| without being stated again", "the diagonal is limited to between |PQ - QR| and PQ + QR, and the stated value lies inside that range"], "branch_facts": ["Q above the diagonal and S below it fixes which way the parallelogram folds"]} {"unique_id": "mech_t1_mech_pantograph_0027", "problem_id": "mech_t1_mech_pantograph_0027", "domain": "PyGeoX-Mech", "template": "mech_pantograph", "tier": 1, "nl_description": "Four bars are pinned together so that they always form a parallelogram, with pins at P, Q, R and S taken in order round the loop: bar PQ stays parallel and equal to bar SR, and bar QR stays parallel and equal to bar PS. Pin P is fixed to the bench. Bar PQ is 14.9 mm long and bar QR is 11.2 mm long. Set up coordinates with P at the origin and with the mechanism opened out so that pin R sits 11 mm from P along the positive x-direction. In this position pin Q lies above the line PR and pin S below it.", "answer_schema": {"points": ["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": "\"\"\"Pantograph parallelogram set to a diagonal\n\nOracle formalization for mech_t1_mech_pantograph_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(19)\n\nP, Q, R, S = scene.add.points([\"P\", \"Q\", \"R\", \"S\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (y)\")\nscene.constraint.eq(R.y, P.y, description=\"datum: the diagonal PR is laid along the x-axis\")\nscene.constraint.gt(R.x, 1.0, description=\"datum: R lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 14.9, description=\"bar PQ is 14.9 long\")\nscene.constraint.eq(Q.distance(R), 11.2, description=\"bar QR is 11.2 long\")\nscene.constraint.eq(P.distance(R), 11, description=\"the mechanism is set so that R sits 11 from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"the four bars always form a parallelogram (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"the four bars always form a parallelogram (y)\")\nscene.constraint.gt(cross(P, R, Q), 0.05, description=\"pin Q lies above the diagonal PR and pin S below it\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [9.889545454545457, 11.14481452077509], "R": [11.0, 0.0], "S": [1.1104545454545427, -11.14481452077509]}, "circles": {}}, "implicit_facts": ["the parallelogram condition is the vector relation R = Q + S - P", "equivalently the two diagonals PR and QS bisect each other", "the opposite bars are equal, so |SR| = |PQ| and |PS| = |QR| without being stated again", "the diagonal is limited to between |PQ - QR| and PQ + QR, and the stated value lies inside that range"], "branch_facts": ["Q above the diagonal and S below it fixes which way the parallelogram folds"]} {"unique_id": "mech_t1_mech_pantograph_0028", "problem_id": "mech_t1_mech_pantograph_0028", "domain": "PyGeoX-Mech", "template": "mech_pantograph", "tier": 1, "nl_description": "Four bars are pinned together so that they always form a parallelogram, with pins at P, Q, R and S taken in order round the loop: bar PQ stays parallel and equal to bar SR, and bar QR stays parallel and equal to bar PS. Pin P is fixed to the bench. Bar PQ is 15.4 mm long and bar QR is 12.9 mm long. Set up coordinates with P at the origin and with the mechanism opened out so that pin R sits 19.2 mm from P along the positive x-direction. In this position pin Q lies above the line PR and pin S below it.", "answer_schema": {"points": ["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": "\"\"\"Pantograph parallelogram set to a diagonal\n\nOracle formalization for mech_t1_mech_pantograph_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(30)\n\nP, Q, R, S = scene.add.points([\"P\", \"Q\", \"R\", \"S\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (y)\")\nscene.constraint.eq(R.y, P.y, description=\"datum: the diagonal PR is laid along the x-axis\")\nscene.constraint.gt(R.x, 1.0, description=\"datum: R lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 15.4, description=\"bar PQ is 15.4 long\")\nscene.constraint.eq(Q.distance(R), 12.9, description=\"bar QR is 12.9 long\")\nscene.constraint.eq(P.distance(R), 19.2, description=\"the mechanism is set so that R sits 19.2 from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"the four bars always form a parallelogram (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"the four bars always form a parallelogram (y)\")\nscene.constraint.gt(cross(P, R, Q), 0.05, description=\"pin Q lies above the diagonal PR and pin S below it\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [11.442447916666667, 10.30681258558487], "R": [19.2, 0.0], "S": [7.757552083333332, -10.30681258558487]}, "circles": {}}, "implicit_facts": ["the parallelogram condition is the vector relation R = Q + S - P", "equivalently the two diagonals PR and QS bisect each other", "the opposite bars are equal, so |SR| = |PQ| and |PS| = |QR| without being stated again", "the diagonal is limited to between |PQ - QR| and PQ + QR, and the stated value lies inside that range"], "branch_facts": ["Q above the diagonal and S below it fixes which way the parallelogram folds"]} {"unique_id": "mech_t1_mech_pantograph_0029", "problem_id": "mech_t1_mech_pantograph_0029", "domain": "PyGeoX-Mech", "template": "mech_pantograph", "tier": 1, "nl_description": "Four bars are pinned together so that they always form a parallelogram, with pins at P, Q, R and S taken in order round the loop: bar PQ stays parallel and equal to bar SR, and bar QR stays parallel and equal to bar PS. Pin P is fixed to the bench. Bar PQ is 16.5 mm long and bar QR is 9.1 mm long. Set up coordinates with P at the origin and with the mechanism opened out so that pin R sits 15.8 mm from P along the positive x-direction. In this position pin Q lies above the line PR and pin S below it.", "answer_schema": {"points": ["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": "\"\"\"Pantograph parallelogram set to a diagonal\n\nOracle formalization for mech_t1_mech_pantograph_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(26)\n\nP, Q, R, S = scene.add.points([\"P\", \"Q\", \"R\", \"S\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (y)\")\nscene.constraint.eq(R.y, P.y, description=\"datum: the diagonal PR is laid along the x-axis\")\nscene.constraint.gt(R.x, 1.0, description=\"datum: R lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 16.5, description=\"bar PQ is 16.5 long\")\nscene.constraint.eq(Q.distance(R), 9.1, description=\"bar QR is 9.1 long\")\nscene.constraint.eq(P.distance(R), 15.8, description=\"the mechanism is set so that R sits 15.8 from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"the four bars always form a parallelogram (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"the four bars always form a parallelogram (y)\")\nscene.constraint.gt(cross(P, R, Q), 0.05, description=\"pin Q lies above the diagonal PR and pin S below it\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [13.89493670886076, 8.898355682751376], "R": [15.8, 0.0], "S": [1.9050632911392409, -8.898355682751376]}, "circles": {}}, "implicit_facts": ["the parallelogram condition is the vector relation R = Q + S - P", "equivalently the two diagonals PR and QS bisect each other", "the opposite bars are equal, so |SR| = |PQ| and |PS| = |QR| without being stated again", "the diagonal is limited to between |PQ - QR| and PQ + QR, and the stated value lies inside that range"], "branch_facts": ["Q above the diagonal and S below it fixes which way the parallelogram folds"]} {"unique_id": "mech_t1_mech_pantograph_0030", "problem_id": "mech_t1_mech_pantograph_0030", "domain": "PyGeoX-Mech", "template": "mech_pantograph", "tier": 1, "nl_description": "Four bars are pinned together so that they always form a parallelogram, with pins at P, Q, R and S taken in order round the loop: bar PQ stays parallel and equal to bar SR, and bar QR stays parallel and equal to bar PS. Pin P is fixed to the bench. Bar PQ is 21.8 mm long and bar QR is 8.8 mm long. Set up coordinates with P at the origin and with the mechanism opened out so that pin R sits 22.8 mm from P along the positive x-direction. In this position pin Q lies above the line PR and pin S below it.", "answer_schema": {"points": ["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": "\"\"\"Pantograph parallelogram set to a diagonal\n\nOracle formalization for mech_t1_mech_pantograph_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(35)\n\nP, Q, R, S = scene.add.points([\"P\", \"Q\", \"R\", \"S\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (y)\")\nscene.constraint.eq(R.y, P.y, description=\"datum: the diagonal PR is laid along the x-axis\")\nscene.constraint.gt(R.x, 1.0, description=\"datum: R lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 21.8, description=\"bar PQ is 21.8 long\")\nscene.constraint.eq(Q.distance(R), 8.8, description=\"bar QR is 8.8 long\")\nscene.constraint.eq(P.distance(R), 22.8, description=\"the mechanism is set so that R sits 22.8 from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"the four bars always form a parallelogram (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"the four bars always form a parallelogram (y)\")\nscene.constraint.gt(cross(P, R, Q), 0.05, description=\"pin Q lies above the diagonal PR and pin S below it\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [20.123684210526317, 8.383157746041393], "R": [22.8, 0.0], "S": [2.6763157894736835, -8.383157746041393]}, "circles": {}}, "implicit_facts": ["the parallelogram condition is the vector relation R = Q + S - P", "equivalently the two diagonals PR and QS bisect each other", "the opposite bars are equal, so |SR| = |PQ| and |PS| = |QR| without being stated again", "the diagonal is limited to between |PQ - QR| and PQ + QR, and the stated value lies inside that range"], "branch_facts": ["Q above the diagonal and S below it fixes which way the parallelogram folds"]} {"unique_id": "mech_t1_mech_pantograph_0031", "problem_id": "mech_t1_mech_pantograph_0031", "domain": "PyGeoX-Mech", "template": "mech_pantograph", "tier": 1, "nl_description": "Four bars are pinned together so that they always form a parallelogram, with pins at P, Q, R and S taken in order round the loop: bar PQ stays parallel and equal to bar SR, and bar QR stays parallel and equal to bar PS. Pin P is fixed to the bench. Bar PQ is 17.9 mm long and bar QR is 8.1 mm long. Set up coordinates with P at the origin and with the mechanism opened out so that pin R sits 22.5 mm from P along the positive x-direction. In this position pin Q lies above the line PR and pin S below it.", "answer_schema": {"points": ["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": "\"\"\"Pantograph parallelogram set to a diagonal\n\nOracle formalization for mech_t1_mech_pantograph_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(34)\n\nP, Q, R, S = scene.add.points([\"P\", \"Q\", \"R\", \"S\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (y)\")\nscene.constraint.eq(R.y, P.y, description=\"datum: the diagonal PR is laid along the x-axis\")\nscene.constraint.gt(R.x, 1.0, description=\"datum: R lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 17.9, description=\"bar PQ is 17.9 long\")\nscene.constraint.eq(Q.distance(R), 8.1, description=\"bar QR is 8.1 long\")\nscene.constraint.eq(P.distance(R), 22.5, description=\"the mechanism is set so that R sits 22.5 from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"the four bars always form a parallelogram (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"the four bars always form a parallelogram (y)\")\nscene.constraint.gt(cross(P, R, Q), 0.05, description=\"pin Q lies above the diagonal PR and pin S below it\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [16.912222222222223, 5.864020762767879], "R": [22.5, 0.0], "S": [5.587777777777777, -5.864020762767879]}, "circles": {}}, "implicit_facts": ["the parallelogram condition is the vector relation R = Q + S - P", "equivalently the two diagonals PR and QS bisect each other", "the opposite bars are equal, so |SR| = |PQ| and |PS| = |QR| without being stated again", "the diagonal is limited to between |PQ - QR| and PQ + QR, and the stated value lies inside that range"], "branch_facts": ["Q above the diagonal and S below it fixes which way the parallelogram folds"]} {"unique_id": "mech_t1_mech_pantograph_0032", "problem_id": "mech_t1_mech_pantograph_0032", "domain": "PyGeoX-Mech", "template": "mech_pantograph", "tier": 1, "nl_description": "Four bars are pinned together so that they always form a parallelogram, with pins at P, Q, R and S taken in order round the loop: bar PQ stays parallel and equal to bar SR, and bar QR stays parallel and equal to bar PS. Pin P is fixed to the bench. Bar PQ is 21.5 mm long and bar QR is 9.1 mm long. Set up coordinates with P at the origin and with the mechanism opened out so that pin R sits 25.2 mm from P along the positive x-direction. In this position pin Q lies above the line PR and pin S below it.", "answer_schema": {"points": ["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": "\"\"\"Pantograph parallelogram set to a diagonal\n\nOracle formalization for mech_t1_mech_pantograph_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(38)\n\nP, Q, R, S = scene.add.points([\"P\", \"Q\", \"R\", \"S\"])\n\nscene.constraint.eq(P.x, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (x)\")\nscene.constraint.eq(P.y, 0, description=\"datum: P is the origin, the fixed pin of the pantograph (y)\")\nscene.constraint.eq(R.y, P.y, description=\"datum: the diagonal PR is laid along the x-axis\")\nscene.constraint.gt(R.x, 1.0, description=\"datum: R lies in the +x direction from P\")\nscene.constraint.eq(P.distance(Q), 21.5, description=\"bar PQ is 21.5 long\")\nscene.constraint.eq(Q.distance(R), 9.1, description=\"bar QR is 9.1 long\")\nscene.constraint.eq(P.distance(R), 25.2, description=\"the mechanism is set so that R sits 25.2 from P\")\nscene.constraint.eq(R.x, Q.x + S.x - P.x, description=\"the four bars always form a parallelogram (x)\")\nscene.constraint.eq(R.y, Q.y + S.y - P.y, description=\"the four bars always form a parallelogram (y)\")\nscene.constraint.gt(cross(P, R, Q), 0.05, description=\"pin Q lies above the diagonal PR and pin S below it\")\n", "possible_solution": {"points": {"P": [0.0, 0.0], "Q": [20.12857142857143, 7.55583299477284], "R": [25.2, 0.0], "S": [5.071428571428569, -7.55583299477284]}, "circles": {}}, "implicit_facts": ["the parallelogram condition is the vector relation R = Q + S - P", "equivalently the two diagonals PR and QS bisect each other", "the opposite bars are equal, so |SR| = |PQ| and |PS| = |QR| without being stated again", "the diagonal is limited to between |PQ - QR| and PQ + QR, and the stated value lies inside that range"], "branch_facts": ["Q above the diagonal and S below it fixes which way the parallelogram folds"]} {"unique_id": "mech_t1_mech_scissor_0033", "problem_id": "mech_t1_mech_scissor_0033", "domain": "PyGeoX-Mech", "template": "mech_scissor", "tier": 1, "nl_description": "Two identical straight bars, each 37.2 mm long, are crossed over each other and pinned together at a single point X that is exactly halfway along both of them. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart from each other. The upper ends are U1 on the first bar and U2 on the second. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 15.6 mm above the floor.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair at a stated pin height\n\nOracle formalization for mech_t1_mech_scissor_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\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 2.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.relate.is_midpoint(X, scene.add.line_segment(F1, U1))\nscene.relate.is_midpoint(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(U1), 37.2, description=\"the first bar is 37.2 long\")\nscene.constraint.eq(F2.distance(U2), 37.2, description=\"the second bar is 37.2 long\")\nscene.constraint.eq(X.y - F1.y, 15.6, description=\"the crossing pin sits 15.6 above the floor\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [20.258331619360963, 0.0], "U1": [20.258331619360963, 31.2], "U2": [0.0, 31.2], "X": [10.129165809680481, 15.6]}, "circles": {}}, "implicit_facts": ["'halfway along both' means X is the midpoint of each bar", "so each half-bar is half the stated bar length", "because X is a midpoint and sits h above the floor, each upper end is 2h above the floor", "each lower end is sqrt((half bar)^2 - h^2) horizontally away from X"], "branch_facts": ["the lower ends rest apart, which rules out the collapsed arrangement with both bars on top of each other"]} {"unique_id": "mech_t1_mech_scissor_0034", "problem_id": "mech_t1_mech_scissor_0034", "domain": "PyGeoX-Mech", "template": "mech_scissor", "tier": 1, "nl_description": "Two identical straight bars, each 34.2 mm long, are crossed over each other and pinned together at a single point X that is exactly halfway along both of them. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart from each other. The upper ends are U1 on the first bar and U2 on the second. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 12.7 mm above the floor.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair at a stated pin height\n\nOracle formalization for mech_t1_mech_scissor_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(38)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 2.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.relate.is_midpoint(X, scene.add.line_segment(F1, U1))\nscene.relate.is_midpoint(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(U1), 34.2, description=\"the first bar is 34.2 long\")\nscene.constraint.eq(F2.distance(U2), 34.2, description=\"the second bar is 34.2 long\")\nscene.constraint.eq(X.y - F1.y, 12.7, description=\"the crossing pin sits 12.7 above the floor\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [22.901528333279423, 0.0], "U1": [22.901528333279423, 25.4], "U2": [0.0, 25.4], "X": [11.450764166639711, 12.7]}, "circles": {}}, "implicit_facts": ["'halfway along both' means X is the midpoint of each bar", "so each half-bar is half the stated bar length", "because X is a midpoint and sits h above the floor, each upper end is 2h above the floor", "each lower end is sqrt((half bar)^2 - h^2) horizontally away from X"], "branch_facts": ["the lower ends rest apart, which rules out the collapsed arrangement with both bars on top of each other"]} {"unique_id": "mech_t1_mech_scissor_0035", "problem_id": "mech_t1_mech_scissor_0035", "domain": "PyGeoX-Mech", "template": "mech_scissor", "tier": 1, "nl_description": "Two identical straight bars, each 29 mm long, are crossed over each other and pinned together at a single point X that is exactly halfway along both of them. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart from each other. The upper ends are U1 on the first bar and U2 on the second. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 6.7 mm above the floor.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair at a stated pin height\n\nOracle formalization for mech_t1_mech_scissor_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(38)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 2.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.relate.is_midpoint(X, scene.add.line_segment(F1, U1))\nscene.relate.is_midpoint(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(U1), 29, description=\"the first bar is 29 long\")\nscene.constraint.eq(F2.distance(U2), 29, description=\"the second bar is 29 long\")\nscene.constraint.eq(X.y - F1.y, 6.7, description=\"the crossing pin sits 6.7 above the floor\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [25.718475849085614, 0.0], "U1": [25.718475849085614, 13.4], "U2": [0.0, 13.4], "X": [12.859237924542807, 6.7]}, "circles": {}}, "implicit_facts": ["'halfway along both' means X is the midpoint of each bar", "so each half-bar is half the stated bar length", "because X is a midpoint and sits h above the floor, each upper end is 2h above the floor", "each lower end is sqrt((half bar)^2 - h^2) horizontally away from X"], "branch_facts": ["the lower ends rest apart, which rules out the collapsed arrangement with both bars on top of each other"]} {"unique_id": "mech_t1_mech_scissor_0036", "problem_id": "mech_t1_mech_scissor_0036", "domain": "PyGeoX-Mech", "template": "mech_scissor", "tier": 1, "nl_description": "Two identical straight bars, each 32.6 mm long, are crossed over each other and pinned together at a single point X that is exactly halfway along both of them. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart from each other. The upper ends are U1 on the first bar and U2 on the second. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 7.2 mm above the floor.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair at a stated pin height\n\nOracle formalization for mech_t1_mech_scissor_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(43)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 2.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.relate.is_midpoint(X, scene.add.line_segment(F1, U1))\nscene.relate.is_midpoint(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(U1), 32.6, description=\"the first bar is 32.6 long\")\nscene.constraint.eq(F2.distance(U2), 32.6, description=\"the second bar is 32.6 long\")\nscene.constraint.eq(X.y - F1.y, 7.2, description=\"the crossing pin sits 7.2 above the floor\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [29.247222090311414, 0.0], "U1": [29.247222090311414, 14.4], "U2": [0.0, 14.4], "X": [14.623611045155707, 7.2]}, "circles": {}}, "implicit_facts": ["'halfway along both' means X is the midpoint of each bar", "so each half-bar is half the stated bar length", "because X is a midpoint and sits h above the floor, each upper end is 2h above the floor", "each lower end is sqrt((half bar)^2 - h^2) horizontally away from X"], "branch_facts": ["the lower ends rest apart, which rules out the collapsed arrangement with both bars on top of each other"]} {"unique_id": "mech_t1_mech_scissor_0037", "problem_id": "mech_t1_mech_scissor_0037", "domain": "PyGeoX-Mech", "template": "mech_scissor", "tier": 1, "nl_description": "Two identical straight bars, each 34.9 mm long, are crossed over each other and pinned together at a single point X that is exactly halfway along both of them. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart from each other. The upper ends are U1 on the first bar and U2 on the second. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 12.5 mm above the floor.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair at a stated pin height\n\nOracle formalization for mech_t1_mech_scissor_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(38)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 2.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.relate.is_midpoint(X, scene.add.line_segment(F1, U1))\nscene.relate.is_midpoint(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(U1), 34.9, description=\"the first bar is 34.9 long\")\nscene.constraint.eq(F2.distance(U2), 34.9, description=\"the second bar is 34.9 long\")\nscene.constraint.eq(X.y - F1.y, 12.5, description=\"the crossing pin sits 12.5 above the floor\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [24.351796648296816, 0.0], "U1": [24.351796648296816, 25.0], "U2": [0.0, 25.0], "X": [12.175898324148408, 12.5]}, "circles": {}}, "implicit_facts": ["'halfway along both' means X is the midpoint of each bar", "so each half-bar is half the stated bar length", "because X is a midpoint and sits h above the floor, each upper end is 2h above the floor", "each lower end is sqrt((half bar)^2 - h^2) horizontally away from X"], "branch_facts": ["the lower ends rest apart, which rules out the collapsed arrangement with both bars on top of each other"]} {"unique_id": "mech_t1_mech_scissor_0038", "problem_id": "mech_t1_mech_scissor_0038", "domain": "PyGeoX-Mech", "template": "mech_scissor", "tier": 1, "nl_description": "Two identical straight bars, each 39.3 mm long, are crossed over each other and pinned together at a single point X that is exactly halfway along both of them. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart from each other. The upper ends are U1 on the first bar and U2 on the second. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 13.3 mm above the floor.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair at a stated pin height\n\nOracle formalization for mech_t1_mech_scissor_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(43)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 2.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.relate.is_midpoint(X, scene.add.line_segment(F1, U1))\nscene.relate.is_midpoint(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(U1), 39.3, description=\"the first bar is 39.3 long\")\nscene.constraint.eq(F2.distance(U2), 39.3, description=\"the second bar is 39.3 long\")\nscene.constraint.eq(X.y - F1.y, 13.3, description=\"the crossing pin sits 13.3 above the floor\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [28.929742480706594, 0.0], "U1": [28.929742480706594, 26.6], "U2": [0.0, 26.6], "X": [14.464871240353297, 13.3]}, "circles": {}}, "implicit_facts": ["'halfway along both' means X is the midpoint of each bar", "so each half-bar is half the stated bar length", "because X is a midpoint and sits h above the floor, each upper end is 2h above the floor", "each lower end is sqrt((half bar)^2 - h^2) horizontally away from X"], "branch_facts": ["the lower ends rest apart, which rules out the collapsed arrangement with both bars on top of each other"]} {"unique_id": "mech_t1_mech_scissor_0039", "problem_id": "mech_t1_mech_scissor_0039", "domain": "PyGeoX-Mech", "template": "mech_scissor", "tier": 1, "nl_description": "Two identical straight bars, each 36.7 mm long, are crossed over each other and pinned together at a single point X that is exactly halfway along both of them. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart from each other. The upper ends are U1 on the first bar and U2 on the second. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 9.1 mm above the floor.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair at a stated pin height\n\nOracle formalization for mech_t1_mech_scissor_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(46)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 2.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.relate.is_midpoint(X, scene.add.line_segment(F1, U1))\nscene.relate.is_midpoint(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(U1), 36.7, description=\"the first bar is 36.7 long\")\nscene.constraint.eq(F2.distance(U2), 36.7, description=\"the second bar is 36.7 long\")\nscene.constraint.eq(X.y - F1.y, 9.1, description=\"the crossing pin sits 9.1 above the floor\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [31.869264189811474, 0.0], "U1": [31.869264189811474, 18.2], "U2": [0.0, 18.2], "X": [15.934632094905737, 9.1]}, "circles": {}}, "implicit_facts": ["'halfway along both' means X is the midpoint of each bar", "so each half-bar is half the stated bar length", "because X is a midpoint and sits h above the floor, each upper end is 2h above the floor", "each lower end is sqrt((half bar)^2 - h^2) horizontally away from X"], "branch_facts": ["the lower ends rest apart, which rules out the collapsed arrangement with both bars on top of each other"]} {"unique_id": "mech_t1_mech_scissor_0040", "problem_id": "mech_t1_mech_scissor_0040", "domain": "PyGeoX-Mech", "template": "mech_scissor", "tier": 1, "nl_description": "Two identical straight bars, each 36.7 mm long, are crossed over each other and pinned together at a single point X that is exactly halfway along both of them. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart from each other. The upper ends are U1 on the first bar and U2 on the second. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 11.2 mm above the floor.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair at a stated pin height\n\nOracle formalization for mech_t1_mech_scissor_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(43)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 2.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.relate.is_midpoint(X, scene.add.line_segment(F1, U1))\nscene.relate.is_midpoint(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(U1), 36.7, description=\"the first bar is 36.7 long\")\nscene.constraint.eq(F2.distance(U2), 36.7, description=\"the second bar is 36.7 long\")\nscene.constraint.eq(X.y - F1.y, 11.2, description=\"the crossing pin sits 11.2 above the floor\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [29.071119689478767, 0.0], "U1": [29.071119689478767, 22.4], "U2": [0.0, 22.4], "X": [14.535559844739383, 11.2]}, "circles": {}}, "implicit_facts": ["'halfway along both' means X is the midpoint of each bar", "so each half-bar is half the stated bar length", "because X is a midpoint and sits h above the floor, each upper end is 2h above the floor", "each lower end is sqrt((half bar)^2 - h^2) horizontally away from X"], "branch_facts": ["the lower ends rest apart, which rules out the collapsed arrangement with both bars on top of each other"]} {"unique_id": "mech_t1_mech_slider_crank_0009", "problem_id": "mech_t1_mech_slider_crank_0009", "domain": "PyGeoX-Mech", "template": "mech_slider_crank", "tier": 1, "nl_description": "A short bar 13.4 mm long is pinned to the bench at a fixed pin O and can spin all the way round. Set up coordinates with O at the origin; the datum direction is marked by a construction point E, which sits 10 mm from O, level with it. The free end of the short bar is the pin P. A second bar, 31.3 mm long, joins P to a block whose centre is Q; the block slides along a straight rail that runs -5.2 mm above the level of O. Show the arrangement at the moment when the short bar stands 35 degrees above the datum direction, turned anticlockwise from it. At that moment the block is on the far side of the crank pin from O, not folded back over it.", "answer_schema": {"points": ["E", "O", "P", "Q"], "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": "\"\"\"Slider-crank at a stated crank angle\n\nOracle formalization for mech_t1_mech_slider_crank_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(56)\n\nO, E, P, Q = scene.add.points([\"O\", \"E\", \"P\", \"Q\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank pin (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank pin (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so angles can be measured from it\")\nscene.constraint.eq(O.distance(P), 13.4, description=\"the short bar OP is 13.4 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 35, description=\"the short bar stands 35 degrees above the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the short bar is turned anticlockwise from the datum direction, so P is above it\")\nscene.constraint.eq(P.distance(Q), 31.3, description=\"the long bar PQ is 31.3 long\")\nscene.constraint.eq(Q.y, -5.2, description=\"the block runs on a straight rail -5.2 above the level of O\")\nscene.constraint.gt(Q.x - P.x, 0.05, description=\"the block sits on the far side of the crank pin from the fixed pin, not folded back over it\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [10.97663739347249, 7.685924247104017], "Q": [39.50106475490951, -5.2]}, "circles": {}}, "implicit_facts": ["the rail is the horizontal line at the stated height, so the block centre has that y", "at the stated instant P is at O plus the crank length times (cos, sin) of the crank angle", "the block is where a circle of the conrod length centred at P cuts the rail line", "that circle cuts the rail line twice, so a choice has to be made"], "branch_facts": ["the block is on the far side of the crank pin, which picks one of the two intersections"]} {"unique_id": "mech_t1_mech_slider_crank_0010", "problem_id": "mech_t1_mech_slider_crank_0010", "domain": "PyGeoX-Mech", "template": "mech_slider_crank", "tier": 1, "nl_description": "A short bar 13.2 mm long is pinned to the bench at a fixed pin O and can spin all the way round. Set up coordinates with O at the origin; the datum direction is marked by a construction point E, which sits 10 mm from O, level with it. The free end of the short bar is the pin P. A second bar, 32.5 mm long, joins P to a block whose centre is Q; the block slides along a straight rail that runs 7.9 mm above the level of O. Show the arrangement at the moment when the short bar stands 70 degrees above the datum direction, turned anticlockwise from it. At that moment the block is on the far side of the crank pin from O, not folded back over it.", "answer_schema": {"points": ["E", "O", "P", "Q"], "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": "\"\"\"Slider-crank at a stated crank angle\n\nOracle formalization for mech_t1_mech_slider_crank_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(53)\n\nO, E, P, Q = scene.add.points([\"O\", \"E\", \"P\", \"Q\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank pin (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank pin (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so angles can be measured from it\")\nscene.constraint.eq(O.distance(P), 13.2, description=\"the short bar OP is 13.2 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 70, description=\"the short bar stands 70 degrees above the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the short bar is turned anticlockwise from the datum direction, so P is above it\")\nscene.constraint.eq(P.distance(Q), 32.5, description=\"the long bar PQ is 32.5 long\")\nscene.constraint.eq(Q.y, 7.9, description=\"the block runs on a straight rail 7.9 above the level of O\")\nscene.constraint.gt(Q.x - P.x, 0.05, description=\"the block sits on the far side of the crank pin from the fixed pin, not folded back over it\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [4.5146658918988285, 12.403942594373989], "Q": [36.70106831755963, 7.9]}, "circles": {}}, "implicit_facts": ["the rail is the horizontal line at the stated height, so the block centre has that y", "at the stated instant P is at O plus the crank length times (cos, sin) of the crank angle", "the block is where a circle of the conrod length centred at P cuts the rail line", "that circle cuts the rail line twice, so a choice has to be made"], "branch_facts": ["the block is on the far side of the crank pin, which picks one of the two intersections"]} {"unique_id": "mech_t1_mech_slider_crank_0011", "problem_id": "mech_t1_mech_slider_crank_0011", "domain": "PyGeoX-Mech", "template": "mech_slider_crank", "tier": 1, "nl_description": "A short bar 9.5 mm long is pinned to the bench at a fixed pin O and can spin all the way round. Set up coordinates with O at the origin; the datum direction is marked by a construction point E, which sits 10 mm from O, level with it. The free end of the short bar is the pin P. A second bar, 25.1 mm long, joins P to a block whose centre is Q; the block slides along a straight rail that runs 6.9 mm above the level of O. Show the arrangement at the moment when the short bar stands 70 degrees above the datum direction, turned anticlockwise from it. At that moment the block is on the far side of the crank pin from O, not folded back over it.", "answer_schema": {"points": ["E", "O", "P", "Q"], "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": "\"\"\"Slider-crank at a stated crank angle\n\nOracle formalization for mech_t1_mech_slider_crank_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(42)\n\nO, E, P, Q = scene.add.points([\"O\", \"E\", \"P\", \"Q\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank pin (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank pin (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so angles can be measured from it\")\nscene.constraint.eq(O.distance(P), 9.5, description=\"the short bar OP is 9.5 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 70, description=\"the short bar stands 70 degrees above the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the short bar is turned anticlockwise from the datum direction, so P is above it\")\nscene.constraint.eq(P.distance(Q), 25.1, description=\"the long bar PQ is 25.1 long\")\nscene.constraint.eq(Q.y, 6.9, description=\"the block runs on a straight rail 6.9 above the level of O\")\nscene.constraint.gt(Q.x - P.x, 0.05, description=\"the block sits on the far side of the crank pin from the fixed pin, not folded back over it\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [3.2491913615938537, 8.927079897466129], "Q": [28.2672038144105, 6.9]}, "circles": {}}, "implicit_facts": ["the rail is the horizontal line at the stated height, so the block centre has that y", "at the stated instant P is at O plus the crank length times (cos, sin) of the crank angle", "the block is where a circle of the conrod length centred at P cuts the rail line", "that circle cuts the rail line twice, so a choice has to be made"], "branch_facts": ["the block is on the far side of the crank pin, which picks one of the two intersections"]} {"unique_id": "mech_t1_mech_slider_crank_0012", "problem_id": "mech_t1_mech_slider_crank_0012", "domain": "PyGeoX-Mech", "template": "mech_slider_crank", "tier": 1, "nl_description": "A short bar 12 mm long is pinned to the bench at a fixed pin O and can spin all the way round. Set up coordinates with O at the origin; the datum direction is marked by a construction point E, which sits 10 mm from O, level with it. The free end of the short bar is the pin P. A second bar, 24.2 mm long, joins P to a block whose centre is Q; the block slides along a straight rail that runs 7.1 mm above the level of O. Show the arrangement at the moment when the short bar stands 55 degrees above the datum direction, turned anticlockwise from it. At that moment the block is on the far side of the crank pin from O, not folded back over it.", "answer_schema": {"points": ["E", "O", "P", "Q"], "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": "\"\"\"Slider-crank at a stated crank angle\n\nOracle formalization for mech_t1_mech_slider_crank_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(45)\n\nO, E, P, Q = scene.add.points([\"O\", \"E\", \"P\", \"Q\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank pin (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank pin (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so angles can be measured from it\")\nscene.constraint.eq(O.distance(P), 12, description=\"the short bar OP is 12 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 55, description=\"the short bar stands 55 degrees above the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the short bar is turned anticlockwise from the datum direction, so P is above it\")\nscene.constraint.eq(P.distance(Q), 24.2, description=\"the long bar PQ is 24.2 long\")\nscene.constraint.eq(Q.y, 7.1, description=\"the block runs on a straight rail 7.1 above the level of O\")\nscene.constraint.gt(Q.x - P.x, 0.05, description=\"the block sits on the far side of the crank pin from the fixed pin, not folded back over it\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [6.882917236212553, 9.829824531467901], "Q": [30.928458569844366, 7.1]}, "circles": {}}, "implicit_facts": ["the rail is the horizontal line at the stated height, so the block centre has that y", "at the stated instant P is at O plus the crank length times (cos, sin) of the crank angle", "the block is where a circle of the conrod length centred at P cuts the rail line", "that circle cuts the rail line twice, so a choice has to be made"], "branch_facts": ["the block is on the far side of the crank pin, which picks one of the two intersections"]} {"unique_id": "mech_t1_mech_slider_crank_0013", "problem_id": "mech_t1_mech_slider_crank_0013", "domain": "PyGeoX-Mech", "template": "mech_slider_crank", "tier": 1, "nl_description": "A short bar 10.3 mm long is pinned to the bench at a fixed pin O and can spin all the way round. Set up coordinates with O at the origin; the datum direction is marked by a construction point E, which sits 10 mm from O, level with it. The free end of the short bar is the pin P. A second bar, 24.1 mm long, joins P to a block whose centre is Q; the block slides along a straight rail that runs -3 mm above the level of O. Show the arrangement at the moment when the short bar stands 70 degrees above the datum direction, turned anticlockwise from it. At that moment the block is on the far side of the crank pin from O, not folded back over it.", "answer_schema": {"points": ["E", "O", "P", "Q"], "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": "\"\"\"Slider-crank at a stated crank angle\n\nOracle formalization for mech_t1_mech_slider_crank_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(36)\n\nO, E, P, Q = scene.add.points([\"O\", \"E\", \"P\", \"Q\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank pin (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank pin (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so angles can be measured from it\")\nscene.constraint.eq(O.distance(P), 10.3, description=\"the short bar OP is 10.3 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 70, description=\"the short bar stands 70 degrees above the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the short bar is turned anticlockwise from the datum direction, so P is above it\")\nscene.constraint.eq(P.distance(Q), 24.1, description=\"the long bar PQ is 24.1 long\")\nscene.constraint.eq(Q.y, -3, description=\"the block runs on a straight rail -3 above the level of O\")\nscene.constraint.gt(Q.x - P.x, 0.05, description=\"the block sits on the far side of the crank pin from the fixed pin, not folded back over it\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [3.522807476254389, 9.678833994094857], "Q": [24.018103730521727, -3.0]}, "circles": {}}, "implicit_facts": ["the rail is the horizontal line at the stated height, so the block centre has that y", "at the stated instant P is at O plus the crank length times (cos, sin) of the crank angle", "the block is where a circle of the conrod length centred at P cuts the rail line", "that circle cuts the rail line twice, so a choice has to be made"], "branch_facts": ["the block is on the far side of the crank pin, which picks one of the two intersections"]} {"unique_id": "mech_t1_mech_slider_crank_0014", "problem_id": "mech_t1_mech_slider_crank_0014", "domain": "PyGeoX-Mech", "template": "mech_slider_crank", "tier": 1, "nl_description": "A short bar 9.5 mm long is pinned to the bench at a fixed pin O and can spin all the way round. Set up coordinates with O at the origin; the datum direction is marked by a construction point E, which sits 10 mm from O, level with it. The free end of the short bar is the pin P. A second bar, 27.4 mm long, joins P to a block whose centre is Q; the block slides along a straight rail that runs 3 mm above the level of O. Show the arrangement at the moment when the short bar stands 35 degrees above the datum direction, turned anticlockwise from it. At that moment the block is on the far side of the crank pin from O, not folded back over it.", "answer_schema": {"points": ["E", "O", "P", "Q"], "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": "\"\"\"Slider-crank at a stated crank angle\n\nOracle formalization for mech_t1_mech_slider_crank_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(51)\n\nO, E, P, Q = scene.add.points([\"O\", \"E\", \"P\", \"Q\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank pin (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank pin (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so angles can be measured from it\")\nscene.constraint.eq(O.distance(P), 9.5, description=\"the short bar OP is 9.5 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 35, description=\"the short bar stands 35 degrees above the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the short bar is turned anticlockwise from the datum direction, so P is above it\")\nscene.constraint.eq(P.distance(Q), 27.4, description=\"the long bar PQ is 27.4 long\")\nscene.constraint.eq(Q.y, 3, description=\"the block runs on a straight rail 3 above the level of O\")\nscene.constraint.gt(Q.x - P.x, 0.05, description=\"the block sits on the far side of the crank pin from the fixed pin, not folded back over it\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [7.781944420745422, 5.448976145334937], "Q": [35.07228182869222, 3.0]}, "circles": {}}, "implicit_facts": ["the rail is the horizontal line at the stated height, so the block centre has that y", "at the stated instant P is at O plus the crank length times (cos, sin) of the crank angle", "the block is where a circle of the conrod length centred at P cuts the rail line", "that circle cuts the rail line twice, so a choice has to be made"], "branch_facts": ["the block is on the far side of the crank pin, which picks one of the two intersections"]} {"unique_id": "mech_t1_mech_slider_crank_0015", "problem_id": "mech_t1_mech_slider_crank_0015", "domain": "PyGeoX-Mech", "template": "mech_slider_crank", "tier": 1, "nl_description": "A short bar 13 mm long is pinned to the bench at a fixed pin O and can spin all the way round. Set up coordinates with O at the origin; the datum direction is marked by a construction point E, which sits 10 mm from O, level with it. The free end of the short bar is the pin P. A second bar, 25.7 mm long, joins P to a block whose centre is Q; the block slides along a straight rail that runs -2.6 mm above the level of O. Show the arrangement at the moment when the short bar stands 50 degrees above the datum direction, turned anticlockwise from it. At that moment the block is on the far side of the crank pin from O, not folded back over it.", "answer_schema": {"points": ["E", "O", "P", "Q"], "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": "\"\"\"Slider-crank at a stated crank angle\n\nOracle formalization for mech_t1_mech_slider_crank_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(45)\n\nO, E, P, Q = scene.add.points([\"O\", \"E\", \"P\", \"Q\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank pin (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank pin (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so angles can be measured from it\")\nscene.constraint.eq(O.distance(P), 13, description=\"the short bar OP is 13 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 50, description=\"the short bar stands 50 degrees above the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the short bar is turned anticlockwise from the datum direction, so P is above it\")\nscene.constraint.eq(P.distance(Q), 25.7, description=\"the long bar PQ is 25.7 long\")\nscene.constraint.eq(Q.y, -2.6, description=\"the block runs on a straight rail -2.6 above the level of O\")\nscene.constraint.gt(Q.x - P.x, 0.05, description=\"the block sits on the far side of the crank pin from the fixed pin, not folded back over it\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [8.356238925925012, 9.958577760546714], "Q": [30.778819612195303, -2.6]}, "circles": {}}, "implicit_facts": ["the rail is the horizontal line at the stated height, so the block centre has that y", "at the stated instant P is at O plus the crank length times (cos, sin) of the crank angle", "the block is where a circle of the conrod length centred at P cuts the rail line", "that circle cuts the rail line twice, so a choice has to be made"], "branch_facts": ["the block is on the far side of the crank pin, which picks one of the two intersections"]} {"unique_id": "mech_t1_mech_slider_crank_0016", "problem_id": "mech_t1_mech_slider_crank_0016", "domain": "PyGeoX-Mech", "template": "mech_slider_crank", "tier": 1, "nl_description": "A short bar 9.5 mm long is pinned to the bench at a fixed pin O and can spin all the way round. Set up coordinates with O at the origin; the datum direction is marked by a construction point E, which sits 10 mm from O, level with it. The free end of the short bar is the pin P. A second bar, 30.7 mm long, joins P to a block whose centre is Q; the block slides along a straight rail that runs -5.6 mm above the level of O. Show the arrangement at the moment when the short bar stands 35 degrees above the datum direction, turned anticlockwise from it. At that moment the block is on the far side of the crank pin from O, not folded back over it.", "answer_schema": {"points": ["E", "O", "P", "Q"], "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": "\"\"\"Slider-crank at a stated crank angle\n\nOracle formalization for mech_t1_mech_slider_crank_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(52)\n\nO, E, P, Q = scene.add.points([\"O\", \"E\", \"P\", \"Q\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank pin (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank pin (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so angles can be measured from it\")\nscene.constraint.eq(O.distance(P), 9.5, description=\"the short bar OP is 9.5 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 35, description=\"the short bar stands 35 degrees above the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the short bar is turned anticlockwise from the datum direction, so P is above it\")\nscene.constraint.eq(P.distance(Q), 30.7, description=\"the long bar PQ is 30.7 long\")\nscene.constraint.eq(Q.y, -5.6, description=\"the block runs on a straight rail -5.6 above the level of O\")\nscene.constraint.gt(Q.x - P.x, 0.05, description=\"the block sits on the far side of the crank pin from the fixed pin, not folded back over it\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [7.781944420745422, 5.448976145334937], "Q": [36.42474676513588, -5.6]}, "circles": {}}, "implicit_facts": ["the rail is the horizontal line at the stated height, so the block centre has that y", "at the stated instant P is at O plus the crank length times (cos, sin) of the crank angle", "the block is where a circle of the conrod length centred at P cuts the rail line", "that circle cuts the rail line twice, so a choice has to be made"], "branch_facts": ["the block is on the far side of the crank pin, which picks one of the two intersections"]} {"unique_id": "mech_t1_mech_trammel_0087", "problem_id": "mech_t1_mech_trammel_0087", "domain": "PyGeoX-Mech", "template": "mech_trammel", "tier": 1, "nl_description": "A straight bar 20.8 mm long has a small block pinned at each end. One block, Bh, slides along a horizontal rail; the other, Bv, slides along a vertical rail. The two rails cross at a point Z, which is the origin of the coordinates. A tracing point T is marked on the bar, 8.9 mm along it from the block Bh. Set the mechanism so that the block Bh is 8.3 mm from the crossing point along its own rail, in the positive direction, with the block Bv on the upper half of the vertical rail.", "answer_schema": {"points": ["Bh", "Bv", "T", "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": "\"\"\"Elliptic trammel with a tracing point\n\nOracle formalization for mech_t1_mech_trammel_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(30)\n\nZ, Bh, Bv, T = scene.add.points([\"Z\", \"Bh\", \"Bv\", \"T\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the crossing point of the two rails (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the crossing point of the two rails (y)\")\nscene.constraint.eq(Bh.y, Z.y, description=\"the first block runs on the horizontal rail through Z\")\nscene.constraint.eq(Bv.x, Z.x, description=\"the second block runs on the vertical rail through Z\")\nscene.constraint.eq(Bh.x - Z.x, 8.3, description=\"the first block is set 8.3 along its rail from the crossing point\")\nscene.constraint.gt(Bv.y - Z.y, 0.05, description=\"the second block sits on the upper half of the vertical rail\")\nscene.constraint.eq(Bh.distance(Bv), 20.8, description=\"the bar joining the two blocks is 20.8 long\")\nscene.relate.point_lies_on(T, scene.add.line_segment(Bh, Bv))\nscene.constraint.eq(Bh.distance(T), 8.9, description=\"the tracing point T is marked 8.9 along the bar from the first block\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh": [8.3, 0.0], "Bv": [0.0, 19.072231122760652], "T": [4.748557692307693, 8.160714278488934]}, "circles": {}}, "implicit_facts": ["each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so the two blocks stay exactly the bar length apart", "with one block's position given, the other follows from the right-angled triangle formed with the rails", "the tracing point lies ON the bar between the blocks, at the stated distance from Bh"], "branch_facts": ["Bv is on the upper half of its rail, which picks the sign of its position"]} {"unique_id": "mech_t1_mech_trammel_0088", "problem_id": "mech_t1_mech_trammel_0088", "domain": "PyGeoX-Mech", "template": "mech_trammel", "tier": 1, "nl_description": "A straight bar 22.2 mm long has a small block pinned at each end. One block, Bh, slides along a horizontal rail; the other, Bv, slides along a vertical rail. The two rails cross at a point Z, which is the origin of the coordinates. A tracing point T is marked on the bar, 11.1 mm along it from the block Bh. Set the mechanism so that the block Bh is 9.2 mm from the crossing point along its own rail, in the positive direction, with the block Bv on the upper half of the vertical rail.", "answer_schema": {"points": ["Bh", "Bv", "T", "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": "\"\"\"Elliptic trammel with a tracing point\n\nOracle formalization for mech_t1_mech_trammel_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(31)\n\nZ, Bh, Bv, T = scene.add.points([\"Z\", \"Bh\", \"Bv\", \"T\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the crossing point of the two rails (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the crossing point of the two rails (y)\")\nscene.constraint.eq(Bh.y, Z.y, description=\"the first block runs on the horizontal rail through Z\")\nscene.constraint.eq(Bv.x, Z.x, description=\"the second block runs on the vertical rail through Z\")\nscene.constraint.eq(Bh.x - Z.x, 9.2, description=\"the first block is set 9.2 along its rail from the crossing point\")\nscene.constraint.gt(Bv.y - Z.y, 0.05, description=\"the second block sits on the upper half of the vertical rail\")\nscene.constraint.eq(Bh.distance(Bv), 22.2, description=\"the bar joining the two blocks is 22.2 long\")\nscene.relate.point_lies_on(T, scene.add.line_segment(Bh, Bv))\nscene.constraint.eq(Bh.distance(T), 11.1, description=\"the tracing point T is marked 11.1 along the bar from the first block\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh": [9.2, 0.0], "Bv": [0.0, 20.203960007879644], "T": [4.6, 10.101980003939822]}, "circles": {}}, "implicit_facts": ["each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so the two blocks stay exactly the bar length apart", "with one block's position given, the other follows from the right-angled triangle formed with the rails", "the tracing point lies ON the bar between the blocks, at the stated distance from Bh"], "branch_facts": ["Bv is on the upper half of its rail, which picks the sign of its position"]} {"unique_id": "mech_t1_mech_trammel_0089", "problem_id": "mech_t1_mech_trammel_0089", "domain": "PyGeoX-Mech", "template": "mech_trammel", "tier": 1, "nl_description": "A straight bar 21.3 mm long has a small block pinned at each end. One block, Bh, slides along a horizontal rail; the other, Bv, slides along a vertical rail. The two rails cross at a point Z, which is the origin of the coordinates. A tracing point T is marked on the bar, 12.3 mm along it from the block Bh. Set the mechanism so that the block Bh is 9.5 mm from the crossing point along its own rail, in the positive direction, with the block Bv on the upper half of the vertical rail.", "answer_schema": {"points": ["Bh", "Bv", "T", "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": "\"\"\"Elliptic trammel with a tracing point\n\nOracle formalization for mech_t1_mech_trammel_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(30)\n\nZ, Bh, Bv, T = scene.add.points([\"Z\", \"Bh\", \"Bv\", \"T\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the crossing point of the two rails (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the crossing point of the two rails (y)\")\nscene.constraint.eq(Bh.y, Z.y, description=\"the first block runs on the horizontal rail through Z\")\nscene.constraint.eq(Bv.x, Z.x, description=\"the second block runs on the vertical rail through Z\")\nscene.constraint.eq(Bh.x - Z.x, 9.5, description=\"the first block is set 9.5 along its rail from the crossing point\")\nscene.constraint.gt(Bv.y - Z.y, 0.05, description=\"the second block sits on the upper half of the vertical rail\")\nscene.constraint.eq(Bh.distance(Bv), 21.3, description=\"the bar joining the two blocks is 21.3 long\")\nscene.relate.point_lies_on(T, scene.add.line_segment(Bh, Bv))\nscene.constraint.eq(Bh.distance(T), 12.3, description=\"the tracing point T is marked 12.3 along the bar from the first block\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh": [9.5, 0.0], "Bv": [0.0, 19.064102391667962], "T": [4.014084507042253, 11.008847859977275]}, "circles": {}}, "implicit_facts": ["each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so the two blocks stay exactly the bar length apart", "with one block's position given, the other follows from the right-angled triangle formed with the rails", "the tracing point lies ON the bar between the blocks, at the stated distance from Bh"], "branch_facts": ["Bv is on the upper half of its rail, which picks the sign of its position"]} {"unique_id": "mech_t1_mech_trammel_0090", "problem_id": "mech_t1_mech_trammel_0090", "domain": "PyGeoX-Mech", "template": "mech_trammel", "tier": 1, "nl_description": "A straight bar 23.8 mm long has a small block pinned at each end. One block, Bh, slides along a horizontal rail; the other, Bv, slides along a vertical rail. The two rails cross at a point Z, which is the origin of the coordinates. A tracing point T is marked on the bar, 17.3 mm along it from the block Bh. Set the mechanism so that the block Bh is 14 mm from the crossing point along its own rail, in the positive direction, with the block Bv on the upper half of the vertical rail.", "answer_schema": {"points": ["Bh", "Bv", "T", "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": "\"\"\"Elliptic trammel with a tracing point\n\nOracle formalization for mech_t1_mech_trammel_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(30)\n\nZ, Bh, Bv, T = scene.add.points([\"Z\", \"Bh\", \"Bv\", \"T\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the crossing point of the two rails (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the crossing point of the two rails (y)\")\nscene.constraint.eq(Bh.y, Z.y, description=\"the first block runs on the horizontal rail through Z\")\nscene.constraint.eq(Bv.x, Z.x, description=\"the second block runs on the vertical rail through Z\")\nscene.constraint.eq(Bh.x - Z.x, 14, description=\"the first block is set 14 along its rail from the crossing point\")\nscene.constraint.gt(Bv.y - Z.y, 0.05, description=\"the second block sits on the upper half of the vertical rail\")\nscene.constraint.eq(Bh.distance(Bv), 23.8, description=\"the bar joining the two blocks is 23.8 long\")\nscene.relate.point_lies_on(T, scene.add.line_segment(Bh, Bv))\nscene.constraint.eq(Bh.distance(T), 17.3, description=\"the tracing point T is marked 17.3 along the bar from the first block\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh": [14.0, 0.0], "Bv": [0.0, 19.24681791881453], "T": [3.8235294117647047, 13.990334033424006]}, "circles": {}}, "implicit_facts": ["each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so the two blocks stay exactly the bar length apart", "with one block's position given, the other follows from the right-angled triangle formed with the rails", "the tracing point lies ON the bar between the blocks, at the stated distance from Bh"], "branch_facts": ["Bv is on the upper half of its rail, which picks the sign of its position"]} {"unique_id": "mech_t1_mech_trammel_0091", "problem_id": "mech_t1_mech_trammel_0091", "domain": "PyGeoX-Mech", "template": "mech_trammel", "tier": 1, "nl_description": "A straight bar 24.2 mm long has a small block pinned at each end. One block, Bh, slides along a horizontal rail; the other, Bv, slides along a vertical rail. The two rails cross at a point Z, which is the origin of the coordinates. A tracing point T is marked on the bar, 16.6 mm along it from the block Bh. Set the mechanism so that the block Bh is 14.9 mm from the crossing point along its own rail, in the positive direction, with the block Bv on the upper half of the vertical rail.", "answer_schema": {"points": ["Bh", "Bv", "T", "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": "\"\"\"Elliptic trammel with a tracing point\n\nOracle formalization for mech_t1_mech_trammel_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\nZ, Bh, Bv, T = scene.add.points([\"Z\", \"Bh\", \"Bv\", \"T\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the crossing point of the two rails (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the crossing point of the two rails (y)\")\nscene.constraint.eq(Bh.y, Z.y, description=\"the first block runs on the horizontal rail through Z\")\nscene.constraint.eq(Bv.x, Z.x, description=\"the second block runs on the vertical rail through Z\")\nscene.constraint.eq(Bh.x - Z.x, 14.9, description=\"the first block is set 14.9 along its rail from the crossing point\")\nscene.constraint.gt(Bv.y - Z.y, 0.05, description=\"the second block sits on the upper half of the vertical rail\")\nscene.constraint.eq(Bh.distance(Bv), 24.2, description=\"the bar joining the two blocks is 24.2 long\")\nscene.relate.point_lies_on(T, scene.add.line_segment(Bh, Bv))\nscene.constraint.eq(Bh.distance(T), 16.6, description=\"the tracing point T is marked 16.6 along the bar from the first block\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh": [14.9, 0.0], "Bv": [0.0, 19.069084928228726], "T": [4.679338842975206, 13.080446686305658]}, "circles": {}}, "implicit_facts": ["each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so the two blocks stay exactly the bar length apart", "with one block's position given, the other follows from the right-angled triangle formed with the rails", "the tracing point lies ON the bar between the blocks, at the stated distance from Bh"], "branch_facts": ["Bv is on the upper half of its rail, which picks the sign of its position"]} {"unique_id": "mech_t1_mech_trammel_0092", "problem_id": "mech_t1_mech_trammel_0092", "domain": "PyGeoX-Mech", "template": "mech_trammel", "tier": 1, "nl_description": "A straight bar 28.1 mm long has a small block pinned at each end. One block, Bh, slides along a horizontal rail; the other, Bv, slides along a vertical rail. The two rails cross at a point Z, which is the origin of the coordinates. A tracing point T is marked on the bar, 16.9 mm along it from the block Bh. Set the mechanism so that the block Bh is 17.3 mm from the crossing point along its own rail, in the positive direction, with the block Bv on the upper half of the vertical rail.", "answer_schema": {"points": ["Bh", "Bv", "T", "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": "\"\"\"Elliptic trammel with a tracing point\n\nOracle formalization for mech_t1_mech_trammel_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(34)\n\nZ, Bh, Bv, T = scene.add.points([\"Z\", \"Bh\", \"Bv\", \"T\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the crossing point of the two rails (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the crossing point of the two rails (y)\")\nscene.constraint.eq(Bh.y, Z.y, description=\"the first block runs on the horizontal rail through Z\")\nscene.constraint.eq(Bv.x, Z.x, description=\"the second block runs on the vertical rail through Z\")\nscene.constraint.eq(Bh.x - Z.x, 17.3, description=\"the first block is set 17.3 along its rail from the crossing point\")\nscene.constraint.gt(Bv.y - Z.y, 0.05, description=\"the second block sits on the upper half of the vertical rail\")\nscene.constraint.eq(Bh.distance(Bv), 28.1, description=\"the bar joining the two blocks is 28.1 long\")\nscene.relate.point_lies_on(T, scene.add.line_segment(Bh, Bv))\nscene.constraint.eq(Bh.distance(T), 16.9, description=\"the tracing point T is marked 16.9 along the bar from the first block\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh": [17.3, 0.0], "Bv": [0.0, 22.143170504695124], "T": [6.895373665480427, 13.317422830225892]}, "circles": {}}, "implicit_facts": ["each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so the two blocks stay exactly the bar length apart", "with one block's position given, the other follows from the right-angled triangle formed with the rails", "the tracing point lies ON the bar between the blocks, at the stated distance from Bh"], "branch_facts": ["Bv is on the upper half of its rail, which picks the sign of its position"]} {"unique_id": "mech_t1_mech_trammel_0093", "problem_id": "mech_t1_mech_trammel_0093", "domain": "PyGeoX-Mech", "template": "mech_trammel", "tier": 1, "nl_description": "A straight bar 24.8 mm long has a small block pinned at each end. One block, Bh, slides along a horizontal rail; the other, Bv, slides along a vertical rail. The two rails cross at a point Z, which is the origin of the coordinates. A tracing point T is marked on the bar, 7.5 mm along it from the block Bh. Set the mechanism so that the block Bh is 13.8 mm from the crossing point along its own rail, in the positive direction, with the block Bv on the upper half of the vertical rail.", "answer_schema": {"points": ["Bh", "Bv", "T", "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": "\"\"\"Elliptic trammel with a tracing point\n\nOracle formalization for mech_t1_mech_trammel_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(32)\n\nZ, Bh, Bv, T = scene.add.points([\"Z\", \"Bh\", \"Bv\", \"T\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the crossing point of the two rails (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the crossing point of the two rails (y)\")\nscene.constraint.eq(Bh.y, Z.y, description=\"the first block runs on the horizontal rail through Z\")\nscene.constraint.eq(Bv.x, Z.x, description=\"the second block runs on the vertical rail through Z\")\nscene.constraint.eq(Bh.x - Z.x, 13.8, description=\"the first block is set 13.8 along its rail from the crossing point\")\nscene.constraint.gt(Bv.y - Z.y, 0.05, description=\"the second block sits on the upper half of the vertical rail\")\nscene.constraint.eq(Bh.distance(Bv), 24.8, description=\"the bar joining the two blocks is 24.8 long\")\nscene.relate.point_lies_on(T, scene.add.line_segment(Bh, Bv))\nscene.constraint.eq(Bh.distance(T), 7.5, description=\"the tracing point T is marked 7.5 along the bar from the first block\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh": [13.8, 0.0], "Bv": [0.0, 20.605824419323774], "T": [9.626612903225809, 6.2316001268116255]}, "circles": {}}, "implicit_facts": ["each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so the two blocks stay exactly the bar length apart", "with one block's position given, the other follows from the right-angled triangle formed with the rails", "the tracing point lies ON the bar between the blocks, at the stated distance from Bh"], "branch_facts": ["Bv is on the upper half of its rail, which picks the sign of its position"]} {"unique_id": "mech_t1_mech_two_cranks_0065", "problem_id": "mech_t1_mech_two_cranks_0065", "domain": "PyGeoX-Mech", "template": "mech_two_cranks", "tier": 1, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 31.8 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 13.5 mm long and ends at a pin W; the bar at V is 23.3 mm long and ends at a pin X. A third bar joins W to X. In the position of interest the bar at U points straight up, and pin X is exactly level with pin W, lying on the side of V that faces U.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks joined by a common link\n\nOracle formalization for mech_t1_mech_two_cranks_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(46)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 31.8, description=\"the two fixed pins are 31.8 apart\")\nscene.constraint.eq(U.distance(W), 13.5, description=\"the bar at U is 13.5 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 23.3, description=\"the bar at V is 23.3 long, ending at pin X\")\nscene.constraint.eq(W.x, U.x, description=\"the bar at U points straight up, so W is directly above U\")\nscene.constraint.gt(W.y - U.y, 0.05, description=\"the bar at U points UP, not down\")\nscene.constraint.eq(X.y, W.y, description=\"pin X is exactly level with pin W\")\nscene.constraint.lt(X.x - V.x, -0.05, description=\"pin X lies on the side of V towards U\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [31.8, 0.0], "W": [0.0, 13.5], "X": [12.809476047249252, 13.5]}, "circles": {}}, "implicit_facts": ["'points straight up' means W is directly above U, at U plus the bar length in the y-direction", "'level with' means X has the same height as W", "with its height fixed and its distance from V fixed, X is determined up to a left/right choice", "the length of the third bar then follows from the two positions"], "branch_facts": ["X lies on the side of V facing U, which picks one of the two possibilities"]} {"unique_id": "mech_t1_mech_two_cranks_0066", "problem_id": "mech_t1_mech_two_cranks_0066", "domain": "PyGeoX-Mech", "template": "mech_two_cranks", "tier": 1, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 21.7 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 13 mm long and ends at a pin W; the bar at V is 21.1 mm long and ends at a pin X. A third bar joins W to X. In the position of interest the bar at U points straight up, and pin X is exactly level with pin W, lying on the side of V that faces U.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks joined by a common link\n\nOracle formalization for mech_t1_mech_two_cranks_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(33)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 21.7, description=\"the two fixed pins are 21.7 apart\")\nscene.constraint.eq(U.distance(W), 13, description=\"the bar at U is 13 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 21.1, description=\"the bar at V is 21.1 long, ending at pin X\")\nscene.constraint.eq(W.x, U.x, description=\"the bar at U points straight up, so W is directly above U\")\nscene.constraint.gt(W.y - U.y, 0.05, description=\"the bar at U points UP, not down\")\nscene.constraint.eq(X.y, W.y, description=\"pin X is exactly level with pin W\")\nscene.constraint.lt(X.x - V.x, -0.05, description=\"pin X lies on the side of V towards U\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [21.7, 0.0], "W": [0.0, 13.0], "X": [5.0804332186425505, 13.0]}, "circles": {}}, "implicit_facts": ["'points straight up' means W is directly above U, at U plus the bar length in the y-direction", "'level with' means X has the same height as W", "with its height fixed and its distance from V fixed, X is determined up to a left/right choice", "the length of the third bar then follows from the two positions"], "branch_facts": ["X lies on the side of V facing U, which picks one of the two possibilities"]} {"unique_id": "mech_t1_mech_two_cranks_0067", "problem_id": "mech_t1_mech_two_cranks_0067", "domain": "PyGeoX-Mech", "template": "mech_two_cranks", "tier": 1, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 25.4 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 9.4 mm long and ends at a pin W; the bar at V is 22.4 mm long and ends at a pin X. A third bar joins W to X. In the position of interest the bar at U points straight up, and pin X is exactly level with pin W, lying on the side of V that faces U.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks joined by a common link\n\nOracle formalization for mech_t1_mech_two_cranks_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(38)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 25.4, description=\"the two fixed pins are 25.4 apart\")\nscene.constraint.eq(U.distance(W), 9.4, description=\"the bar at U is 9.4 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 22.4, description=\"the bar at V is 22.4 long, ending at pin X\")\nscene.constraint.eq(W.x, U.x, description=\"the bar at U points straight up, so W is directly above U\")\nscene.constraint.gt(W.y - U.y, 0.05, description=\"the bar at U points UP, not down\")\nscene.constraint.eq(X.y, W.y, description=\"pin X is exactly level with pin W\")\nscene.constraint.lt(X.x - V.x, -0.05, description=\"pin X lies on the side of V towards U\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [25.4, 0.0], "W": [0.0, 9.4], "X": [5.067759592214145, 9.4]}, "circles": {}}, "implicit_facts": ["'points straight up' means W is directly above U, at U plus the bar length in the y-direction", "'level with' means X has the same height as W", "with its height fixed and its distance from V fixed, X is determined up to a left/right choice", "the length of the third bar then follows from the two positions"], "branch_facts": ["X lies on the side of V facing U, which picks one of the two possibilities"]} {"unique_id": "mech_t1_mech_two_cranks_0068", "problem_id": "mech_t1_mech_two_cranks_0068", "domain": "PyGeoX-Mech", "template": "mech_two_cranks", "tier": 1, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 29.2 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 9.7 mm long and ends at a pin W; the bar at V is 23.3 mm long and ends at a pin X. A third bar joins W to X. In the position of interest the bar at U points straight up, and pin X is exactly level with pin W, lying on the side of V that faces U.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks joined by a common link\n\nOracle formalization for mech_t1_mech_two_cranks_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(43)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 29.2, description=\"the two fixed pins are 29.2 apart\")\nscene.constraint.eq(U.distance(W), 9.7, description=\"the bar at U is 9.7 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 23.3, description=\"the bar at V is 23.3 long, ending at pin X\")\nscene.constraint.eq(W.x, U.x, description=\"the bar at U points straight up, so W is directly above U\")\nscene.constraint.gt(W.y - U.y, 0.05, description=\"the bar at U points UP, not down\")\nscene.constraint.eq(X.y, W.y, description=\"pin X is exactly level with pin W\")\nscene.constraint.lt(X.x - V.x, -0.05, description=\"pin X lies on the side of V towards U\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [29.2, 0.0], "W": [0.0, 9.7], "X": [8.01509971701542, 9.7]}, "circles": {}}, "implicit_facts": ["'points straight up' means W is directly above U, at U plus the bar length in the y-direction", "'level with' means X has the same height as W", "with its height fixed and its distance from V fixed, X is determined up to a left/right choice", "the length of the third bar then follows from the two positions"], "branch_facts": ["X lies on the side of V facing U, which picks one of the two possibilities"]} {"unique_id": "mech_t1_mech_two_cranks_0069", "problem_id": "mech_t1_mech_two_cranks_0069", "domain": "PyGeoX-Mech", "template": "mech_two_cranks", "tier": 1, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 23.2 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 11.1 mm long and ends at a pin W; the bar at V is 22.7 mm long and ends at a pin X. A third bar joins W to X. In the position of interest the bar at U points straight up, and pin X is exactly level with pin W, lying on the side of V that faces U.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks joined by a common link\n\nOracle formalization for mech_t1_mech_two_cranks_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(35)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 23.2, description=\"the two fixed pins are 23.2 apart\")\nscene.constraint.eq(U.distance(W), 11.1, description=\"the bar at U is 11.1 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 22.7, description=\"the bar at V is 22.7 long, ending at pin X\")\nscene.constraint.eq(W.x, U.x, description=\"the bar at U points straight up, so W is directly above U\")\nscene.constraint.gt(W.y - U.y, 0.05, description=\"the bar at U points UP, not down\")\nscene.constraint.eq(X.y, W.y, description=\"pin X is exactly level with pin W\")\nscene.constraint.lt(X.x - V.x, -0.05, description=\"pin X lies on the side of V towards U\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [23.2, 0.0], "W": [0.0, 11.1], "X": [3.398989924753838, 11.1]}, "circles": {}}, "implicit_facts": ["'points straight up' means W is directly above U, at U plus the bar length in the y-direction", "'level with' means X has the same height as W", "with its height fixed and its distance from V fixed, X is determined up to a left/right choice", "the length of the third bar then follows from the two positions"], "branch_facts": ["X lies on the side of V facing U, which picks one of the two possibilities"]} {"unique_id": "mech_t1_mech_two_cranks_0070", "problem_id": "mech_t1_mech_two_cranks_0070", "domain": "PyGeoX-Mech", "template": "mech_two_cranks", "tier": 1, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 23.3 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 10.2 mm long and ends at a pin W; the bar at V is 20 mm long and ends at a pin X. A third bar joins W to X. In the position of interest the bar at U points straight up, and pin X is exactly level with pin W, lying on the side of V that faces U.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks joined by a common link\n\nOracle formalization for mech_t1_mech_two_cranks_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(35)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 23.3, description=\"the two fixed pins are 23.3 apart\")\nscene.constraint.eq(U.distance(W), 10.2, description=\"the bar at U is 10.2 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 20, description=\"the bar at V is 20 long, ending at pin X\")\nscene.constraint.eq(W.x, U.x, description=\"the bar at U points straight up, so W is directly above U\")\nscene.constraint.gt(W.y - U.y, 0.05, description=\"the bar at U points UP, not down\")\nscene.constraint.eq(X.y, W.y, description=\"pin X is exactly level with pin W\")\nscene.constraint.lt(X.x - V.x, -0.05, description=\"pin X lies on the side of V towards U\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [23.3, 0.0], "W": [0.0, 10.2], "X": [6.096511981577688, 10.2]}, "circles": {}}, "implicit_facts": ["'points straight up' means W is directly above U, at U plus the bar length in the y-direction", "'level with' means X has the same height as W", "with its height fixed and its distance from V fixed, X is determined up to a left/right choice", "the length of the third bar then follows from the two positions"], "branch_facts": ["X lies on the side of V facing U, which picks one of the two possibilities"]} {"unique_id": "mech_t1_mech_two_cranks_0071", "problem_id": "mech_t1_mech_two_cranks_0071", "domain": "PyGeoX-Mech", "template": "mech_two_cranks", "tier": 1, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 27.3 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 13.6 mm long and ends at a pin W; the bar at V is 26.3 mm long and ends at a pin X. A third bar joins W to X. In the position of interest the bar at U points straight up, and pin X is exactly level with pin W, lying on the side of V that faces U.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks joined by a common link\n\nOracle formalization for mech_t1_mech_two_cranks_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(40)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 27.3, description=\"the two fixed pins are 27.3 apart\")\nscene.constraint.eq(U.distance(W), 13.6, description=\"the bar at U is 13.6 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 26.3, description=\"the bar at V is 26.3 long, ending at pin X\")\nscene.constraint.eq(W.x, U.x, description=\"the bar at U points straight up, so W is directly above U\")\nscene.constraint.gt(W.y - U.y, 0.05, description=\"the bar at U points UP, not down\")\nscene.constraint.eq(X.y, W.y, description=\"pin X is exactly level with pin W\")\nscene.constraint.lt(X.x - V.x, -0.05, description=\"pin X lies on the side of V towards U\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [27.3, 0.0], "W": [0.0, 13.6], "X": [4.789335860530457, 13.6]}, "circles": {}}, "implicit_facts": ["'points straight up' means W is directly above U, at U plus the bar length in the y-direction", "'level with' means X has the same height as W", "with its height fixed and its distance from V fixed, X is determined up to a left/right choice", "the length of the third bar then follows from the two positions"], "branch_facts": ["X lies on the side of V facing U, which picks one of the two possibilities"]} {"unique_id": "mech_t1_mech_two_cranks_0072", "problem_id": "mech_t1_mech_two_cranks_0072", "domain": "PyGeoX-Mech", "template": "mech_two_cranks", "tier": 1, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 27.6 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 10.5 mm long and ends at a pin W; the bar at V is 16.3 mm long and ends at a pin X. A third bar joins W to X. In the position of interest the bar at U points straight up, and pin X is exactly level with pin W, lying on the side of V that faces U.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks joined by a common link\n\nOracle formalization for mech_t1_mech_two_cranks_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(41)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 27.6, description=\"the two fixed pins are 27.6 apart\")\nscene.constraint.eq(U.distance(W), 10.5, description=\"the bar at U is 10.5 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 16.3, description=\"the bar at V is 16.3 long, ending at pin X\")\nscene.constraint.eq(W.x, U.x, description=\"the bar at U points straight up, so W is directly above U\")\nscene.constraint.gt(W.y - U.y, 0.05, description=\"the bar at U points UP, not down\")\nscene.constraint.eq(X.y, W.y, description=\"pin X is exactly level with pin W\")\nscene.constraint.lt(X.x - V.x, -0.05, description=\"pin X lies on the side of V towards U\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [27.6, 0.0], "W": [0.0, 10.5], "X": [15.132442099593042, 10.5]}, "circles": {}}, "implicit_facts": ["'points straight up' means W is directly above U, at U plus the bar length in the y-direction", "'level with' means X has the same height as W", "with its height fixed and its distance from V fixed, X is determined up to a left/right choice", "the length of the third bar then follows from the two positions"], "branch_facts": ["X lies on the side of V facing U, which picks one of the two possibilities"]} {"unique_id": "mech_t2_mech_bell_crank_0041", "problem_id": "mech_t2_mech_bell_crank_0041", "domain": "PyGeoX-Mech", "template": "mech_bell_crank", "tier": 2, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 14.3 mm from F out to a pin G, the other arm runs 10.2 mm from F out to a pin H, and the two arms are set at a right angle to each other at F. A straight pull-rod 30.7 mm long connects pin G to a block whose centre is K; the block slides along a straight rail that runs 25 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block sits below the level of pin G.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with a pull-rod to a rail\n\nOracle formalization for mech_t2_mech_bell_crank_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(38)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(G), 14.3, description=\"the arm FG is 14.3 long\")\nscene.constraint.eq(F.distance(H), 10.2, description=\"the arm FH is 10.2 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 25, description=\"the block runs on a straight rail 25 to the right of F\")\nscene.constraint.eq(G.distance(K), 30.7, description=\"the pull-rod from G to the block is 30.7 long\")\nscene.constraint.lt(K.y - G.y, -0.05, description=\"the block sits below the level of pin G\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 14.3], "H": [10.2, 0.0], "K": [25.0, -3.518249072229292]}, "circles": {}}, "implicit_facts": ["'H is level with F' means H has the same height as F, so the arm FH lies along the datum direction", "the right angle at F then forces the arm FG to stand straight up", "the block lies on the vertical line at the stated offset from F", "the block is where a circle of the rod length centred at G cuts that vertical line - two crossings"], "branch_facts": ["the block sits below the level of G, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_bell_crank_0042", "problem_id": "mech_t2_mech_bell_crank_0042", "domain": "PyGeoX-Mech", "template": "mech_bell_crank", "tier": 2, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 18.9 mm from F out to a pin G, the other arm runs 9.4 mm from F out to a pin H, and the two arms are set at a right angle to each other at F. A straight pull-rod 31.4 mm long connects pin G to a block whose centre is K; the block slides along a straight rail that runs 21.7 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block sits below the level of pin G.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with a pull-rod to a rail\n\nOracle formalization for mech_t2_mech_bell_crank_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(33)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(G), 18.9, description=\"the arm FG is 18.9 long\")\nscene.constraint.eq(F.distance(H), 9.4, description=\"the arm FH is 9.4 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 21.7, description=\"the block runs on a straight rail 21.7 to the right of F\")\nscene.constraint.eq(G.distance(K), 31.4, description=\"the pull-rod from G to the block is 31.4 long\")\nscene.constraint.lt(K.y - G.y, -0.05, description=\"the block sits below the level of pin G\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 18.9], "H": [9.4, 0.0], "K": [21.7, -3.7951536676886164]}, "circles": {}}, "implicit_facts": ["'H is level with F' means H has the same height as F, so the arm FH lies along the datum direction", "the right angle at F then forces the arm FG to stand straight up", "the block lies on the vertical line at the stated offset from F", "the block is where a circle of the rod length centred at G cuts that vertical line - two crossings"], "branch_facts": ["the block sits below the level of G, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_bell_crank_0043", "problem_id": "mech_t2_mech_bell_crank_0043", "domain": "PyGeoX-Mech", "template": "mech_bell_crank", "tier": 2, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 16.2 mm from F out to a pin G, the other arm runs 12.8 mm from F out to a pin H, and the two arms are set at a right angle to each other at F. A straight pull-rod 28.9 mm long connects pin G to a block whose centre is K; the block slides along a straight rail that runs 22.7 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block sits below the level of pin G.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with a pull-rod to a rail\n\nOracle formalization for mech_t2_mech_bell_crank_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(35)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(G), 16.2, description=\"the arm FG is 16.2 long\")\nscene.constraint.eq(F.distance(H), 12.8, description=\"the arm FH is 12.8 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 22.7, description=\"the block runs on a straight rail 22.7 to the right of F\")\nscene.constraint.eq(G.distance(K), 28.9, description=\"the pull-rod from G to the block is 28.9 long\")\nscene.constraint.lt(K.y - G.y, -0.05, description=\"the block sits below the level of pin G\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 16.2], "H": [12.8, 0.0], "K": [22.7, -1.6863076122490952]}, "circles": {}}, "implicit_facts": ["'H is level with F' means H has the same height as F, so the arm FH lies along the datum direction", "the right angle at F then forces the arm FG to stand straight up", "the block lies on the vertical line at the stated offset from F", "the block is where a circle of the rod length centred at G cuts that vertical line - two crossings"], "branch_facts": ["the block sits below the level of G, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_bell_crank_0044", "problem_id": "mech_t2_mech_bell_crank_0044", "domain": "PyGeoX-Mech", "template": "mech_bell_crank", "tier": 2, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 15.7 mm from F out to a pin G, the other arm runs 12.8 mm from F out to a pin H, and the two arms are set at a right angle to each other at F. A straight pull-rod 28.5 mm long connects pin G to a block whose centre is K; the block slides along a straight rail that runs 18.9 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block sits below the level of pin G.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with a pull-rod to a rail\n\nOracle formalization for mech_t2_mech_bell_crank_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(30)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(G), 15.7, description=\"the arm FG is 15.7 long\")\nscene.constraint.eq(F.distance(H), 12.8, description=\"the arm FH is 12.8 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 18.9, description=\"the block runs on a straight rail 18.9 to the right of F\")\nscene.constraint.eq(G.distance(K), 28.5, description=\"the pull-rod from G to the block is 28.5 long\")\nscene.constraint.lt(K.y - G.y, -0.05, description=\"the block sits below the level of pin G\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 15.7], "H": [12.8, 0.0], "K": [18.9, -5.631666601557416]}, "circles": {}}, "implicit_facts": ["'H is level with F' means H has the same height as F, so the arm FH lies along the datum direction", "the right angle at F then forces the arm FG to stand straight up", "the block lies on the vertical line at the stated offset from F", "the block is where a circle of the rod length centred at G cuts that vertical line - two crossings"], "branch_facts": ["the block sits below the level of G, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_bell_crank_0045", "problem_id": "mech_t2_mech_bell_crank_0045", "domain": "PyGeoX-Mech", "template": "mech_bell_crank", "tier": 2, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 16.2 mm from F out to a pin G, the other arm runs 11.3 mm from F out to a pin H, and the two arms are set at a right angle to each other at F. A straight pull-rod 21.2 mm long connects pin G to a block whose centre is K; the block slides along a straight rail that runs 16.5 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block sits below the level of pin G.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with a pull-rod to a rail\n\nOracle formalization for mech_t2_mech_bell_crank_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(26)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(G), 16.2, description=\"the arm FG is 16.2 long\")\nscene.constraint.eq(F.distance(H), 11.3, description=\"the arm FH is 11.3 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 16.5, description=\"the block runs on a straight rail 16.5 to the right of F\")\nscene.constraint.eq(G.distance(K), 21.2, description=\"the pull-rod from G to the block is 21.2 long\")\nscene.constraint.lt(K.y - G.y, -0.05, description=\"the block sits below the level of pin G\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 16.2], "H": [11.3, 0.0], "K": [16.5, 2.888726582328795]}, "circles": {}}, "implicit_facts": ["'H is level with F' means H has the same height as F, so the arm FH lies along the datum direction", "the right angle at F then forces the arm FG to stand straight up", "the block lies on the vertical line at the stated offset from F", "the block is where a circle of the rod length centred at G cuts that vertical line - two crossings"], "branch_facts": ["the block sits below the level of G, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_bell_crank_0046", "problem_id": "mech_t2_mech_bell_crank_0046", "domain": "PyGeoX-Mech", "template": "mech_bell_crank", "tier": 2, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 16.2 mm from F out to a pin G, the other arm runs 12.5 mm from F out to a pin H, and the two arms are set at a right angle to each other at F. A straight pull-rod 30.8 mm long connects pin G to a block whose centre is K; the block slides along a straight rail that runs 20.4 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block sits below the level of pin G.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with a pull-rod to a rail\n\nOracle formalization for mech_t2_mech_bell_crank_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(32)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(G), 16.2, description=\"the arm FG is 16.2 long\")\nscene.constraint.eq(F.distance(H), 12.5, description=\"the arm FH is 12.5 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 20.4, description=\"the block runs on a straight rail 20.4 to the right of F\")\nscene.constraint.eq(G.distance(K), 30.8, description=\"the pull-rod from G to the block is 30.8 long\")\nscene.constraint.lt(K.y - G.y, -0.05, description=\"the block sits below the level of pin G\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 16.2], "H": [12.5, 0.0], "K": [20.4, -6.875528162969534]}, "circles": {}}, "implicit_facts": ["'H is level with F' means H has the same height as F, so the arm FH lies along the datum direction", "the right angle at F then forces the arm FG to stand straight up", "the block lies on the vertical line at the stated offset from F", "the block is where a circle of the rod length centred at G cuts that vertical line - two crossings"], "branch_facts": ["the block sits below the level of G, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_bell_crank_0047", "problem_id": "mech_t2_mech_bell_crank_0047", "domain": "PyGeoX-Mech", "template": "mech_bell_crank", "tier": 2, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 13.8 mm from F out to a pin G, the other arm runs 9.2 mm from F out to a pin H, and the two arms are set at a right angle to each other at F. A straight pull-rod 29.3 mm long connects pin G to a block whose centre is K; the block slides along a straight rail that runs 17.4 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block sits below the level of pin G.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with a pull-rod to a rail\n\nOracle formalization for mech_t2_mech_bell_crank_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(28)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(G), 13.8, description=\"the arm FG is 13.8 long\")\nscene.constraint.eq(F.distance(H), 9.2, description=\"the arm FH is 9.2 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 17.4, description=\"the block runs on a straight rail 17.4 to the right of F\")\nscene.constraint.eq(G.distance(K), 29.3, description=\"the pull-rod from G to the block is 29.3 long\")\nscene.constraint.lt(K.y - G.y, -0.05, description=\"the block sits below the level of pin G\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 13.8], "H": [9.2, 0.0], "K": [17.4, -9.773926274594142]}, "circles": {}}, "implicit_facts": ["'H is level with F' means H has the same height as F, so the arm FH lies along the datum direction", "the right angle at F then forces the arm FG to stand straight up", "the block lies on the vertical line at the stated offset from F", "the block is where a circle of the rod length centred at G cuts that vertical line - two crossings"], "branch_facts": ["the block sits below the level of G, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_bell_crank_0048", "problem_id": "mech_t2_mech_bell_crank_0048", "domain": "PyGeoX-Mech", "template": "mech_bell_crank", "tier": 2, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 18 mm from F out to a pin G, the other arm runs 11 mm from F out to a pin H, and the two arms are set at a right angle to each other at F. A straight pull-rod 27.8 mm long connects pin G to a block whose centre is K; the block slides along a straight rail that runs 23.5 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block sits below the level of pin G.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with a pull-rod to a rail\n\nOracle formalization for mech_t2_mech_bell_crank_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(36)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(G), 18, description=\"the arm FG is 18 long\")\nscene.constraint.eq(F.distance(H), 11, description=\"the arm FH is 11 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 23.5, description=\"the block runs on a straight rail 23.5 to the right of F\")\nscene.constraint.eq(G.distance(K), 27.8, description=\"the pull-rod from G to the block is 27.8 long\")\nscene.constraint.lt(K.y - G.y, -0.05, description=\"the block sits below the level of pin G\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 18.0], "H": [11.0, 0.0], "K": [23.5, 3.1477274466161234]}, "circles": {}}, "implicit_facts": ["'H is level with F' means H has the same height as F, so the arm FH lies along the datum direction", "the right angle at F then forces the arm FG to stand straight up", "the block lies on the vertical line at the stated offset from F", "the block is where a circle of the rod length centred at G cuts that vertical line - two crossings"], "branch_facts": ["the block sits below the level of G, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_rocker_two_positions_0057", "problem_id": "mech_t2_mech_rocker_two_positions_0057", "domain": "PyGeoX-Mech", "template": "mech_rocker_two_positions", "tier": 2, "nl_description": "A bar is pinned to the bench at a fixed pin M and swings in the plane; its free end is N. Working on the bench grid, the end N has to be able to reach both (21.9, 8.3) - call that position N1 - and (9.3, 23.2) - call that N2. The fixed pin M has to be mounted somewhere on the bench base line, which is the line at height 0. A second bar 28.3 mm long connects the end of the first bar to a pin that slides along a straight rail at height 29.6; call that sliding pin Q1 when the end is at N1 and Q2 when it is at N2. In both positions the sliding pin lies to the right of the bar end. Both bars are rigid, so neither changes length as the mechanism moves.", "answer_schema": {"points": ["M", "N1", "N2", "Q1", "Q2"], "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": "\"\"\"Rocker reaching two required positions\n\nOracle formalization for mech_t2_mech_rocker_two_positions_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(58)\n\nM, N1, N2, Q1, Q2 = scene.add.points([\"M\", \"N1\", \"N2\", \"Q1\", \"Q2\"])\n\nscene.constraint.eq(N1.x, 21.9, description=\"the first required position of the bar end (x)\")\nscene.constraint.eq(N1.y, 8.3, description=\"the first required position of the bar end (y)\")\nscene.constraint.eq(N2.x, 9.3, description=\"the second required position of the bar end (x)\")\nscene.constraint.eq(N2.y, 23.2, description=\"the second required position of the bar end (y)\")\nscene.constraint.eq(M.y, 0, description=\"the fixed pin M is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(M.distance(N1), M.distance(N2), description=\"the bar is rigid, so its end is the same distance from M in both positions\")\nscene.constraint.eq(Q1.y, 29.6, description=\"the sliding pin runs on a straight rail at height 29.6 (first position)\")\nscene.constraint.eq(Q2.y, 29.6, description=\"the sliding pin runs on a straight rail at height 29.6 (second position)\")\nscene.constraint.eq(N1.distance(Q1), 28.3, description=\"the connecting bar is 28.3 long, the same in both positions (first)\")\nscene.constraint.eq(N2.distance(Q2), 28.3, description=\"the connecting bar is 28.3 long, the same in both positions (second)\")\nscene.constraint.gt(Q1.x - N1.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the first position\")\nscene.constraint.gt(Q2.x - N2.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the second position\")\n", "possible_solution": {"points": {"M": [-3.025000000000004, 0.0], "N1": [21.9, 8.3], "N2": [9.3, 23.2], "Q1": [40.53330351816338, 29.6], "Q2": [36.866827891507576, 29.6]}, "circles": {}}, "implicit_facts": ["the bar is rigid and M is fixed, so both required positions are the SAME distance from M", "therefore M lies on the perpendicular bisector of the segment joining the two positions", "combined with 'M is on the base line', that pins M completely", "the connecting bar has the same length in both positions, so each sliding pin is that far from its bar end"], "branch_facts": ["the sliding pin is to the right of the bar end in each position, picking one crossing of each circle with the rail"]} {"unique_id": "mech_t2_mech_rocker_two_positions_0058", "problem_id": "mech_t2_mech_rocker_two_positions_0058", "domain": "PyGeoX-Mech", "template": "mech_rocker_two_positions", "tier": 2, "nl_description": "A bar is pinned to the bench at a fixed pin M and swings in the plane; its free end is N. Working on the bench grid, the end N has to be able to reach both (19.7, 7.9) - call that position N1 - and (7.5, 17.5) - call that N2. The fixed pin M has to be mounted somewhere on the bench base line, which is the line at height 0. A second bar 28.4 mm long connects the end of the first bar to a pin that slides along a straight rail at height 26.3; call that sliding pin Q1 when the end is at N1 and Q2 when it is at N2. In both positions the sliding pin lies to the right of the bar end. Both bars are rigid, so neither changes length as the mechanism moves.", "answer_schema": {"points": ["M", "N1", "N2", "Q1", "Q2"], "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": "\"\"\"Rocker reaching two required positions\n\nOracle formalization for mech_t2_mech_rocker_two_positions_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(59)\n\nM, N1, N2, Q1, Q2 = scene.add.points([\"M\", \"N1\", \"N2\", \"Q1\", \"Q2\"])\n\nscene.constraint.eq(N1.x, 19.7, description=\"the first required position of the bar end (x)\")\nscene.constraint.eq(N1.y, 7.9, description=\"the first required position of the bar end (y)\")\nscene.constraint.eq(N2.x, 7.5, description=\"the second required position of the bar end (x)\")\nscene.constraint.eq(N2.y, 17.5, description=\"the second required position of the bar end (y)\")\nscene.constraint.eq(M.y, 0, description=\"the fixed pin M is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(M.distance(N1), M.distance(N2), description=\"the bar is rigid, so its end is the same distance from M in both positions\")\nscene.constraint.eq(Q1.y, 26.3, description=\"the sliding pin runs on a straight rail at height 26.3 (first position)\")\nscene.constraint.eq(Q2.y, 26.3, description=\"the sliding pin runs on a straight rail at height 26.3 (second position)\")\nscene.constraint.eq(N1.distance(Q1), 28.4, description=\"the connecting bar is 28.4 long, the same in both positions (first)\")\nscene.constraint.eq(N2.distance(Q2), 28.4, description=\"the connecting bar is 28.4 long, the same in both positions (second)\")\nscene.constraint.gt(Q1.x - N1.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the first position\")\nscene.constraint.gt(Q2.x - N2.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the second position\")\n", "possible_solution": {"points": {"M": [3.60655737704918, 0.0], "N1": [19.7, 7.9], "N2": [7.5, 17.5], "Q1": [41.33330765278394, 26.3], "Q2": [34.50222213078027, 26.3]}, "circles": {}}, "implicit_facts": ["the bar is rigid and M is fixed, so both required positions are the SAME distance from M", "therefore M lies on the perpendicular bisector of the segment joining the two positions", "combined with 'M is on the base line', that pins M completely", "the connecting bar has the same length in both positions, so each sliding pin is that far from its bar end"], "branch_facts": ["the sliding pin is to the right of the bar end in each position, picking one crossing of each circle with the rail"]} {"unique_id": "mech_t2_mech_rocker_two_positions_0059", "problem_id": "mech_t2_mech_rocker_two_positions_0059", "domain": "PyGeoX-Mech", "template": "mech_rocker_two_positions", "tier": 2, "nl_description": "A bar is pinned to the bench at a fixed pin M and swings in the plane; its free end is N. Working on the bench grid, the end N has to be able to reach both (18.4, 10) - call that position N1 - and (8, 19.3) - call that N2. The fixed pin M has to be mounted somewhere on the bench base line, which is the line at height 0. A second bar 23.8 mm long connects the end of the first bar to a pin that slides along a straight rail at height 25.5; call that sliding pin Q1 when the end is at N1 and Q2 when it is at N2. In both positions the sliding pin lies to the right of the bar end. Both bars are rigid, so neither changes length as the mechanism moves.", "answer_schema": {"points": ["M", "N1", "N2", "Q1", "Q2"], "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": "\"\"\"Rocker reaching two required positions\n\nOracle formalization for mech_t2_mech_rocker_two_positions_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(52)\n\nM, N1, N2, Q1, Q2 = scene.add.points([\"M\", \"N1\", \"N2\", \"Q1\", \"Q2\"])\n\nscene.constraint.eq(N1.x, 18.4, description=\"the first required position of the bar end (x)\")\nscene.constraint.eq(N1.y, 10, description=\"the first required position of the bar end (y)\")\nscene.constraint.eq(N2.x, 8, description=\"the second required position of the bar end (x)\")\nscene.constraint.eq(N2.y, 19.3, description=\"the second required position of the bar end (y)\")\nscene.constraint.eq(M.y, 0, description=\"the fixed pin M is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(M.distance(N1), M.distance(N2), description=\"the bar is rigid, so its end is the same distance from M in both positions\")\nscene.constraint.eq(Q1.y, 25.5, description=\"the sliding pin runs on a straight rail at height 25.5 (first position)\")\nscene.constraint.eq(Q2.y, 25.5, description=\"the sliding pin runs on a straight rail at height 25.5 (second position)\")\nscene.constraint.eq(N1.distance(Q1), 23.8, description=\"the connecting bar is 23.8 long, the same in both positions (first)\")\nscene.constraint.eq(N2.distance(Q2), 23.8, description=\"the connecting bar is 23.8 long, the same in both positions (second)\")\nscene.constraint.gt(Q1.x - N1.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the first position\")\nscene.constraint.gt(Q2.x - N2.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the second position\")\n", "possible_solution": {"points": {"M": [0.09951923076922853, 0.0], "N1": [18.4, 10.0], "N2": [8.0, 19.3], "Q1": [36.46073088222069, 25.5], "Q2": [30.978250586152118, 25.5]}, "circles": {}}, "implicit_facts": ["the bar is rigid and M is fixed, so both required positions are the SAME distance from M", "therefore M lies on the perpendicular bisector of the segment joining the two positions", "combined with 'M is on the base line', that pins M completely", "the connecting bar has the same length in both positions, so each sliding pin is that far from its bar end"], "branch_facts": ["the sliding pin is to the right of the bar end in each position, picking one crossing of each circle with the rail"]} {"unique_id": "mech_t2_mech_rocker_two_positions_0060", "problem_id": "mech_t2_mech_rocker_two_positions_0060", "domain": "PyGeoX-Mech", "template": "mech_rocker_two_positions", "tier": 2, "nl_description": "A bar is pinned to the bench at a fixed pin M and swings in the plane; its free end is N. Working on the bench grid, the end N has to be able to reach both (21.7, 11) - call that position N1 - and (10.3, 22.7) - call that N2. The fixed pin M has to be mounted somewhere on the bench base line, which is the line at height 0. A second bar 24.3 mm long connects the end of the first bar to a pin that slides along a straight rail at height 33.3; call that sliding pin Q1 when the end is at N1 and Q2 when it is at N2. In both positions the sliding pin lies to the right of the bar end. Both bars are rigid, so neither changes length as the mechanism moves.", "answer_schema": {"points": ["M", "N1", "N2", "Q1", "Q2"], "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": "\"\"\"Rocker reaching two required positions\n\nOracle formalization for mech_t2_mech_rocker_two_positions_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(48)\n\nM, N1, N2, Q1, Q2 = scene.add.points([\"M\", \"N1\", \"N2\", \"Q1\", \"Q2\"])\n\nscene.constraint.eq(N1.x, 21.7, description=\"the first required position of the bar end (x)\")\nscene.constraint.eq(N1.y, 11, description=\"the first required position of the bar end (y)\")\nscene.constraint.eq(N2.x, 10.3, description=\"the second required position of the bar end (x)\")\nscene.constraint.eq(N2.y, 22.7, description=\"the second required position of the bar end (y)\")\nscene.constraint.eq(M.y, 0, description=\"the fixed pin M is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(M.distance(N1), M.distance(N2), description=\"the bar is rigid, so its end is the same distance from M in both positions\")\nscene.constraint.eq(Q1.y, 33.3, description=\"the sliding pin runs on a straight rail at height 33.3 (first position)\")\nscene.constraint.eq(Q2.y, 33.3, description=\"the sliding pin runs on a straight rail at height 33.3 (second position)\")\nscene.constraint.eq(N1.distance(Q1), 24.3, description=\"the connecting bar is 24.3 long, the same in both positions (first)\")\nscene.constraint.eq(N2.distance(Q2), 24.3, description=\"the connecting bar is 24.3 long, the same in both positions (second)\")\nscene.constraint.gt(Q1.x - N1.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the first position\")\nscene.constraint.gt(Q2.x - N2.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the second position\")\n", "possible_solution": {"points": {"M": [-1.293421052631583, 0.0], "N1": [21.7, 11.0], "N2": [10.3, 22.7], "Q1": [31.354014708917745, 33.3], "Q2": [32.16618393776106, 33.3]}, "circles": {}}, "implicit_facts": ["the bar is rigid and M is fixed, so both required positions are the SAME distance from M", "therefore M lies on the perpendicular bisector of the segment joining the two positions", "combined with 'M is on the base line', that pins M completely", "the connecting bar has the same length in both positions, so each sliding pin is that far from its bar end"], "branch_facts": ["the sliding pin is to the right of the bar end in each position, picking one crossing of each circle with the rail"]} {"unique_id": "mech_t2_mech_rocker_two_positions_0061", "problem_id": "mech_t2_mech_rocker_two_positions_0061", "domain": "PyGeoX-Mech", "template": "mech_rocker_two_positions", "tier": 2, "nl_description": "A bar is pinned to the bench at a fixed pin M and swings in the plane; its free end is N. Working on the bench grid, the end N has to be able to reach both (21.3, 10.9) - call that position N1 - and (7.1, 17.4) - call that N2. The fixed pin M has to be mounted somewhere on the bench base line, which is the line at height 0. A second bar 29.7 mm long connects the end of the first bar to a pin that slides along a straight rail at height 23.5; call that sliding pin Q1 when the end is at N1 and Q2 when it is at N2. In both positions the sliding pin lies to the right of the bar end. Both bars are rigid, so neither changes length as the mechanism moves.", "answer_schema": {"points": ["M", "N1", "N2", "Q1", "Q2"], "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": "\"\"\"Rocker reaching two required positions\n\nOracle formalization for mech_t2_mech_rocker_two_positions_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(68)\n\nM, N1, N2, Q1, Q2 = scene.add.points([\"M\", \"N1\", \"N2\", \"Q1\", \"Q2\"])\n\nscene.constraint.eq(N1.x, 21.3, description=\"the first required position of the bar end (x)\")\nscene.constraint.eq(N1.y, 10.9, description=\"the first required position of the bar end (y)\")\nscene.constraint.eq(N2.x, 7.1, description=\"the second required position of the bar end (x)\")\nscene.constraint.eq(N2.y, 17.4, description=\"the second required position of the bar end (y)\")\nscene.constraint.eq(M.y, 0, description=\"the fixed pin M is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(M.distance(N1), M.distance(N2), description=\"the bar is rigid, so its end is the same distance from M in both positions\")\nscene.constraint.eq(Q1.y, 23.5, description=\"the sliding pin runs on a straight rail at height 23.5 (first position)\")\nscene.constraint.eq(Q2.y, 23.5, description=\"the sliding pin runs on a straight rail at height 23.5 (second position)\")\nscene.constraint.eq(N1.distance(Q1), 29.7, description=\"the connecting bar is 29.7 long, the same in both positions (first)\")\nscene.constraint.eq(N2.distance(Q2), 29.7, description=\"the connecting bar is 29.7 long, the same in both positions (second)\")\nscene.constraint.gt(Q1.x - N1.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the first position\")\nscene.constraint.gt(Q2.x - N2.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the second position\")\n", "possible_solution": {"points": {"M": [7.722887323943664, 0.0], "N1": [21.3, 10.9], "N2": [7.1, 17.4], "Q1": [48.19479503547108, 23.5], "Q2": [36.166819571463265, 23.5]}, "circles": {}}, "implicit_facts": ["the bar is rigid and M is fixed, so both required positions are the SAME distance from M", "therefore M lies on the perpendicular bisector of the segment joining the two positions", "combined with 'M is on the base line', that pins M completely", "the connecting bar has the same length in both positions, so each sliding pin is that far from its bar end"], "branch_facts": ["the sliding pin is to the right of the bar end in each position, picking one crossing of each circle with the rail"]} {"unique_id": "mech_t2_mech_rocker_two_positions_0062", "problem_id": "mech_t2_mech_rocker_two_positions_0062", "domain": "PyGeoX-Mech", "template": "mech_rocker_two_positions", "tier": 2, "nl_description": "A bar is pinned to the bench at a fixed pin M and swings in the plane; its free end is N. Working on the bench grid, the end N has to be able to reach both (20.2, 8.9) - call that position N1 - and (10.9, 16.2) - call that N2. The fixed pin M has to be mounted somewhere on the bench base line, which is the line at height 0. A second bar 25.4 mm long connects the end of the first bar to a pin that slides along a straight rail at height 27.3; call that sliding pin Q1 when the end is at N1 and Q2 when it is at N2. In both positions the sliding pin lies to the right of the bar end. Both bars are rigid, so neither changes length as the mechanism moves.", "answer_schema": {"points": ["M", "N1", "N2", "Q1", "Q2"], "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": "\"\"\"Rocker reaching two required positions\n\nOracle formalization for mech_t2_mech_rocker_two_positions_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(54)\n\nM, N1, N2, Q1, Q2 = scene.add.points([\"M\", \"N1\", \"N2\", \"Q1\", \"Q2\"])\n\nscene.constraint.eq(N1.x, 20.2, description=\"the first required position of the bar end (x)\")\nscene.constraint.eq(N1.y, 8.9, description=\"the first required position of the bar end (y)\")\nscene.constraint.eq(N2.x, 10.9, description=\"the second required position of the bar end (x)\")\nscene.constraint.eq(N2.y, 16.2, description=\"the second required position of the bar end (y)\")\nscene.constraint.eq(M.y, 0, description=\"the fixed pin M is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(M.distance(N1), M.distance(N2), description=\"the bar is rigid, so its end is the same distance from M in both positions\")\nscene.constraint.eq(Q1.y, 27.3, description=\"the sliding pin runs on a straight rail at height 27.3 (first position)\")\nscene.constraint.eq(Q2.y, 27.3, description=\"the sliding pin runs on a straight rail at height 27.3 (second position)\")\nscene.constraint.eq(N1.distance(Q1), 25.4, description=\"the connecting bar is 25.4 long, the same in both positions (first)\")\nscene.constraint.eq(N2.distance(Q2), 25.4, description=\"the connecting bar is 25.4 long, the same in both positions (second)\")\nscene.constraint.gt(Q1.x - N1.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the first position\")\nscene.constraint.gt(Q2.x - N2.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the second position\")\n", "possible_solution": {"points": {"M": [5.698924731182796, 0.0], "N1": [20.2, 8.9], "N2": [10.9, 16.2], "Q1": [37.70999714448863, 27.3], "Q2": [33.74622507111405, 27.3]}, "circles": {}}, "implicit_facts": ["the bar is rigid and M is fixed, so both required positions are the SAME distance from M", "therefore M lies on the perpendicular bisector of the segment joining the two positions", "combined with 'M is on the base line', that pins M completely", "the connecting bar has the same length in both positions, so each sliding pin is that far from its bar end"], "branch_facts": ["the sliding pin is to the right of the bar end in each position, picking one crossing of each circle with the rail"]} {"unique_id": "mech_t2_mech_rocker_two_positions_0063", "problem_id": "mech_t2_mech_rocker_two_positions_0063", "domain": "PyGeoX-Mech", "template": "mech_rocker_two_positions", "tier": 2, "nl_description": "A bar is pinned to the bench at a fixed pin M and swings in the plane; its free end is N. Working on the bench grid, the end N has to be able to reach both (19.2, 9.6) - call that position N1 - and (11.6, 19.4) - call that N2. The fixed pin M has to be mounted somewhere on the bench base line, which is the line at height 0. A second bar 21 mm long connects the end of the first bar to a pin that slides along a straight rail at height 27.7; call that sliding pin Q1 when the end is at N1 and Q2 when it is at N2. In both positions the sliding pin lies to the right of the bar end. Both bars are rigid, so neither changes length as the mechanism moves.", "answer_schema": {"points": ["M", "N1", "N2", "Q1", "Q2"], "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": "\"\"\"Rocker reaching two required positions\n\nOracle formalization for mech_t2_mech_rocker_two_positions_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(45)\n\nM, N1, N2, Q1, Q2 = scene.add.points([\"M\", \"N1\", \"N2\", \"Q1\", \"Q2\"])\n\nscene.constraint.eq(N1.x, 19.2, description=\"the first required position of the bar end (x)\")\nscene.constraint.eq(N1.y, 9.6, description=\"the first required position of the bar end (y)\")\nscene.constraint.eq(N2.x, 11.6, description=\"the second required position of the bar end (x)\")\nscene.constraint.eq(N2.y, 19.4, description=\"the second required position of the bar end (y)\")\nscene.constraint.eq(M.y, 0, description=\"the fixed pin M is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(M.distance(N1), M.distance(N2), description=\"the bar is rigid, so its end is the same distance from M in both positions\")\nscene.constraint.eq(Q1.y, 27.7, description=\"the sliding pin runs on a straight rail at height 27.7 (first position)\")\nscene.constraint.eq(Q2.y, 27.7, description=\"the sliding pin runs on a straight rail at height 27.7 (second position)\")\nscene.constraint.eq(N1.distance(Q1), 21, description=\"the connecting bar is 21 long, the same in both positions (first)\")\nscene.constraint.eq(N2.distance(Q2), 21, description=\"the connecting bar is 21 long, the same in both positions (second)\")\nscene.constraint.gt(Q1.x - N1.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the first position\")\nscene.constraint.gt(Q2.x - N2.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the second position\")\n", "possible_solution": {"points": {"M": [-3.2973684210526315, 0.0], "N1": [19.2, 9.6], "N2": [11.6, 19.4], "Q1": [29.848474069086137, 27.7], "Q2": [30.89015292837255, 27.7]}, "circles": {}}, "implicit_facts": ["the bar is rigid and M is fixed, so both required positions are the SAME distance from M", "therefore M lies on the perpendicular bisector of the segment joining the two positions", "combined with 'M is on the base line', that pins M completely", "the connecting bar has the same length in both positions, so each sliding pin is that far from its bar end"], "branch_facts": ["the sliding pin is to the right of the bar end in each position, picking one crossing of each circle with the rail"]} {"unique_id": "mech_t2_mech_rocker_two_positions_0064", "problem_id": "mech_t2_mech_rocker_two_positions_0064", "domain": "PyGeoX-Mech", "template": "mech_rocker_two_positions", "tier": 2, "nl_description": "A bar is pinned to the bench at a fixed pin M and swings in the plane; its free end is N. Working on the bench grid, the end N has to be able to reach both (20.8, 11.2) - call that position N1 - and (10.5, 18.1) - call that N2. The fixed pin M has to be mounted somewhere on the bench base line, which is the line at height 0. A second bar 27.6 mm long connects the end of the first bar to a pin that slides along a straight rail at height 25.3; call that sliding pin Q1 when the end is at N1 and Q2 when it is at N2. In both positions the sliding pin lies to the right of the bar end. Both bars are rigid, so neither changes length as the mechanism moves.", "answer_schema": {"points": ["M", "N1", "N2", "Q1", "Q2"], "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": "\"\"\"Rocker reaching two required positions\n\nOracle formalization for mech_t2_mech_rocker_two_positions_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(63)\n\nM, N1, N2, Q1, Q2 = scene.add.points([\"M\", \"N1\", \"N2\", \"Q1\", \"Q2\"])\n\nscene.constraint.eq(N1.x, 20.8, description=\"the first required position of the bar end (x)\")\nscene.constraint.eq(N1.y, 11.2, description=\"the first required position of the bar end (y)\")\nscene.constraint.eq(N2.x, 10.5, description=\"the second required position of the bar end (x)\")\nscene.constraint.eq(N2.y, 18.1, description=\"the second required position of the bar end (y)\")\nscene.constraint.eq(M.y, 0, description=\"the fixed pin M is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(M.distance(N1), M.distance(N2), description=\"the bar is rigid, so its end is the same distance from M in both positions\")\nscene.constraint.eq(Q1.y, 25.3, description=\"the sliding pin runs on a straight rail at height 25.3 (first position)\")\nscene.constraint.eq(Q2.y, 25.3, description=\"the sliding pin runs on a straight rail at height 25.3 (second position)\")\nscene.constraint.eq(N1.distance(Q1), 27.6, description=\"the connecting bar is 27.6 long, the same in both positions (first)\")\nscene.constraint.eq(N2.distance(Q2), 27.6, description=\"the connecting bar is 27.6 long, the same in both positions (second)\")\nscene.constraint.gt(Q1.x - N1.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the first position\")\nscene.constraint.gt(Q2.x - N2.x, 0.05, description=\"the sliding pin lies to the right of the bar end in the second position\")\n", "possible_solution": {"points": {"M": [5.835922330097086, 0.0], "N1": [20.8, 11.2], "N2": [10.5, 18.1], "Q1": [44.526567387635325, 25.3], "Q2": [37.14432397340943, 25.3]}, "circles": {}}, "implicit_facts": ["the bar is rigid and M is fixed, so both required positions are the SAME distance from M", "therefore M lies on the perpendicular bisector of the segment joining the two positions", "combined with 'M is on the base line', that pins M completely", "the connecting bar has the same length in both positions, so each sliding pin is that far from its bar end"], "branch_facts": ["the sliding pin is to the right of the bar end in each position, picking one crossing of each circle with the rail"]} {"unique_id": "mech_t2_mech_three_rod_reach_0017", "problem_id": "mech_t2_mech_three_rod_reach_0017", "domain": "PyGeoX-Mech", "template": "mech_three_rod_reach", "tier": 2, "nl_description": "Three rigid rods are joined end to end by pins to make a chain. The first rod is 15.8 mm long, the second 14.2 mm and the third 11 mm. Set up coordinates with the base pin at the origin J0; the joint between the first and second rods is J1, the joint between the second and third is J2, and the far tip of the last rod is J3. The tip must sit exactly at (27.2, 15.8), and at that moment the last rod must point 35 degrees above the horizontal, running upwards from J2 to the tip. A construction point E, 10 mm to the right of the tip and level with it, is provided so that angle can be measured. The chain must fold with the middle joint J1 standing above the straight line from J0 to J2. A round post stands on the bench with its centre W at (8.9, 0.7) and a radius of 3.3 mm; no joint of the chain may come within 3 mm of the post.", "answer_schema": {"points": ["E", "J0", "J1", "J2", "J3", "W"], "circles": ["W"], "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-rod chain reaching a required pose\n\nOracle formalization for mech_t2_mech_three_rod_reach_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(53)\n\nJ0, J1, J2, J3, E, W = scene.add.points([\"J0\", \"J1\", \"J2\", \"J3\", \"E\", \"W\"])\n\nscene.constraint.eq(J0.x, 0, description=\"datum: J0 is the origin, the fixed base pin (x)\")\nscene.constraint.eq(J0.y, 0, description=\"datum: J0 is the origin, the fixed base pin (y)\")\nscene.constraint.eq(J3.x, 27.2, description=\"the tip of the chain must sit at the required point (x)\")\nscene.constraint.eq(J3.y, 15.8, description=\"the tip of the chain must sit at the required point (y)\")\nscene.constraint.eq(E.y, J3.y, description=\"construction point E is level with the tip\")\nscene.constraint.eq(E.x - J3.x, 10, description=\"construction point E is 10 mm from the tip in the datum direction, so the last rod's angle can be measured\")\nscene.constraint.eq(J0.distance(J1), 15.8, description=\"the first rod is 15.8 long\")\nscene.constraint.eq(J1.distance(J2), 14.2, description=\"the second rod is 14.2 long\")\nscene.constraint.eq(J2.distance(J3), 11, description=\"the third rod is 11 long\")\nscene.constraint.eq(scene.add.angle(J2, J3, E).value, 145, description=\"the last rod points 35 degrees above the datum direction, running up towards the tip\")\nscene.constraint.lt(cross(J3, E, J2), -0.05, description=\"the last rod comes up from below-left, so J2 is below the datum direction seen from the tip\")\nscene.constraint.gt(cross(J0, J2, J1), 0.05, description=\"the middle joint J1 stands above the straight line from the base pin to J2 (elbow up)\")\nscene.constraint.eq(W.x, 8.9, description=\"the post centre is fixed on the bench (x)\")\nscene.constraint.eq(W.y, 0.7, description=\"the post centre is fixed on the bench (y)\")\npost = scene.add.circle(W)\nscene.constraint.eq(post.radius, 3.3, description=\"the post is 3.3 in radius\")\nscene.constraint.geq(J1.distance(W) - 3.3, 3, description=\"joint J1 keeps at least 3 clear of the post\")\nscene.constraint.geq(J2.distance(W) - 3.3, 3, description=\"joint J2 keeps at least 3 clear of the post\")\n", "possible_solution": {"points": {"J0": [0.0, 0.0], "J1": [5.084672741663311, 14.959482046855303], "J2": [18.189327512821087, 9.490659200138495], "J3": [27.2, 15.8], "E": [37.2, 15.8], "W": [8.9, 0.7]}, "circles": {"W": 3.3}}, "implicit_facts": ["consecutive pins are exactly one rod length apart", "the stated direction of the last rod fixes J2 at the tip minus the rod length times (cos, sin) of that angle", "with J0 and J2 both known, J1 is where two circles cross - two solutions", "the clearance is measured from the post SURFACE, so each joint is at least (post radius + clearance) from W", "the clearance statements are inequalities, unlike the rod lengths"], "branch_facts": ["'elbow up' picks one of the two circle intersections for J1"]} {"unique_id": "mech_t2_mech_three_rod_reach_0018", "problem_id": "mech_t2_mech_three_rod_reach_0018", "domain": "PyGeoX-Mech", "template": "mech_three_rod_reach", "tier": 2, "nl_description": "Three rigid rods are joined end to end by pins to make a chain. The first rod is 17.4 mm long, the second 12.4 mm and the third 11 mm. Set up coordinates with the base pin at the origin J0; the joint between the first and second rods is J1, the joint between the second and third is J2, and the far tip of the last rod is J3. The tip must sit exactly at (21.2, 11.2), and at that moment the last rod must point 35 degrees above the horizontal, running upwards from J2 to the tip. A construction point E, 10 mm to the right of the tip and level with it, is provided so that angle can be measured. The chain must fold with the middle joint J1 standing above the straight line from J0 to J2. A round post stands on the bench with its centre W at (13.1, -6.7) and a radius of 4.4 mm; no joint of the chain may come within 2.4 mm of the post.", "answer_schema": {"points": ["E", "J0", "J1", "J2", "J3", "W"], "circles": ["W"], "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-rod chain reaching a required pose\n\nOracle formalization for mech_t2_mech_three_rod_reach_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(46)\n\nJ0, J1, J2, J3, E, W = scene.add.points([\"J0\", \"J1\", \"J2\", \"J3\", \"E\", \"W\"])\n\nscene.constraint.eq(J0.x, 0, description=\"datum: J0 is the origin, the fixed base pin (x)\")\nscene.constraint.eq(J0.y, 0, description=\"datum: J0 is the origin, the fixed base pin (y)\")\nscene.constraint.eq(J3.x, 21.2, description=\"the tip of the chain must sit at the required point (x)\")\nscene.constraint.eq(J3.y, 11.2, description=\"the tip of the chain must sit at the required point (y)\")\nscene.constraint.eq(E.y, J3.y, description=\"construction point E is level with the tip\")\nscene.constraint.eq(E.x - J3.x, 10, description=\"construction point E is 10 mm from the tip in the datum direction, so the last rod's angle can be measured\")\nscene.constraint.eq(J0.distance(J1), 17.4, description=\"the first rod is 17.4 long\")\nscene.constraint.eq(J1.distance(J2), 12.4, description=\"the second rod is 12.4 long\")\nscene.constraint.eq(J2.distance(J3), 11, description=\"the third rod is 11 long\")\nscene.constraint.eq(scene.add.angle(J2, J3, E).value, 145, description=\"the last rod points 35 degrees above the datum direction, running up towards the tip\")\nscene.constraint.lt(cross(J3, E, J2), -0.05, description=\"the last rod comes up from below-left, so J2 is below the datum direction seen from the tip\")\nscene.constraint.gt(cross(J0, J2, J1), 0.05, description=\"the middle joint J1 stands above the straight line from the base pin to J2 (elbow up)\")\nscene.constraint.eq(W.x, 13.1, description=\"the post centre is fixed on the bench (x)\")\nscene.constraint.eq(W.y, -6.7, description=\"the post centre is fixed on the bench (y)\")\npost = scene.add.circle(W)\nscene.constraint.eq(post.radius, 4.4, description=\"the post is 4.4 in radius\")\nscene.constraint.geq(J1.distance(W) - 4.4, 2.4, description=\"joint J1 keeps at least 2.4 clear of the post\")\nscene.constraint.geq(J2.distance(W) - 4.4, 2.4, description=\"joint J2 keeps at least 2.4 clear of the post\")\n", "possible_solution": {"points": {"J0": [0.0, 0.0], "J1": [6.753732820456977, 16.035806589937476], "J2": [12.18932751282109, 4.890659200138493], "J3": [21.2, 11.2], "E": [31.2, 11.2], "W": [13.1, -6.7]}, "circles": {"W": 4.4}}, "implicit_facts": ["consecutive pins are exactly one rod length apart", "the stated direction of the last rod fixes J2 at the tip minus the rod length times (cos, sin) of that angle", "with J0 and J2 both known, J1 is where two circles cross - two solutions", "the clearance is measured from the post SURFACE, so each joint is at least (post radius + clearance) from W", "the clearance statements are inequalities, unlike the rod lengths"], "branch_facts": ["'elbow up' picks one of the two circle intersections for J1"]} {"unique_id": "mech_t2_mech_three_rod_reach_0019", "problem_id": "mech_t2_mech_three_rod_reach_0019", "domain": "PyGeoX-Mech", "template": "mech_three_rod_reach", "tier": 2, "nl_description": "Three rigid rods are joined end to end by pins to make a chain. The first rod is 17.6 mm long, the second 13.5 mm and the third 7.3 mm. Set up coordinates with the base pin at the origin J0; the joint between the first and second rods is J1, the joint between the second and third is J2, and the far tip of the last rod is J3. The tip must sit exactly at (22.3, 17.8), and at that moment the last rod must point 35 degrees above the horizontal, running upwards from J2 to the tip. A construction point E, 10 mm to the right of the tip and level with it, is provided so that angle can be measured. The chain must fold with the middle joint J1 standing above the straight line from J0 to J2. A round post stands on the bench with its centre W at (17.5, -5.2) and a radius of 5.4 mm; no joint of the chain may come within 1.9 mm of the post.", "answer_schema": {"points": ["E", "J0", "J1", "J2", "J3", "W"], "circles": ["W"], "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-rod chain reaching a required pose\n\nOracle formalization for mech_t2_mech_three_rod_reach_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(47)\n\nJ0, J1, J2, J3, E, W = scene.add.points([\"J0\", \"J1\", \"J2\", \"J3\", \"E\", \"W\"])\n\nscene.constraint.eq(J0.x, 0, description=\"datum: J0 is the origin, the fixed base pin (x)\")\nscene.constraint.eq(J0.y, 0, description=\"datum: J0 is the origin, the fixed base pin (y)\")\nscene.constraint.eq(J3.x, 22.3, description=\"the tip of the chain must sit at the required point (x)\")\nscene.constraint.eq(J3.y, 17.8, description=\"the tip of the chain must sit at the required point (y)\")\nscene.constraint.eq(E.y, J3.y, description=\"construction point E is level with the tip\")\nscene.constraint.eq(E.x - J3.x, 10, description=\"construction point E is 10 mm from the tip in the datum direction, so the last rod's angle can be measured\")\nscene.constraint.eq(J0.distance(J1), 17.6, description=\"the first rod is 17.6 long\")\nscene.constraint.eq(J1.distance(J2), 13.5, description=\"the second rod is 13.5 long\")\nscene.constraint.eq(J2.distance(J3), 7.3, description=\"the third rod is 7.3 long\")\nscene.constraint.eq(scene.add.angle(J2, J3, E).value, 145, description=\"the last rod points 35 degrees above the datum direction, running up towards the tip\")\nscene.constraint.lt(cross(J3, E, J2), -0.05, description=\"the last rod comes up from below-left, so J2 is below the datum direction seen from the tip\")\nscene.constraint.gt(cross(J0, J2, J1), 0.05, description=\"the middle joint J1 stands above the straight line from the base pin to J2 (elbow up)\")\nscene.constraint.eq(W.x, 17.5, description=\"the post centre is fixed on the bench (x)\")\nscene.constraint.eq(W.y, -5.2, description=\"the post centre is fixed on the bench (y)\")\npost = scene.add.circle(W)\nscene.constraint.eq(post.radius, 5.4, description=\"the post is 5.4 in radius\")\nscene.constraint.geq(J1.distance(W) - 5.4, 1.9, description=\"joint J1 keeps at least 1.9 clear of the post\")\nscene.constraint.geq(J2.distance(W) - 5.4, 1.9, description=\"joint J2 keeps at least 1.9 clear of the post\")\n", "possible_solution": {"points": {"J0": [0.0, 0.0], "J1": [3.3284724155159333, 17.282397732350383], "J2": [16.32019007669036, 13.612892014637364], "J3": [22.3, 17.8], "E": [32.3, 17.8], "W": [17.5, -5.2]}, "circles": {"W": 5.4}}, "implicit_facts": ["consecutive pins are exactly one rod length apart", "the stated direction of the last rod fixes J2 at the tip minus the rod length times (cos, sin) of that angle", "with J0 and J2 both known, J1 is where two circles cross - two solutions", "the clearance is measured from the post SURFACE, so each joint is at least (post radius + clearance) from W", "the clearance statements are inequalities, unlike the rod lengths"], "branch_facts": ["'elbow up' picks one of the two circle intersections for J1"]} {"unique_id": "mech_t2_mech_three_rod_reach_0020", "problem_id": "mech_t2_mech_three_rod_reach_0020", "domain": "PyGeoX-Mech", "template": "mech_three_rod_reach", "tier": 2, "nl_description": "Three rigid rods are joined end to end by pins to make a chain. The first rod is 19.8 mm long, the second 14 mm and the third 11.2 mm. Set up coordinates with the base pin at the origin J0; the joint between the first and second rods is J1, the joint between the second and third is J2, and the far tip of the last rod is J3. The tip must sit exactly at (21.4, 12.1), and at that moment the last rod must point 30 degrees above the horizontal, running upwards from J2 to the tip. A construction point E, 10 mm to the right of the tip and level with it, is provided so that angle can be measured. The chain must fold with the middle joint J1 standing above the straight line from J0 to J2. A round post stands on the bench with its centre W at (8.1, -3) and a radius of 4.5 mm; no joint of the chain may come within 3 mm of the post.", "answer_schema": {"points": ["E", "J0", "J1", "J2", "J3", "W"], "circles": ["W"], "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-rod chain reaching a required pose\n\nOracle formalization for mech_t2_mech_three_rod_reach_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(46)\n\nJ0, J1, J2, J3, E, W = scene.add.points([\"J0\", \"J1\", \"J2\", \"J3\", \"E\", \"W\"])\n\nscene.constraint.eq(J0.x, 0, description=\"datum: J0 is the origin, the fixed base pin (x)\")\nscene.constraint.eq(J0.y, 0, description=\"datum: J0 is the origin, the fixed base pin (y)\")\nscene.constraint.eq(J3.x, 21.4, description=\"the tip of the chain must sit at the required point (x)\")\nscene.constraint.eq(J3.y, 12.1, description=\"the tip of the chain must sit at the required point (y)\")\nscene.constraint.eq(E.y, J3.y, description=\"construction point E is level with the tip\")\nscene.constraint.eq(E.x - J3.x, 10, description=\"construction point E is 10 mm from the tip in the datum direction, so the last rod's angle can be measured\")\nscene.constraint.eq(J0.distance(J1), 19.8, description=\"the first rod is 19.8 long\")\nscene.constraint.eq(J1.distance(J2), 14, description=\"the second rod is 14 long\")\nscene.constraint.eq(J2.distance(J3), 11.2, description=\"the third rod is 11.2 long\")\nscene.constraint.eq(scene.add.angle(J2, J3, E).value, 150, description=\"the last rod points 30 degrees above the datum direction, running up towards the tip\")\nscene.constraint.lt(cross(J3, E, J2), -0.05, description=\"the last rod comes up from below-left, so J2 is below the datum direction seen from the tip\")\nscene.constraint.gt(cross(J0, J2, J1), 0.05, description=\"the middle joint J1 stands above the straight line from the base pin to J2 (elbow up)\")\nscene.constraint.eq(W.x, 8.1, description=\"the post centre is fixed on the bench (x)\")\nscene.constraint.eq(W.y, -3, description=\"the post centre is fixed on the bench (y)\")\npost = scene.add.circle(W)\nscene.constraint.eq(post.radius, 4.5, description=\"the post is 4.5 in radius\")\nscene.constraint.geq(J1.distance(W) - 4.5, 3, description=\"joint J1 keeps at least 3 clear of the post\")\nscene.constraint.geq(J2.distance(W) - 4.5, 3, description=\"joint J2 keeps at least 3 clear of the post\")\n", "possible_solution": {"points": {"J0": [0.0, 0.0], "J1": [5.460129537992273, 19.03226170029049], "J2": [11.700515477614285, 6.500000000000001], "J3": [21.4, 12.1], "E": [31.4, 12.1], "W": [8.1, -3.0]}, "circles": {"W": 4.5}}, "implicit_facts": ["consecutive pins are exactly one rod length apart", "the stated direction of the last rod fixes J2 at the tip minus the rod length times (cos, sin) of that angle", "with J0 and J2 both known, J1 is where two circles cross - two solutions", "the clearance is measured from the post SURFACE, so each joint is at least (post radius + clearance) from W", "the clearance statements are inequalities, unlike the rod lengths"], "branch_facts": ["'elbow up' picks one of the two circle intersections for J1"]} {"unique_id": "mech_t2_mech_three_rod_reach_0021", "problem_id": "mech_t2_mech_three_rod_reach_0021", "domain": "PyGeoX-Mech", "template": "mech_three_rod_reach", "tier": 2, "nl_description": "Three rigid rods are joined end to end by pins to make a chain. The first rod is 16.8 mm long, the second 14.3 mm and the third 9.9 mm. Set up coordinates with the base pin at the origin J0; the joint between the first and second rods is J1, the joint between the second and third is J2, and the far tip of the last rod is J3. The tip must sit exactly at (24.4, 16.2), and at that moment the last rod must point 35 degrees above the horizontal, running upwards from J2 to the tip. A construction point E, 10 mm to the right of the tip and level with it, is provided so that angle can be measured. The chain must fold with the middle joint J1 standing above the straight line from J0 to J2. A round post stands on the bench with its centre W at (12.7, 1.1) and a radius of 4.3 mm; no joint of the chain may come within 2.9 mm of the post.", "answer_schema": {"points": ["E", "J0", "J1", "J2", "J3", "W"], "circles": ["W"], "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-rod chain reaching a required pose\n\nOracle formalization for mech_t2_mech_three_rod_reach_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(50)\n\nJ0, J1, J2, J3, E, W = scene.add.points([\"J0\", \"J1\", \"J2\", \"J3\", \"E\", \"W\"])\n\nscene.constraint.eq(J0.x, 0, description=\"datum: J0 is the origin, the fixed base pin (x)\")\nscene.constraint.eq(J0.y, 0, description=\"datum: J0 is the origin, the fixed base pin (y)\")\nscene.constraint.eq(J3.x, 24.4, description=\"the tip of the chain must sit at the required point (x)\")\nscene.constraint.eq(J3.y, 16.2, description=\"the tip of the chain must sit at the required point (y)\")\nscene.constraint.eq(E.y, J3.y, description=\"construction point E is level with the tip\")\nscene.constraint.eq(E.x - J3.x, 10, description=\"construction point E is 10 mm from the tip in the datum direction, so the last rod's angle can be measured\")\nscene.constraint.eq(J0.distance(J1), 16.8, description=\"the first rod is 16.8 long\")\nscene.constraint.eq(J1.distance(J2), 14.3, description=\"the second rod is 14.3 long\")\nscene.constraint.eq(J2.distance(J3), 9.9, description=\"the third rod is 9.9 long\")\nscene.constraint.eq(scene.add.angle(J2, J3, E).value, 145, description=\"the last rod points 35 degrees above the datum direction, running up towards the tip\")\nscene.constraint.lt(cross(J3, E, J2), -0.05, description=\"the last rod comes up from below-left, so J2 is below the datum direction seen from the tip\")\nscene.constraint.gt(cross(J0, J2, J1), 0.05, description=\"the middle joint J1 stands above the straight line from the base pin to J2 (elbow up)\")\nscene.constraint.eq(W.x, 12.7, description=\"the post centre is fixed on the bench (x)\")\nscene.constraint.eq(W.y, 1.1, description=\"the post centre is fixed on the bench (y)\")\npost = scene.add.circle(W)\nscene.constraint.eq(post.radius, 4.3, description=\"the post is 4.3 in radius\")\nscene.constraint.geq(J1.distance(W) - 4.3, 2.9, description=\"joint J1 keeps at least 2.9 clear of the post\")\nscene.constraint.geq(J2.distance(W) - 4.3, 2.9, description=\"joint J2 keeps at least 2.9 clear of the post\")\n", "possible_solution": {"points": {"J0": [0.0, 0.0], "J1": [3.2885835275942634, 16.474987659541227], "J2": [16.29039476153898, 10.521593280124643], "J3": [24.4, 16.2], "E": [34.4, 16.2], "W": [12.7, 1.1]}, "circles": {"W": 4.3}}, "implicit_facts": ["consecutive pins are exactly one rod length apart", "the stated direction of the last rod fixes J2 at the tip minus the rod length times (cos, sin) of that angle", "with J0 and J2 both known, J1 is where two circles cross - two solutions", "the clearance is measured from the post SURFACE, so each joint is at least (post radius + clearance) from W", "the clearance statements are inequalities, unlike the rod lengths"], "branch_facts": ["'elbow up' picks one of the two circle intersections for J1"]} {"unique_id": "mech_t2_mech_three_rod_reach_0022", "problem_id": "mech_t2_mech_three_rod_reach_0022", "domain": "PyGeoX-Mech", "template": "mech_three_rod_reach", "tier": 2, "nl_description": "Three rigid rods are joined end to end by pins to make a chain. The first rod is 18.2 mm long, the second 14.3 mm and the third 10.1 mm. Set up coordinates with the base pin at the origin J0; the joint between the first and second rods is J1, the joint between the second and third is J2, and the far tip of the last rod is J3. The tip must sit exactly at (21.9, 15.8), and at that moment the last rod must point 25 degrees above the horizontal, running upwards from J2 to the tip. A construction point E, 10 mm to the right of the tip and level with it, is provided so that angle can be measured. The chain must fold with the middle joint J1 standing above the straight line from J0 to J2. A round post stands on the bench with its centre W at (8.1, -5.9) and a radius of 3.7 mm; no joint of the chain may come within 1.9 mm of the post.", "answer_schema": {"points": ["E", "J0", "J1", "J2", "J3", "W"], "circles": ["W"], "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-rod chain reaching a required pose\n\nOracle formalization for mech_t2_mech_three_rod_reach_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(46)\n\nJ0, J1, J2, J3, E, W = scene.add.points([\"J0\", \"J1\", \"J2\", \"J3\", \"E\", \"W\"])\n\nscene.constraint.eq(J0.x, 0, description=\"datum: J0 is the origin, the fixed base pin (x)\")\nscene.constraint.eq(J0.y, 0, description=\"datum: J0 is the origin, the fixed base pin (y)\")\nscene.constraint.eq(J3.x, 21.9, description=\"the tip of the chain must sit at the required point (x)\")\nscene.constraint.eq(J3.y, 15.8, description=\"the tip of the chain must sit at the required point (y)\")\nscene.constraint.eq(E.y, J3.y, description=\"construction point E is level with the tip\")\nscene.constraint.eq(E.x - J3.x, 10, description=\"construction point E is 10 mm from the tip in the datum direction, so the last rod's angle can be measured\")\nscene.constraint.eq(J0.distance(J1), 18.2, description=\"the first rod is 18.2 long\")\nscene.constraint.eq(J1.distance(J2), 14.3, description=\"the second rod is 14.3 long\")\nscene.constraint.eq(J2.distance(J3), 10.1, description=\"the third rod is 10.1 long\")\nscene.constraint.eq(scene.add.angle(J2, J3, E).value, 155, description=\"the last rod points 25 degrees above the datum direction, running up towards the tip\")\nscene.constraint.lt(cross(J3, E, J2), -0.05, description=\"the last rod comes up from below-left, so J2 is below the datum direction seen from the tip\")\nscene.constraint.gt(cross(J0, J2, J1), 0.05, description=\"the middle joint J1 stands above the straight line from the base pin to J2 (elbow up)\")\nscene.constraint.eq(W.x, 8.1, description=\"the post centre is fixed on the bench (x)\")\nscene.constraint.eq(W.y, -5.9, description=\"the post centre is fixed on the bench (y)\")\npost = scene.add.circle(W)\nscene.constraint.eq(post.radius, 3.7, description=\"the post is 3.7 in radius\")\nscene.constraint.geq(J1.distance(W) - 3.7, 1.9, description=\"joint J1 keeps at least 1.9 clear of the post\")\nscene.constraint.geq(J2.distance(W) - 3.7, 1.9, description=\"joint J2 keeps at least 1.9 clear of the post\")\n", "possible_solution": {"points": {"J0": [0.0, 0.0], "J1": [0.09618311895187048, 18.19974584458884], "J2": [12.746291350929834, 11.531555556418937], "J3": [21.9, 15.8], "E": [31.9, 15.8], "W": [8.1, -5.9]}, "circles": {"W": 3.7}}, "implicit_facts": ["consecutive pins are exactly one rod length apart", "the stated direction of the last rod fixes J2 at the tip minus the rod length times (cos, sin) of that angle", "with J0 and J2 both known, J1 is where two circles cross - two solutions", "the clearance is measured from the post SURFACE, so each joint is at least (post radius + clearance) from W", "the clearance statements are inequalities, unlike the rod lengths"], "branch_facts": ["'elbow up' picks one of the two circle intersections for J1"]} {"unique_id": "mech_t2_mech_three_rod_reach_0023", "problem_id": "mech_t2_mech_three_rod_reach_0023", "domain": "PyGeoX-Mech", "template": "mech_three_rod_reach", "tier": 2, "nl_description": "Three rigid rods are joined end to end by pins to make a chain. The first rod is 14.9 mm long, the second 12.4 mm and the third 7.7 mm. Set up coordinates with the base pin at the origin J0; the joint between the first and second rods is J1, the joint between the second and third is J2, and the far tip of the last rod is J3. The tip must sit exactly at (28.6, 13.4), and at that moment the last rod must point 45 degrees above the horizontal, running upwards from J2 to the tip. A construction point E, 10 mm to the right of the tip and level with it, is provided so that angle can be measured. The chain must fold with the middle joint J1 standing above the straight line from J0 to J2. A round post stands on the bench with its centre W at (15.1, -6.3) and a radius of 4.9 mm; no joint of the chain may come within 2.8 mm of the post.", "answer_schema": {"points": ["E", "J0", "J1", "J2", "J3", "W"], "circles": ["W"], "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-rod chain reaching a required pose\n\nOracle formalization for mech_t2_mech_three_rod_reach_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(55)\n\nJ0, J1, J2, J3, E, W = scene.add.points([\"J0\", \"J1\", \"J2\", \"J3\", \"E\", \"W\"])\n\nscene.constraint.eq(J0.x, 0, description=\"datum: J0 is the origin, the fixed base pin (x)\")\nscene.constraint.eq(J0.y, 0, description=\"datum: J0 is the origin, the fixed base pin (y)\")\nscene.constraint.eq(J3.x, 28.6, description=\"the tip of the chain must sit at the required point (x)\")\nscene.constraint.eq(J3.y, 13.4, description=\"the tip of the chain must sit at the required point (y)\")\nscene.constraint.eq(E.y, J3.y, description=\"construction point E is level with the tip\")\nscene.constraint.eq(E.x - J3.x, 10, description=\"construction point E is 10 mm from the tip in the datum direction, so the last rod's angle can be measured\")\nscene.constraint.eq(J0.distance(J1), 14.9, description=\"the first rod is 14.9 long\")\nscene.constraint.eq(J1.distance(J2), 12.4, description=\"the second rod is 12.4 long\")\nscene.constraint.eq(J2.distance(J3), 7.7, description=\"the third rod is 7.7 long\")\nscene.constraint.eq(scene.add.angle(J2, J3, E).value, 135, description=\"the last rod points 45 degrees above the datum direction, running up towards the tip\")\nscene.constraint.lt(cross(J3, E, J2), -0.05, description=\"the last rod comes up from below-left, so J2 is below the datum direction seen from the tip\")\nscene.constraint.gt(cross(J0, J2, J1), 0.05, description=\"the middle joint J1 stands above the straight line from the base pin to J2 (elbow up)\")\nscene.constraint.eq(W.x, 15.1, description=\"the post centre is fixed on the bench (x)\")\nscene.constraint.eq(W.y, -6.3, description=\"the post centre is fixed on the bench (y)\")\npost = scene.add.circle(W)\nscene.constraint.eq(post.radius, 4.9, description=\"the post is 4.9 in radius\")\nscene.constraint.geq(J1.distance(W) - 4.9, 2.8, description=\"joint J1 keeps at least 2.8 clear of the post\")\nscene.constraint.geq(J2.distance(W) - 4.9, 2.8, description=\"joint J2 keeps at least 2.8 clear of the post\")\n", "possible_solution": {"points": {"J0": [0.0, 0.0], "J1": [10.944131954531294, 10.111180730350291], "J2": [23.155277784863586, 7.955277784863584], "J3": [28.6, 13.4], "E": [38.6, 13.4], "W": [15.1, -6.3]}, "circles": {"W": 4.9}}, "implicit_facts": ["consecutive pins are exactly one rod length apart", "the stated direction of the last rod fixes J2 at the tip minus the rod length times (cos, sin) of that angle", "with J0 and J2 both known, J1 is where two circles cross - two solutions", "the clearance is measured from the post SURFACE, so each joint is at least (post radius + clearance) from W", "the clearance statements are inequalities, unlike the rod lengths"], "branch_facts": ["'elbow up' picks one of the two circle intersections for J1"]} {"unique_id": "mech_t2_mech_three_rod_reach_0024", "problem_id": "mech_t2_mech_three_rod_reach_0024", "domain": "PyGeoX-Mech", "template": "mech_three_rod_reach", "tier": 2, "nl_description": "Three rigid rods are joined end to end by pins to make a chain. The first rod is 14.5 mm long, the second 15.6 mm and the third 9.7 mm. Set up coordinates with the base pin at the origin J0; the joint between the first and second rods is J1, the joint between the second and third is J2, and the far tip of the last rod is J3. The tip must sit exactly at (28.7, 13.9), and at that moment the last rod must point 35 degrees above the horizontal, running upwards from J2 to the tip. A construction point E, 10 mm to the right of the tip and level with it, is provided so that angle can be measured. The chain must fold with the middle joint J1 standing above the straight line from J0 to J2. A round post stands on the bench with its centre W at (13.9, -2.4) and a radius of 4.3 mm; no joint of the chain may come within 2.6 mm of the post.", "answer_schema": {"points": ["E", "J0", "J1", "J2", "J3", "W"], "circles": ["W"], "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-rod chain reaching a required pose\n\nOracle formalization for mech_t2_mech_three_rod_reach_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(55)\n\nJ0, J1, J2, J3, E, W = scene.add.points([\"J0\", \"J1\", \"J2\", \"J3\", \"E\", \"W\"])\n\nscene.constraint.eq(J0.x, 0, description=\"datum: J0 is the origin, the fixed base pin (x)\")\nscene.constraint.eq(J0.y, 0, description=\"datum: J0 is the origin, the fixed base pin (y)\")\nscene.constraint.eq(J3.x, 28.7, description=\"the tip of the chain must sit at the required point (x)\")\nscene.constraint.eq(J3.y, 13.9, description=\"the tip of the chain must sit at the required point (y)\")\nscene.constraint.eq(E.y, J3.y, description=\"construction point E is level with the tip\")\nscene.constraint.eq(E.x - J3.x, 10, description=\"construction point E is 10 mm from the tip in the datum direction, so the last rod's angle can be measured\")\nscene.constraint.eq(J0.distance(J1), 14.5, description=\"the first rod is 14.5 long\")\nscene.constraint.eq(J1.distance(J2), 15.6, description=\"the second rod is 15.6 long\")\nscene.constraint.eq(J2.distance(J3), 9.7, description=\"the third rod is 9.7 long\")\nscene.constraint.eq(scene.add.angle(J2, J3, E).value, 145, description=\"the last rod points 35 degrees above the datum direction, running up towards the tip\")\nscene.constraint.lt(cross(J3, E, J2), -0.05, description=\"the last rod comes up from below-left, so J2 is below the datum direction seen from the tip\")\nscene.constraint.gt(cross(J0, J2, J1), 0.05, description=\"the middle joint J1 stands above the straight line from the base pin to J2 (elbow up)\")\nscene.constraint.eq(W.x, 13.9, description=\"the post centre is fixed on the bench (x)\")\nscene.constraint.eq(W.y, -2.4, description=\"the post centre is fixed on the bench (y)\")\npost = scene.add.circle(W)\nscene.constraint.eq(post.radius, 4.3, description=\"the post is 4.3 in radius\")\nscene.constraint.geq(J1.distance(W) - 4.3, 2.6, description=\"joint J1 keeps at least 2.6 clear of the post\")\nscene.constraint.geq(J2.distance(W) - 4.3, 2.6, description=\"joint J2 keeps at least 2.6 clear of the post\")\n", "possible_solution": {"points": {"J0": [0.0, 0.0], "J1": [5.940731590408513, 13.22715797783947], "J2": [20.75422517039678, 8.336308567394855], "J3": [28.7, 13.9], "E": [38.7, 13.9], "W": [13.9, -2.4]}, "circles": {"W": 4.3}}, "implicit_facts": ["consecutive pins are exactly one rod length apart", "the stated direction of the last rod fixes J2 at the tip minus the rod length times (cos, sin) of that angle", "with J0 and J2 both known, J1 is where two circles cross - two solutions", "the clearance is measured from the post SURFACE, so each joint is at least (post radius + clearance) from W", "the clearance statements are inequalities, unlike the rod lengths"], "branch_facts": ["'elbow up' picks one of the two circle intersections for J1"]} {"unique_id": "mech_t2_mech_toggle_clamp_0049", "problem_id": "mech_t2_mech_toggle_clamp_0049", "domain": "PyGeoX-Mech", "template": "mech_toggle_clamp", "tier": 2, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 10.5 mm long, ending at a pin K. The second runs from K to a clamping pad at L and is 22.4 mm long. Set up coordinates with J at the origin. The pad at L presses on a flat workpiece surface, which is the straight line 8.7 mm below J, and the pad lies to the right of J. In the clamped position the two bars are very nearly straight: the angle at pin K, measured between the two bars, is 172 degrees. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp near its straight position\n\nOracle formalization for mech_t2_mech_toggle_clamp_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(46)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.y, -8.7, description=\"the clamping pad presses on a flat workpiece surface 8.7 below J\")\nscene.constraint.gt(L.x, 1.0, description=\"datum: the pad lies in the +x direction from J\")\nscene.constraint.eq(J.distance(K), 10.5, description=\"the first bar JK is 10.5 long\")\nscene.constraint.eq(K.distance(L), 22.4, description=\"the second bar KL is 22.4 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 172, description=\"in the clamped position the angle at pin K, between the two bars, is 172 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [10.343079673427573, -1.8085084653186945], "L": [31.656628050631586, -8.7]}, "circles": {}}, "implicit_facts": ["the workpiece surface is the horizontal line at the stated depth, so L's height is fixed", "'very nearly straight' means J, K and L are close to collinear but not exactly so", "the distance from J to the pad follows from the cosine rule on the angle at K", "K is then where two circles cross - two solutions"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_toggle_clamp_0050", "problem_id": "mech_t2_mech_toggle_clamp_0050", "domain": "PyGeoX-Mech", "template": "mech_toggle_clamp", "tier": 2, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 12 mm long, ending at a pin K. The second runs from K to a clamping pad at L and is 22.8 mm long. Set up coordinates with J at the origin. The pad at L presses on a flat workpiece surface, which is the straight line 13.4 mm below J, and the pad lies to the right of J. In the clamped position the two bars are very nearly straight: the angle at pin K, measured between the two bars, is 172 degrees. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp near its straight position\n\nOracle formalization for mech_t2_mech_toggle_clamp_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(47)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.y, -13.4, description=\"the clamping pad presses on a flat workpiece surface 13.4 below J\")\nscene.constraint.gt(L.x, 1.0, description=\"datum: the pad lies in the +x direction from J\")\nscene.constraint.eq(J.distance(K), 12, description=\"the first bar JK is 12 long\")\nscene.constraint.eq(K.distance(L), 22.8, description=\"the second bar KL is 22.8 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 172, description=\"in the clamped position the angle at pin K, between the two bars, is 172 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [11.447317911054645, -3.5998489750609783], "L": [32.03364929594172, -13.4]}, "circles": {}}, "implicit_facts": ["the workpiece surface is the horizontal line at the stated depth, so L's height is fixed", "'very nearly straight' means J, K and L are close to collinear but not exactly so", "the distance from J to the pad follows from the cosine rule on the angle at K", "K is then where two circles cross - two solutions"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_toggle_clamp_0051", "problem_id": "mech_t2_mech_toggle_clamp_0051", "domain": "PyGeoX-Mech", "template": "mech_toggle_clamp", "tier": 2, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 13.6 mm long, ending at a pin K. The second runs from K to a clamping pad at L and is 18.4 mm long. Set up coordinates with J at the origin. The pad at L presses on a flat workpiece surface, which is the straight line 13.1 mm below J, and the pad lies to the right of J. In the clamped position the two bars are very nearly straight: the angle at pin K, measured between the two bars, is 172 degrees. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp near its straight position\n\nOracle formalization for mech_t2_mech_toggle_clamp_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\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.y, -13.1, description=\"the clamping pad presses on a flat workpiece surface 13.1 below J\")\nscene.constraint.gt(L.x, 1.0, description=\"datum: the pad lies in the +x direction from J\")\nscene.constraint.eq(J.distance(K), 13.6, description=\"the first bar JK is 13.6 long\")\nscene.constraint.eq(K.distance(L), 18.4, description=\"the second bar KL is 18.4 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 172, description=\"in the clamped position the angle at pin K, between the two bars, is 172 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [12.809909872859505, -4.567954580467781], "L": [29.11218581700421, -13.1]}, "circles": {}}, "implicit_facts": ["the workpiece surface is the horizontal line at the stated depth, so L's height is fixed", "'very nearly straight' means J, K and L are close to collinear but not exactly so", "the distance from J to the pad follows from the cosine rule on the angle at K", "K is then where two circles cross - two solutions"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_toggle_clamp_0052", "problem_id": "mech_t2_mech_toggle_clamp_0052", "domain": "PyGeoX-Mech", "template": "mech_toggle_clamp", "tier": 2, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 15.9 mm long, ending at a pin K. The second runs from K to a clamping pad at L and is 25.2 mm long. Set up coordinates with J at the origin. The pad at L presses on a flat workpiece surface, which is the straight line 8.9 mm below J, and the pad lies to the right of J. In the clamped position the two bars are very nearly straight: the angle at pin K, measured between the two bars, is 172 degrees. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp near its straight position\n\nOracle formalization for mech_t2_mech_toggle_clamp_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(57)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.y, -8.9, description=\"the clamping pad presses on a flat workpiece surface 8.9 below J\")\nscene.constraint.gt(L.x, 1.0, description=\"datum: the pad lies in the +x direction from J\")\nscene.constraint.eq(J.distance(K), 15.9, description=\"the first bar JK is 15.9 long\")\nscene.constraint.eq(K.distance(L), 25.2, description=\"the second bar KL is 25.2 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 172, description=\"in the clamped position the angle at pin K, between the two bars, is 172 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [15.759256331386457, -2.110886041844952], "L": [40.027505787479996, -8.9]}, "circles": {}}, "implicit_facts": ["the workpiece surface is the horizontal line at the stated depth, so L's height is fixed", "'very nearly straight' means J, K and L are close to collinear but not exactly so", "the distance from J to the pad follows from the cosine rule on the angle at K", "K is then where two circles cross - two solutions"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_toggle_clamp_0053", "problem_id": "mech_t2_mech_toggle_clamp_0053", "domain": "PyGeoX-Mech", "template": "mech_toggle_clamp", "tier": 2, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 14.7 mm long, ending at a pin K. The second runs from K to a clamping pad at L and is 19.6 mm long. Set up coordinates with J at the origin. The pad at L presses on a flat workpiece surface, which is the straight line 11.8 mm below J, and the pad lies to the right of J. In the clamped position the two bars are very nearly straight: the angle at pin K, measured between the two bars, is 170 degrees. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp near its straight position\n\nOracle formalization for mech_t2_mech_toggle_clamp_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(47)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.y, -11.8, description=\"the clamping pad presses on a flat workpiece surface 11.8 below J\")\nscene.constraint.gt(L.x, 1.0, description=\"datum: the pad lies in the +x direction from J\")\nscene.constraint.eq(J.distance(K), 14.7, description=\"the first bar JK is 14.7 long\")\nscene.constraint.eq(K.distance(L), 19.6, description=\"the second bar KL is 19.6 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 170, description=\"in the clamped position the angle at pin K, between the two bars, is 170 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [14.232753818591199, -3.6767810295661665], "L": [32.070167127655495, -11.8]}, "circles": {}}, "implicit_facts": ["the workpiece surface is the horizontal line at the stated depth, so L's height is fixed", "'very nearly straight' means J, K and L are close to collinear but not exactly so", "the distance from J to the pad follows from the cosine rule on the angle at K", "K is then where two circles cross - two solutions"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_toggle_clamp_0054", "problem_id": "mech_t2_mech_toggle_clamp_0054", "domain": "PyGeoX-Mech", "template": "mech_toggle_clamp", "tier": 2, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 11.1 mm long, ending at a pin K. The second runs from K to a clamping pad at L and is 22.6 mm long. Set up coordinates with J at the origin. The pad at L presses on a flat workpiece surface, which is the straight line 12 mm below J, and the pad lies to the right of J. In the clamped position the two bars are very nearly straight: the angle at pin K, measured between the two bars, is 170 degrees. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp near its straight position\n\nOracle formalization for mech_t2_mech_toggle_clamp_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(46)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.y, -12, description=\"the clamping pad presses on a flat workpiece surface 12 below J\")\nscene.constraint.gt(L.x, 1.0, description=\"datum: the pad lies in the +x direction from J\")\nscene.constraint.eq(J.distance(K), 11.1, description=\"the first bar JK is 11.1 long\")\nscene.constraint.eq(K.distance(L), 22.6, description=\"the second bar KL is 22.6 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 170, description=\"in the clamped position the angle at pin K, between the two bars, is 170 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [10.759729145034703, -2.7273116296988795], "L": [31.369854093401283, -12.0]}, "circles": {}}, "implicit_facts": ["the workpiece surface is the horizontal line at the stated depth, so L's height is fixed", "'very nearly straight' means J, K and L are close to collinear but not exactly so", "the distance from J to the pad follows from the cosine rule on the angle at K", "K is then where two circles cross - two solutions"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_toggle_clamp_0055", "problem_id": "mech_t2_mech_toggle_clamp_0055", "domain": "PyGeoX-Mech", "template": "mech_toggle_clamp", "tier": 2, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 11.1 mm long, ending at a pin K. The second runs from K to a clamping pad at L and is 24.2 mm long. Set up coordinates with J at the origin. The pad at L presses on a flat workpiece surface, which is the straight line 7.1 mm below J, and the pad lies to the right of J. In the clamped position the two bars are very nearly straight: the angle at pin K, measured between the two bars, is 168 degrees. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp near its straight position\n\nOracle formalization for mech_t2_mech_toggle_clamp_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(50)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.y, -7.1, description=\"the clamping pad presses on a flat workpiece surface 7.1 below J\")\nscene.constraint.gt(L.x, 1.0, description=\"datum: the pad lies in the +x direction from J\")\nscene.constraint.eq(J.distance(K), 11.1, description=\"the first bar JK is 11.1 long\")\nscene.constraint.eq(K.distance(L), 24.2, description=\"the second bar KL is 24.2 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 168, description=\"in the clamped position the angle at pin K, between the two bars, is 168 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [11.08016942211392, -0.6632085473300882], "L": [34.40842944713155, -7.1]}, "circles": {}}, "implicit_facts": ["the workpiece surface is the horizontal line at the stated depth, so L's height is fixed", "'very nearly straight' means J, K and L are close to collinear but not exactly so", "the distance from J to the pad follows from the cosine rule on the angle at K", "K is then where two circles cross - two solutions"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_toggle_clamp_0056", "problem_id": "mech_t2_mech_toggle_clamp_0056", "domain": "PyGeoX-Mech", "template": "mech_toggle_clamp", "tier": 2, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 15.8 mm long, ending at a pin K. The second runs from K to a clamping pad at L and is 21.8 mm long. Set up coordinates with J at the origin. The pad at L presses on a flat workpiece surface, which is the straight line 7.3 mm below J, and the pad lies to the right of J. In the clamped position the two bars are very nearly straight: the angle at pin K, measured between the two bars, is 170 degrees. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp near its straight position\n\nOracle formalization for mech_t2_mech_toggle_clamp_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(53)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.y, -7.3, description=\"the clamping pad presses on a flat workpiece surface 7.3 below J\")\nscene.constraint.gt(L.x, 1.0, description=\"datum: the pad lies in the +x direction from J\")\nscene.constraint.eq(J.distance(K), 15.8, description=\"the first bar JK is 15.8 long\")\nscene.constraint.eq(K.distance(L), 21.8, description=\"the second bar KL is 21.8 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 170, description=\"in the clamped position the angle at pin K, between the two bars, is 170 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [15.728905917870012, -1.4971702063538699], "L": [36.74240554039773, -7.3]}, "circles": {}}, "implicit_facts": ["the workpiece surface is the horizontal line at the stated depth, so L's height is fixed", "'very nearly straight' means J, K and L are close to collinear but not exactly so", "the distance from J to the pad follows from the cosine rule on the angle at K", "K is then where two circles cross - two solutions"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two crossings"]} {"unique_id": "mech_t2_mech_walking_beam_0073", "problem_id": "mech_t2_mech_walking_beam_0073", "domain": "PyGeoX-Mech", "template": "mech_walking_beam", "tier": 2, "nl_description": "A straight beam 36 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 and E2. One end, E1, is connected by a rod 23.1 mm long down to a block whose centre is K; the block slides on a straight vertical rail 23.6 mm to the left of the pivot, and it hangs below the beam end it is connected to. Arrange the beam so that the other end, E2, sits 7.2 mm above the height of the pivot and lies on the right of it.", "answer_schema": {"points": ["E1", "E2", "K", "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": "\"\"\"Walking beam rocking about its midpoint\n\nOracle formalization for mech_t2_mech_walking_beam_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(43)\n\nZ, E1, E2, K = scene.add.points([\"Z\", \"E1\", \"E2\", \"K\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 36, description=\"the beam is 36 long overall\")\nscene.constraint.eq(E2.y - Z.y, 7.2, description=\"the far end E2 sits 7.2 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"the far end E2 is on the right of the pivot\")\nscene.constraint.eq(K.x - Z.x, -23.6, description=\"the block runs on a straight rail 23.6 to the left of the pivot\")\nscene.constraint.eq(E1.distance(K), 23.1, description=\"the rod from the beam end E1 down to the block is 23.1 long\")\nscene.constraint.lt(K.y - E1.y, -0.05, description=\"the block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-16.497272501841024, -7.2], "E2": [16.497272501841024, 7.2], "K": [-23.6, -29.180929509165356]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length away from it", "the two ends are diametrically opposite about the pivot: whatever offset one end has, the other has the exact opposite", "so if E2 is dy above the pivot then E1 is dy BELOW it, and their x-offsets are opposite too", "the block is where a circle of the rod length centred at E1 cuts the rail line"], "branch_facts": ["E2 is on the right of the pivot, fixing which way the beam is tilted", "the block hangs below E1, picking one of the two crossings with the rail"]} {"unique_id": "mech_t2_mech_walking_beam_0074", "problem_id": "mech_t2_mech_walking_beam_0074", "domain": "PyGeoX-Mech", "template": "mech_walking_beam", "tier": 2, "nl_description": "A straight beam 30.7 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 and E2. One end, E1, is connected by a rod 21.4 mm long down to a block whose centre is K; the block slides on a straight vertical rail 19 mm to the left of the pivot, and it hangs below the beam end it is connected to. Arrange the beam so that the other end, E2, sits 10.5 mm above the height of the pivot and lies on the right of it.", "answer_schema": {"points": ["E1", "E2", "K", "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": "\"\"\"Walking beam rocking about its midpoint\n\nOracle formalization for mech_t2_mech_walking_beam_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(45)\n\nZ, E1, E2, K = scene.add.points([\"Z\", \"E1\", \"E2\", \"K\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 30.7, description=\"the beam is 30.7 long overall\")\nscene.constraint.eq(E2.y - Z.y, 10.5, description=\"the far end E2 sits 10.5 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"the far end E2 is on the right of the pivot\")\nscene.constraint.eq(K.x - Z.x, -19, description=\"the block runs on a straight rail 19 to the left of the pivot\")\nscene.constraint.eq(E1.distance(K), 21.4, description=\"the rod from the beam end E1 down to the block is 21.4 long\")\nscene.constraint.lt(K.y - E1.y, -0.05, description=\"the block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-11.196986201652658, -10.5], "E2": [11.196986201652658, 10.5], "K": [-19.0, -30.426690032787707]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length away from it", "the two ends are diametrically opposite about the pivot: whatever offset one end has, the other has the exact opposite", "so if E2 is dy above the pivot then E1 is dy BELOW it, and their x-offsets are opposite too", "the block is where a circle of the rod length centred at E1 cuts the rail line"], "branch_facts": ["E2 is on the right of the pivot, fixing which way the beam is tilted", "the block hangs below E1, picking one of the two crossings with the rail"]} {"unique_id": "mech_t2_mech_walking_beam_0075", "problem_id": "mech_t2_mech_walking_beam_0075", "domain": "PyGeoX-Mech", "template": "mech_walking_beam", "tier": 2, "nl_description": "A straight beam 31.2 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 and E2. One end, E1, is connected by a rod 23.2 mm long down to a block whose centre is K; the block slides on a straight vertical rail 23 mm to the left of the pivot, and it hangs below the beam end it is connected to. Arrange the beam so that the other end, E2, sits 6.4 mm above the height of the pivot and lies on the right of it.", "answer_schema": {"points": ["E1", "E2", "K", "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": "\"\"\"Walking beam rocking about its midpoint\n\nOracle formalization for mech_t2_mech_walking_beam_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(41)\n\nZ, E1, E2, K = scene.add.points([\"Z\", \"E1\", \"E2\", \"K\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 31.2, description=\"the beam is 31.2 long overall\")\nscene.constraint.eq(E2.y - Z.y, 6.4, description=\"the far end E2 sits 6.4 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"the far end E2 is on the right of the pivot\")\nscene.constraint.eq(K.x - Z.x, -23, description=\"the block runs on a straight rail 23 to the left of the pivot\")\nscene.constraint.eq(E1.distance(K), 23.2, description=\"the rod from the beam end E1 down to the block is 23.2 long\")\nscene.constraint.lt(K.y - E1.y, -0.05, description=\"the block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-14.226735395022992, -6.4], "E2": [14.226735395022992, 6.4], "K": [-23.0, -27.8771932097995]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length away from it", "the two ends are diametrically opposite about the pivot: whatever offset one end has, the other has the exact opposite", "so if E2 is dy above the pivot then E1 is dy BELOW it, and their x-offsets are opposite too", "the block is where a circle of the rod length centred at E1 cuts the rail line"], "branch_facts": ["E2 is on the right of the pivot, fixing which way the beam is tilted", "the block hangs below E1, picking one of the two crossings with the rail"]} {"unique_id": "mech_t2_mech_walking_beam_0076", "problem_id": "mech_t2_mech_walking_beam_0076", "domain": "PyGeoX-Mech", "template": "mech_walking_beam", "tier": 2, "nl_description": "A straight beam 31.9 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 and E2. One end, E1, is connected by a rod 18.7 mm long down to a block whose centre is K; the block slides on a straight vertical rail 20.5 mm to the left of the pivot, and it hangs below the beam end it is connected to. Arrange the beam so that the other end, E2, sits 10.4 mm above the height of the pivot and lies on the right of it.", "answer_schema": {"points": ["E1", "E2", "K", "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": "\"\"\"Walking beam rocking about its midpoint\n\nOracle formalization for mech_t2_mech_walking_beam_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(40)\n\nZ, E1, E2, K = scene.add.points([\"Z\", \"E1\", \"E2\", \"K\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 31.9, description=\"the beam is 31.9 long overall\")\nscene.constraint.eq(E2.y - Z.y, 10.4, description=\"the far end E2 sits 10.4 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"the far end E2 is on the right of the pivot\")\nscene.constraint.eq(K.x - Z.x, -20.5, description=\"the block runs on a straight rail 20.5 to the left of the pivot\")\nscene.constraint.eq(E1.distance(K), 18.7, description=\"the rod from the beam end E1 down to the block is 18.7 long\")\nscene.constraint.lt(K.y - E1.y, -0.05, description=\"the block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-12.093076531635775, -10.4], "E2": [12.093076531635775, 10.4], "K": [-20.5, -27.103701320278297]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length away from it", "the two ends are diametrically opposite about the pivot: whatever offset one end has, the other has the exact opposite", "so if E2 is dy above the pivot then E1 is dy BELOW it, and their x-offsets are opposite too", "the block is where a circle of the rod length centred at E1 cuts the rail line"], "branch_facts": ["E2 is on the right of the pivot, fixing which way the beam is tilted", "the block hangs below E1, picking one of the two crossings with the rail"]} {"unique_id": "mech_t2_mech_walking_beam_0077", "problem_id": "mech_t2_mech_walking_beam_0077", "domain": "PyGeoX-Mech", "template": "mech_walking_beam", "tier": 2, "nl_description": "A straight beam 37.4 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 and E2. One end, E1, is connected by a rod 16.6 mm long down to a block whose centre is K; the block slides on a straight vertical rail 22.8 mm to the left of the pivot, and it hangs below the beam end it is connected to. Arrange the beam so that the other end, E2, sits 8.6 mm above the height of the pivot and lies on the right of it.", "answer_schema": {"points": ["E1", "E2", "K", "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": "\"\"\"Walking beam rocking about its midpoint\n\nOracle formalization for mech_t2_mech_walking_beam_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(36)\n\nZ, E1, E2, K = scene.add.points([\"Z\", \"E1\", \"E2\", \"K\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 37.4, description=\"the beam is 37.4 long overall\")\nscene.constraint.eq(E2.y - Z.y, 8.6, description=\"the far end E2 sits 8.6 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"the far end E2 is on the right of the pivot\")\nscene.constraint.eq(K.x - Z.x, -22.8, description=\"the block runs on a straight rail 22.8 to the left of the pivot\")\nscene.constraint.eq(E1.distance(K), 16.6, description=\"the rod from the beam end E1 down to the block is 16.6 long\")\nscene.constraint.lt(K.y - E1.y, -0.05, description=\"the block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-16.60511969243221, -8.6], "E2": [16.60511969243221, 8.6], "K": [-22.8, -24.0007616037295]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length away from it", "the two ends are diametrically opposite about the pivot: whatever offset one end has, the other has the exact opposite", "so if E2 is dy above the pivot then E1 is dy BELOW it, and their x-offsets are opposite too", "the block is where a circle of the rod length centred at E1 cuts the rail line"], "branch_facts": ["E2 is on the right of the pivot, fixing which way the beam is tilted", "the block hangs below E1, picking one of the two crossings with the rail"]} {"unique_id": "mech_t2_mech_walking_beam_0078", "problem_id": "mech_t2_mech_walking_beam_0078", "domain": "PyGeoX-Mech", "template": "mech_walking_beam", "tier": 2, "nl_description": "A straight beam 30.8 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 and E2. One end, E1, is connected by a rod 20.2 mm long down to a block whose centre is K; the block slides on a straight vertical rail 27 mm to the left of the pivot, and it hangs below the beam end it is connected to. Arrange the beam so that the other end, E2, sits 8 mm above the height of the pivot and lies on the right of it.", "answer_schema": {"points": ["E1", "E2", "K", "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": "\"\"\"Walking beam rocking about its midpoint\n\nOracle formalization for mech_t2_mech_walking_beam_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(40)\n\nZ, E1, E2, K = scene.add.points([\"Z\", \"E1\", \"E2\", \"K\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 30.8, description=\"the beam is 30.8 long overall\")\nscene.constraint.eq(E2.y - Z.y, 8, description=\"the far end E2 sits 8 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"the far end E2 is on the right of the pivot\")\nscene.constraint.eq(K.x - Z.x, -27, description=\"the block runs on a straight rail 27 to the left of the pivot\")\nscene.constraint.eq(E1.distance(K), 20.2, description=\"the rod from the beam end E1 down to the block is 20.2 long\")\nscene.constraint.lt(K.y - E1.y, -0.05, description=\"the block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-13.159027319676786, -8.0], "E2": [13.159027319676786, 8.0], "K": [-27.0, -22.712833692479037]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length away from it", "the two ends are diametrically opposite about the pivot: whatever offset one end has, the other has the exact opposite", "so if E2 is dy above the pivot then E1 is dy BELOW it, and their x-offsets are opposite too", "the block is where a circle of the rod length centred at E1 cuts the rail line"], "branch_facts": ["E2 is on the right of the pivot, fixing which way the beam is tilted", "the block hangs below E1, picking one of the two crossings with the rail"]} {"unique_id": "mech_t2_mech_walking_beam_0079", "problem_id": "mech_t2_mech_walking_beam_0079", "domain": "PyGeoX-Mech", "template": "mech_walking_beam", "tier": 2, "nl_description": "A straight beam 30.9 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 and E2. One end, E1, is connected by a rod 16.3 mm long down to a block whose centre is K; the block slides on a straight vertical rail 20.5 mm to the left of the pivot, and it hangs below the beam end it is connected to. Arrange the beam so that the other end, E2, sits 7.6 mm above the height of the pivot and lies on the right of it.", "answer_schema": {"points": ["E1", "E2", "K", "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": "\"\"\"Walking beam rocking about its midpoint\n\nOracle formalization for mech_t2_mech_walking_beam_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(34)\n\nZ, E1, E2, K = scene.add.points([\"Z\", \"E1\", \"E2\", \"K\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 30.9, description=\"the beam is 30.9 long overall\")\nscene.constraint.eq(E2.y - Z.y, 7.6, description=\"the far end E2 sits 7.6 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"the far end E2 is on the right of the pivot\")\nscene.constraint.eq(K.x - Z.x, -20.5, description=\"the block runs on a straight rail 20.5 to the left of the pivot\")\nscene.constraint.eq(E1.distance(K), 16.3, description=\"the rod from the beam end E1 down to the block is 16.3 long\")\nscene.constraint.lt(K.y - E1.y, -0.05, description=\"the block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-13.451486906658312, -7.6], "E2": [13.451486906658312, 7.6], "K": [-20.5, -22.29722637687094]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length away from it", "the two ends are diametrically opposite about the pivot: whatever offset one end has, the other has the exact opposite", "so if E2 is dy above the pivot then E1 is dy BELOW it, and their x-offsets are opposite too", "the block is where a circle of the rod length centred at E1 cuts the rail line"], "branch_facts": ["E2 is on the right of the pivot, fixing which way the beam is tilted", "the block hangs below E1, picking one of the two crossings with the rail"]} {"unique_id": "mech_t3_mech_fourbar_three_positions_0001", "problem_id": "mech_t3_mech_fourbar_three_positions_0001", "domain": "PyGeoX-Mech", "template": "mech_fourbar_three_positions", "tier": 3, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid, so no bar changes length as the mechanism moves, and the two fixed pins never move. The fixed pin B sits at the same height as A and 24 mm to its right. Bar 2 is 14.5 mm long and bar 3 is 19.6315 mm long. As the mechanism is worked, pin C has to pass through three required positions in turn: C1 at (17.6852, 18.5506), C2 at (12.7465, 21.7798) and C3 at (6.9531, 22.8999), measured on the bench grid. Write down where the fixed pin A must be placed, and where pin D sits in each of the three positions - call those D1, D2 and D3. In every one of the three positions the pin D lies to the left of the straight line running from C to B.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar linkage through three required positions\n\nOracle formalization for mech_t3_mech_fourbar_three_positions_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(53)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 17.6852, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 18.5506, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 12.7465, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 21.7798, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 6.9531, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 22.8999, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.y, A.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.eq(B.x - A.x, 24, description=\"the fixed pin B is 24 to the right of the fixed pin A\")\nscene.constraint.eq(B.distance(D1), 14.5, description=\"bar BD is 14.5 long, the same in position 1 as in every other\")\nscene.constraint.eq(B.distance(D2), 14.5, description=\"bar BD is 14.5 long, the same in position 2 as in every other\")\nscene.constraint.eq(B.distance(D3), 14.5, description=\"bar BD is 14.5 long, the same in position 3 as in every other\")\nscene.constraint.eq(C1.distance(D1), 19.6315, description=\"the connecting bar CD is 19.6315 long, the same in position 1 as in every other\")\nscene.constraint.eq(C2.distance(D2), 19.6315, description=\"the connecting bar CD is 19.6315 long, the same in position 2 as in every other\")\nscene.constraint.eq(C3.distance(D3), 19.6315, description=\"the connecting bar CD is 19.6315 long, the same in position 3 as in every other\")\nscene.constraint.gt(cross(C1, B, D1), 0.05, description=\"in position 1 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C2, B, D2), 0.05, description=\"in position 2 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C3, B, D3), 0.05, description=\"in position 3 the pin D lies to the left of the line from C to B\")\n", "possible_solution": {"points": {"A": [7.0, 7.6], "B": [31.0, 7.6], "C1": [17.68519551758085, 18.55064366834526], "C2": [12.746508008552311, 21.779832358234852], "C3": [6.953091101517706, 22.899928089871636], "D1": [37.19769363405256, 20.708722043678563], "D2": [32.37633259279934, 22.034531810696123], "D3": [26.526623865364414, 21.39271205956511]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the FIXED pin A, so the distance AC cannot change", "therefore the three required positions of C all lie on one circle centred at A", "so A is the CIRCUMCENTRE of C1, C2, C3 and bar 1's length is that circumradius", "likewise D always lies on a circle of radius |BD| centred at B", "the three positions must share one set of bar lengths - they are three poses of the same mechanism"], "branch_facts": ["in each pose D is on the stated side of the line CB, which picks one of the two assemblies"]} {"unique_id": "mech_t3_mech_fourbar_three_positions_0002", "problem_id": "mech_t3_mech_fourbar_three_positions_0002", "domain": "PyGeoX-Mech", "template": "mech_fourbar_three_positions", "tier": 3, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid, so no bar changes length as the mechanism moves, and the two fixed pins never move. The fixed pin B sits at the same height as A and 21.3 mm to its right. Bar 2 is 16.3 mm long and bar 3 is 14.2472 mm long. As the mechanism is worked, pin C has to pass through three required positions in turn: C1 at (18.1529, 19.5684), C2 at (10.8607, 23.7195) and C3 at (2.4761, 24.0457), measured on the bench grid. Write down where the fixed pin A must be placed, and where pin D sits in each of the three positions - call those D1, D2 and D3. In every one of the three positions the pin D lies to the left of the straight line running from C to B.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar linkage through three required positions\n\nOracle formalization for mech_t3_mech_fourbar_three_positions_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\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 18.1529, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 19.5684, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 10.8607, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 23.7195, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 2.4761, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 24.0457, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.y, A.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.eq(B.x - A.x, 21.3, description=\"the fixed pin B is 21.3 to the right of the fixed pin A\")\nscene.constraint.eq(B.distance(D1), 16.3, description=\"bar BD is 16.3 long, the same in position 1 as in every other\")\nscene.constraint.eq(B.distance(D2), 16.3, description=\"bar BD is 16.3 long, the same in position 2 as in every other\")\nscene.constraint.eq(B.distance(D3), 16.3, description=\"bar BD is 16.3 long, the same in position 3 as in every other\")\nscene.constraint.eq(C1.distance(D1), 14.2472, description=\"the connecting bar CD is 14.2472 long, the same in position 1 as in every other\")\nscene.constraint.eq(C2.distance(D2), 14.2472, description=\"the connecting bar CD is 14.2472 long, the same in position 2 as in every other\")\nscene.constraint.eq(C3.distance(D3), 14.2472, description=\"the connecting bar CD is 14.2472 long, the same in position 3 as in every other\")\nscene.constraint.gt(cross(C1, B, D1), 0.05, description=\"in position 1 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C2, B, D2), 0.05, description=\"in position 2 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C3, B, D3), 0.05, description=\"in position 3 the pin D lies to the left of the line from C to B\")\n", "possible_solution": {"points": {"A": [6.0, 6.7], "B": [27.3, 6.7], "C1": [18.152938584086662, 19.568414190234073], "C2": [10.860706954869759, 23.71950433763806], "C3": [2.4760894010221004, 24.045663841156703], "D1": [32.142768866121386, 22.263983735192138], "D2": [25.081199684351816, 22.848279325032728], "D3": [15.183338721959633, 17.603050925004304]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the FIXED pin A, so the distance AC cannot change", "therefore the three required positions of C all lie on one circle centred at A", "so A is the CIRCUMCENTRE of C1, C2, C3 and bar 1's length is that circumradius", "likewise D always lies on a circle of radius |BD| centred at B", "the three positions must share one set of bar lengths - they are three poses of the same mechanism"], "branch_facts": ["in each pose D is on the stated side of the line CB, which picks one of the two assemblies"]} {"unique_id": "mech_t3_mech_fourbar_three_positions_0003", "problem_id": "mech_t3_mech_fourbar_three_positions_0003", "domain": "PyGeoX-Mech", "template": "mech_fourbar_three_positions", "tier": 3, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid, so no bar changes length as the mechanism moves, and the two fixed pins never move. The fixed pin B sits at the same height as A and 19.4 mm to its right. Bar 2 is 13.4 mm long and bar 3 is 20.2128 mm long. As the mechanism is worked, pin C has to pass through three required positions in turn: C1 at (15.1774, 21.7525), C2 at (6.6341, 23.9958) and C3 at (-1.7976, 21.3642), measured on the bench grid. Write down where the fixed pin A must be placed, and where pin D sits in each of the three positions - call those D1, D2 and D3. In every one of the three positions the pin D lies to the left of the straight line running from C to B.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar linkage through three required positions\n\nOracle formalization for mech_t3_mech_fourbar_three_positions_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(51)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 15.1774, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 21.7525, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 6.6341, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 23.9958, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, -1.7976, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 21.3642, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.y, A.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.eq(B.x - A.x, 19.4, description=\"the fixed pin B is 19.4 to the right of the fixed pin A\")\nscene.constraint.eq(B.distance(D1), 13.4, description=\"bar BD is 13.4 long, the same in position 1 as in every other\")\nscene.constraint.eq(B.distance(D2), 13.4, description=\"bar BD is 13.4 long, the same in position 2 as in every other\")\nscene.constraint.eq(B.distance(D3), 13.4, description=\"bar BD is 13.4 long, the same in position 3 as in every other\")\nscene.constraint.eq(C1.distance(D1), 20.2128, description=\"the connecting bar CD is 20.2128 long, the same in position 1 as in every other\")\nscene.constraint.eq(C2.distance(D2), 20.2128, description=\"the connecting bar CD is 20.2128 long, the same in position 2 as in every other\")\nscene.constraint.eq(C3.distance(D3), 20.2128, description=\"the connecting bar CD is 20.2128 long, the same in position 3 as in every other\")\nscene.constraint.gt(cross(C1, B, D1), 0.05, description=\"in position 1 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C2, B, D2), 0.05, description=\"in position 2 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C3, B, D3), 0.05, description=\"in position 3 the pin D lies to the left of the line from C to B\")\n", "possible_solution": {"points": {"A": [7.0, 8.0], "B": [26.4, 8.0], "C1": [15.177403918294162, 21.752456695335102], "C2": [6.634125491048222, 23.99581619810941], "C3": [-1.7976465795745895, 21.364184025255696], "D1": [35.077503361431965, 18.210824423734692], "D2": [26.679168463648715, 21.397091660838335], "D3": [18.2279582372153, 18.61968612659069]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the FIXED pin A, so the distance AC cannot change", "therefore the three required positions of C all lie on one circle centred at A", "so A is the CIRCUMCENTRE of C1, C2, C3 and bar 1's length is that circumradius", "likewise D always lies on a circle of radius |BD| centred at B", "the three positions must share one set of bar lengths - they are three poses of the same mechanism"], "branch_facts": ["in each pose D is on the stated side of the line CB, which picks one of the two assemblies"]} {"unique_id": "mech_t3_mech_fourbar_three_positions_0004", "problem_id": "mech_t3_mech_fourbar_three_positions_0004", "domain": "PyGeoX-Mech", "template": "mech_fourbar_three_positions", "tier": 3, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid, so no bar changes length as the mechanism moves, and the two fixed pins never move. The fixed pin B sits at the same height as A and 25.3 mm to its right. Bar 2 is 14 mm long and bar 3 is 21.4652 mm long. As the mechanism is worked, pin C has to pass through three required positions in turn: C1 at (19.1686, 13.2059), C2 at (13.7589, 17.7673) and C3 at (6.8498, 19.2959), measured on the bench grid. Write down where the fixed pin A must be placed, and where pin D sits in each of the three positions - call those D1, D2 and D3. In every one of the three positions the pin D lies to the left of the straight line running from C to B.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar linkage through three required positions\n\nOracle formalization for mech_t3_mech_fourbar_three_positions_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(58)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 19.1686, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 13.2059, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 13.7589, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 17.7673, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 6.8498, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 19.2959, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.y, A.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.eq(B.x - A.x, 25.3, description=\"the fixed pin B is 25.3 to the right of the fixed pin A\")\nscene.constraint.eq(B.distance(D1), 14, description=\"bar BD is 14 long, the same in position 1 as in every other\")\nscene.constraint.eq(B.distance(D2), 14, description=\"bar BD is 14 long, the same in position 2 as in every other\")\nscene.constraint.eq(B.distance(D3), 14, description=\"bar BD is 14 long, the same in position 3 as in every other\")\nscene.constraint.eq(C1.distance(D1), 21.4652, description=\"the connecting bar CD is 21.4652 long, the same in position 1 as in every other\")\nscene.constraint.eq(C2.distance(D2), 21.4652, description=\"the connecting bar CD is 21.4652 long, the same in position 2 as in every other\")\nscene.constraint.eq(C3.distance(D3), 21.4652, description=\"the connecting bar CD is 21.4652 long, the same in position 3 as in every other\")\nscene.constraint.gt(cross(C1, B, D1), 0.05, description=\"in position 1 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C2, B, D2), 0.05, description=\"in position 2 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C3, B, D3), 0.05, description=\"in position 3 the pin D lies to the left of the line from C to B\")\n", "possible_solution": {"points": {"A": [7.2, 4.5], "B": [32.5, 4.5], "C1": [19.168621436401132, 13.205865891002416], "C2": [13.75889988257427, 17.767284286181827], "C3": [6.84984133201389, 19.29585715351545], "D1": [40.447846860894586, 16.025264867922477], "D2": [35.21902944699987, 18.233421964912733], "D3": [28.265901949728065, 17.844377606343556]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the FIXED pin A, so the distance AC cannot change", "therefore the three required positions of C all lie on one circle centred at A", "so A is the CIRCUMCENTRE of C1, C2, C3 and bar 1's length is that circumradius", "likewise D always lies on a circle of radius |BD| centred at B", "the three positions must share one set of bar lengths - they are three poses of the same mechanism"], "branch_facts": ["in each pose D is on the stated side of the line CB, which picks one of the two assemblies"]} {"unique_id": "mech_t3_mech_fourbar_three_positions_0005", "problem_id": "mech_t3_mech_fourbar_three_positions_0005", "domain": "PyGeoX-Mech", "template": "mech_fourbar_three_positions", "tier": 3, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid, so no bar changes length as the mechanism moves, and the two fixed pins never move. The fixed pin B sits at the same height as A and 20.7 mm to its right. Bar 2 is 12.2 mm long and bar 3 is 16.4874 mm long. As the mechanism is worked, pin C has to pass through three required positions in turn: C1 at (19.1496, 18.2329), C2 at (13.4335, 22.2335) and C3 at (6.6121, 23.6987), measured on the bench grid. Write down where the fixed pin A must be placed, and where pin D sits in each of the three positions - call those D1, D2 and D3. In every one of the three positions the pin D lies to the left of the straight line running from C to B.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar linkage through three required positions\n\nOracle formalization for mech_t3_mech_fourbar_three_positions_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(51)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 19.1496, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 18.2329, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 13.4335, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 22.2335, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 6.6121, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 23.6987, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.y, A.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.eq(B.x - A.x, 20.7, description=\"the fixed pin B is 20.7 to the right of the fixed pin A\")\nscene.constraint.eq(B.distance(D1), 12.2, description=\"bar BD is 12.2 long, the same in position 1 as in every other\")\nscene.constraint.eq(B.distance(D2), 12.2, description=\"bar BD is 12.2 long, the same in position 2 as in every other\")\nscene.constraint.eq(B.distance(D3), 12.2, description=\"bar BD is 12.2 long, the same in position 3 as in every other\")\nscene.constraint.eq(C1.distance(D1), 16.4874, description=\"the connecting bar CD is 16.4874 long, the same in position 1 as in every other\")\nscene.constraint.eq(C2.distance(D2), 16.4874, description=\"the connecting bar CD is 16.4874 long, the same in position 2 as in every other\")\nscene.constraint.eq(C3.distance(D3), 16.4874, description=\"the connecting bar CD is 16.4874 long, the same in position 3 as in every other\")\nscene.constraint.gt(cross(C1, B, D1), 0.05, description=\"in position 1 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C2, B, D2), 0.05, description=\"in position 2 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C3, B, D3), 0.05, description=\"in position 3 the pin D lies to the left of the line from C to B\")\n", "possible_solution": {"points": {"A": [6.4, 6.1], "B": [27.1, 6.1], "C1": [19.149598073764153, 18.232920050732613], "C2": [13.433504826834675, 22.233499615734132], "C3": [6.612105949646856, 23.698721858877263], "D1": [35.33661244561562, 15.099900856161131], "D2": [29.390003252979007, 18.08315004918763], "D3": [21.693638156924646, 17.036692901500835]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the FIXED pin A, so the distance AC cannot change", "therefore the three required positions of C all lie on one circle centred at A", "so A is the CIRCUMCENTRE of C1, C2, C3 and bar 1's length is that circumradius", "likewise D always lies on a circle of radius |BD| centred at B", "the three positions must share one set of bar lengths - they are three poses of the same mechanism"], "branch_facts": ["in each pose D is on the stated side of the line CB, which picks one of the two assemblies"]} {"unique_id": "mech_t3_mech_fourbar_three_positions_0006", "problem_id": "mech_t3_mech_fourbar_three_positions_0006", "domain": "PyGeoX-Mech", "template": "mech_fourbar_three_positions", "tier": 3, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid, so no bar changes length as the mechanism moves, and the two fixed pins never move. The fixed pin B sits at the same height as A and 27.8 mm to its right. Bar 2 is 11.7 mm long and bar 3 is 26.0514 mm long. As the mechanism is worked, pin C has to pass through three required positions in turn: C1 at (19.0025, 19.1887), C2 at (13.3624, 21.6984) and C3 at (7.192, 21.886), measured on the bench grid. Write down where the fixed pin A must be placed, and where pin D sits in each of the three positions - call those D1, D2 and D3. In every one of the three positions the pin D lies to the left of the straight line running from C to B.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar linkage through three required positions\n\nOracle formalization for mech_t3_mech_fourbar_three_positions_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(63)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 19.0025, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 19.1887, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 13.3624, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 21.6984, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 7.192, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 21.886, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.y, A.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.eq(B.x - A.x, 27.8, description=\"the fixed pin B is 27.8 to the right of the fixed pin A\")\nscene.constraint.eq(B.distance(D1), 11.7, description=\"bar BD is 11.7 long, the same in position 1 as in every other\")\nscene.constraint.eq(B.distance(D2), 11.7, description=\"bar BD is 11.7 long, the same in position 2 as in every other\")\nscene.constraint.eq(B.distance(D3), 11.7, description=\"bar BD is 11.7 long, the same in position 3 as in every other\")\nscene.constraint.eq(C1.distance(D1), 26.0514, description=\"the connecting bar CD is 26.0514 long, the same in position 1 as in every other\")\nscene.constraint.eq(C2.distance(D2), 26.0514, description=\"the connecting bar CD is 26.0514 long, the same in position 2 as in every other\")\nscene.constraint.eq(C3.distance(D3), 26.0514, description=\"the connecting bar CD is 26.0514 long, the same in position 3 as in every other\")\nscene.constraint.gt(cross(C1, B, D1), 0.05, description=\"in position 1 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C2, B, D2), 0.05, description=\"in position 2 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C3, B, D3), 0.05, description=\"in position 3 the pin D lies to the left of the line from C to B\")\n", "possible_solution": {"points": {"A": [9.8, 6.1], "B": [37.6, 6.1], "C1": [19.002478724311896, 19.188712134071363], "C2": [13.362405226680044, 21.698373921692053], "C3": [7.192022744778736, 21.886020861390264], "D1": [44.76901113626498, 15.346365736228947], "D2": [39.105609723111314, 17.702721205030855], "D3": [32.729791811212635, 16.738189328916288]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the FIXED pin A, so the distance AC cannot change", "therefore the three required positions of C all lie on one circle centred at A", "so A is the CIRCUMCENTRE of C1, C2, C3 and bar 1's length is that circumradius", "likewise D always lies on a circle of radius |BD| centred at B", "the three positions must share one set of bar lengths - they are three poses of the same mechanism"], "branch_facts": ["in each pose D is on the stated side of the line CB, which picks one of the two assemblies"]} {"unique_id": "mech_t3_mech_fourbar_three_positions_0007", "problem_id": "mech_t3_mech_fourbar_three_positions_0007", "domain": "PyGeoX-Mech", "template": "mech_fourbar_three_positions", "tier": 3, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid, so no bar changes length as the mechanism moves, and the two fixed pins never move. The fixed pin B sits at the same height as A and 26.2 mm to its right. Bar 2 is 17.6 mm long and bar 3 is 20.5739 mm long. As the mechanism is worked, pin C has to pass through three required positions in turn: C1 at (20.0161, 13.5466), C2 at (15.3901, 17.228) and C3 at (9.6836, 18.7732), measured on the bench grid. Write down where the fixed pin A must be placed, and where pin D sits in each of the three positions - call those D1, D2 and D3. In every one of the three positions the pin D lies to the left of the straight line running from C to B.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar linkage through three required positions\n\nOracle formalization for mech_t3_mech_fourbar_three_positions_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(56)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 20.0161, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 13.5466, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 15.3901, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 17.228, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 9.6836, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 18.7732, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.y, A.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.eq(B.x - A.x, 26.2, description=\"the fixed pin B is 26.2 to the right of the fixed pin A\")\nscene.constraint.eq(B.distance(D1), 17.6, description=\"bar BD is 17.6 long, the same in position 1 as in every other\")\nscene.constraint.eq(B.distance(D2), 17.6, description=\"bar BD is 17.6 long, the same in position 2 as in every other\")\nscene.constraint.eq(B.distance(D3), 17.6, description=\"bar BD is 17.6 long, the same in position 3 as in every other\")\nscene.constraint.eq(C1.distance(D1), 20.5739, description=\"the connecting bar CD is 20.5739 long, the same in position 1 as in every other\")\nscene.constraint.eq(C2.distance(D2), 20.5739, description=\"the connecting bar CD is 20.5739 long, the same in position 2 as in every other\")\nscene.constraint.eq(C3.distance(D3), 20.5739, description=\"the connecting bar CD is 20.5739 long, the same in position 3 as in every other\")\nscene.constraint.gt(cross(C1, B, D1), 0.05, description=\"in position 1 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C2, B, D2), 0.05, description=\"in position 2 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C3, B, D3), 0.05, description=\"in position 3 the pin D lies to the left of the line from C to B\")\n", "possible_solution": {"points": {"A": [8.8, 4.2], "B": [35.0, 4.2], "C1": [20.01610827118541, 13.546599127439148], "C2": [15.39013994354019, 17.228048799592212], "C3": [9.683561343856685, 18.773239837168745], "D1": [39.06305674188423, 21.324589627556893], "D2": [35.450883447952634, 21.794223600840205], "D3": [30.12415045444051, 21.1111232982634]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the FIXED pin A, so the distance AC cannot change", "therefore the three required positions of C all lie on one circle centred at A", "so A is the CIRCUMCENTRE of C1, C2, C3 and bar 1's length is that circumradius", "likewise D always lies on a circle of radius |BD| centred at B", "the three positions must share one set of bar lengths - they are three poses of the same mechanism"], "branch_facts": ["in each pose D is on the stated side of the line CB, which picks one of the two assemblies"]} {"unique_id": "mech_t3_mech_fourbar_three_positions_0008", "problem_id": "mech_t3_mech_fourbar_three_positions_0008", "domain": "PyGeoX-Mech", "template": "mech_fourbar_three_positions", "tier": 3, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid, so no bar changes length as the mechanism moves, and the two fixed pins never move. The fixed pin B sits at the same height as A and 26.1 mm to its right. Bar 2 is 17.2 mm long and bar 3 is 26.4432 mm long. As the mechanism is worked, pin C has to pass through three required positions in turn: C1 at (18.4183, 22.5027), C2 at (9.1016, 26.6292) and C3 at (-0.9824, 25.1651), measured on the bench grid. Write down where the fixed pin A must be placed, and where pin D sits in each of the three positions - call those D1, D2 and D3. In every one of the three positions the pin D lies to the left of the straight line running from C to B.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar linkage through three required positions\n\nOracle formalization for mech_t3_mech_fourbar_three_positions_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(63)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 18.4183, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 22.5027, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 9.1016, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 26.6292, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, -0.9824, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 25.1651, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.y, A.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.eq(B.x - A.x, 26.1, description=\"the fixed pin B is 26.1 to the right of the fixed pin A\")\nscene.constraint.eq(B.distance(D1), 17.2, description=\"bar BD is 17.2 long, the same in position 1 as in every other\")\nscene.constraint.eq(B.distance(D2), 17.2, description=\"bar BD is 17.2 long, the same in position 2 as in every other\")\nscene.constraint.eq(B.distance(D3), 17.2, description=\"bar BD is 17.2 long, the same in position 3 as in every other\")\nscene.constraint.eq(C1.distance(D1), 26.4432, description=\"the connecting bar CD is 26.4432 long, the same in position 1 as in every other\")\nscene.constraint.eq(C2.distance(D2), 26.4432, description=\"the connecting bar CD is 26.4432 long, the same in position 2 as in every other\")\nscene.constraint.eq(C3.distance(D3), 26.4432, description=\"the connecting bar CD is 26.4432 long, the same in position 3 as in every other\")\nscene.constraint.gt(cross(C1, B, D1), 0.05, description=\"in position 1 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C2, B, D2), 0.05, description=\"in position 2 the pin D lies to the left of the line from C to B\")\nscene.constraint.gt(cross(C3, B, D3), 0.05, description=\"in position 3 the pin D lies to the left of the line from C to B\")\n", "possible_solution": {"points": {"A": [6.6, 8.4], "B": [32.7, 8.4], "C1": [18.41833977168408, 22.502724738185535], "C2": [9.101557140177132, 26.629158287601456], "C3": [-0.9824053377332405, 25.16505679365013], "D1": [44.795043897713796, 20.62906018925315], "D2": [35.51464052756859, 25.36814069662815], "D3": [25.43465125279505, 23.990211915862062]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the FIXED pin A, so the distance AC cannot change", "therefore the three required positions of C all lie on one circle centred at A", "so A is the CIRCUMCENTRE of C1, C2, C3 and bar 1's length is that circumradius", "likewise D always lies on a circle of radius |BD| centred at B", "the three positions must share one set of bar lengths - they are three poses of the same mechanism"], "branch_facts": ["in each pose D is on the stated side of the line CB, which picks one of the two assemblies"]} {"unique_id": "mech_t3_mech_quick_return_0080", "problem_id": "mech_t3_mech_quick_return_0080", "domain": "PyGeoX-Mech", "template": "mech_quick_return", "tier": 3, "nl_description": "A quick-return drive works like this. A crank 10.6 mm long turns about a fixed centre O; set up coordinates with O at the origin and mark the datum direction with a construction point E, 10 mm from O and level with it. The crank pin at its far end is P. A long lever is pivoted to the bench at S, directly below O and 16.4 mm down from it. The lever has a straight slot cut along it, and the crank pin P slides in that slot, so P always sits somewhere along the lever between the pivot S and the lever's tip T. The lever measures 31.5 mm from S to T. A connecting rod 13.4 mm long runs from the tip T to a block whose centre is R, sliding on a straight rail at height 15.8; the block lies to the right of the lever tip. Show the arrangement at the moment when the crank stands 40 degrees round from the datum direction, turned anticlockwise.", "answer_schema": {"points": ["E", "O", "P", "R", "S", "T"], "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": "\"\"\"Quick-return drive with a slotted lever\n\nOracle formalization for mech_t3_mech_quick_return_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(36)\n\nO, E, P, S, T, R = scene.add.points([\"O\", \"E\", \"P\", \"S\", \"T\", \"R\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so the crank angle can be measured\")\nscene.constraint.eq(O.distance(P), 10.6, description=\"the crank OP is 10.6 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 40, description=\"the crank stands 40 degrees round from the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the crank is turned anticlockwise from the datum direction\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 16.4, description=\"the lever pivot S is 16.4 below the crank centre\")\nscene.constraint.eq(S.distance(T), 31.5, description=\"the slotted lever is 31.5 long from its pivot S to its tip T\")\nscene.relate.point_lies_on(P, scene.add.line_segment(S, T))\nscene.constraint.eq(T.distance(R), 13.4, description=\"the connecting rod from the lever tip to the block is 13.4 long\")\nscene.constraint.eq(R.y, 15.8, description=\"the block runs on a straight rail at height 15.8\")\nscene.constraint.gt(R.x - T.x, 0.05, description=\"the block lies to the right of the lever tip\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [8.120071097061167, 6.813548662677316], "S": [0.0, -16.4], "T": [10.400708293490128, 13.333403219169611], "R": [23.571733322727496, 15.8]}, "circles": {}}, "implicit_facts": ["the crank pin position follows from the crank length and the stated angle", "'slides in the slot' means S, P and T are collinear, with P between S and T", "so the lever's direction is set by the line from S through P, and T is that far along it", "the block is where a circle of the rod length centred at T cuts the rail line"], "branch_facts": ["the block is to the right of the tip, picking one of the two crossings"]} {"unique_id": "mech_t3_mech_quick_return_0081", "problem_id": "mech_t3_mech_quick_return_0081", "domain": "PyGeoX-Mech", "template": "mech_quick_return", "tier": 3, "nl_description": "A quick-return drive works like this. A crank 10.1 mm long turns about a fixed centre O; set up coordinates with O at the origin and mark the datum direction with a construction point E, 10 mm from O and level with it. The crank pin at its far end is P. A long lever is pivoted to the bench at S, directly below O and 19.9 mm down from it. The lever has a straight slot cut along it, and the crank pin P slides in that slot, so P always sits somewhere along the lever between the pivot S and the lever's tip T. The lever measures 40.9 mm from S to T. A connecting rod 11.8 mm long runs from the tip T to a block whose centre is R, sliding on a straight rail at height 27.9; the block lies to the right of the lever tip. Show the arrangement at the moment when the crank stands 130 degrees round from the datum direction, turned anticlockwise.", "answer_schema": {"points": ["E", "O", "P", "R", "S", "T"], "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": "\"\"\"Quick-return drive with a slotted lever\n\nOracle formalization for mech_t3_mech_quick_return_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(41)\n\nO, E, P, S, T, R = scene.add.points([\"O\", \"E\", \"P\", \"S\", \"T\", \"R\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so the crank angle can be measured\")\nscene.constraint.eq(O.distance(P), 10.1, description=\"the crank OP is 10.1 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 130, description=\"the crank stands 130 degrees round from the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the crank is turned anticlockwise from the datum direction\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 19.9, description=\"the lever pivot S is 19.9 below the crank centre\")\nscene.constraint.eq(S.distance(T), 40.9, description=\"the slotted lever is 40.9 long from its pivot S to its tip T\")\nscene.relate.point_lies_on(P, scene.add.line_segment(S, T))\nscene.constraint.eq(T.distance(R), 11.8, description=\"the connecting rod from the lever tip to the block is 11.8 long\")\nscene.constraint.eq(R.y, 27.9, description=\"the block runs on a straight rail at height 27.9\")\nscene.constraint.gt(R.x - T.x, 0.05, description=\"the block lies to the right of the lever tip\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [-6.4921548578340476, 7.737048875501678], "S": [0.0, -19.9], "T": [-9.353128566149435, 19.916189998856993], "R": [-0.6641252938719457, 27.9]}, "circles": {}}, "implicit_facts": ["the crank pin position follows from the crank length and the stated angle", "'slides in the slot' means S, P and T are collinear, with P between S and T", "so the lever's direction is set by the line from S through P, and T is that far along it", "the block is where a circle of the rod length centred at T cuts the rail line"], "branch_facts": ["the block is to the right of the tip, picking one of the two crossings"]} {"unique_id": "mech_t3_mech_quick_return_0082", "problem_id": "mech_t3_mech_quick_return_0082", "domain": "PyGeoX-Mech", "template": "mech_quick_return", "tier": 3, "nl_description": "A quick-return drive works like this. A crank 11.8 mm long turns about a fixed centre O; set up coordinates with O at the origin and mark the datum direction with a construction point E, 10 mm from O and level with it. The crank pin at its far end is P. A long lever is pivoted to the bench at S, directly below O and 14.8 mm down from it. The lever has a straight slot cut along it, and the crank pin P slides in that slot, so P always sits somewhere along the lever between the pivot S and the lever's tip T. The lever measures 34.5 mm from S to T. A connecting rod 10.8 mm long runs from the tip T to a block whose centre is R, sliding on a straight rail at height 26.4; the block lies to the right of the lever tip. Show the arrangement at the moment when the crank stands 55 degrees round from the datum direction, turned anticlockwise.", "answer_schema": {"points": ["E", "O", "P", "R", "S", "T"], "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": "\"\"\"Quick-return drive with a slotted lever\n\nOracle formalization for mech_t3_mech_quick_return_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(39)\n\nO, E, P, S, T, R = scene.add.points([\"O\", \"E\", \"P\", \"S\", \"T\", \"R\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so the crank angle can be measured\")\nscene.constraint.eq(O.distance(P), 11.8, description=\"the crank OP is 11.8 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 55, description=\"the crank stands 55 degrees round from the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the crank is turned anticlockwise from the datum direction\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 14.8, description=\"the lever pivot S is 14.8 below the crank centre\")\nscene.constraint.eq(S.distance(T), 34.5, description=\"the slotted lever is 34.5 long from its pivot S to its tip T\")\nscene.relate.point_lies_on(P, scene.add.line_segment(S, T))\nscene.constraint.eq(T.distance(R), 10.8, description=\"the connecting rod from the lever tip to the block is 10.8 long\")\nscene.constraint.eq(R.y, 26.4, description=\"the block runs on a straight rail at height 26.4\")\nscene.constraint.gt(R.x - T.x, 0.05, description=\"the block lies to the right of the lever tip\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [6.768201948942344, 9.665994122610103], "S": [0.0, -14.8], "T": [9.198496620701837, 18.45113020513644], "R": [16.509821328598882, 26.4]}, "circles": {}}, "implicit_facts": ["the crank pin position follows from the crank length and the stated angle", "'slides in the slot' means S, P and T are collinear, with P between S and T", "so the lever's direction is set by the line from S through P, and T is that far along it", "the block is where a circle of the rod length centred at T cuts the rail line"], "branch_facts": ["the block is to the right of the tip, picking one of the two crossings"]} {"unique_id": "mech_t3_mech_quick_return_0083", "problem_id": "mech_t3_mech_quick_return_0083", "domain": "PyGeoX-Mech", "template": "mech_quick_return", "tier": 3, "nl_description": "A quick-return drive works like this. A crank 11.8 mm long turns about a fixed centre O; set up coordinates with O at the origin and mark the datum direction with a construction point E, 10 mm from O and level with it. The crank pin at its far end is P. A long lever is pivoted to the bench at S, directly below O and 20.1 mm down from it. The lever has a straight slot cut along it, and the crank pin P slides in that slot, so P always sits somewhere along the lever between the pivot S and the lever's tip T. The lever measures 34.6 mm from S to T. A connecting rod 18.2 mm long runs from the tip T to a block whose centre is R, sliding on a straight rail at height 16.8; the block lies to the right of the lever tip. Show the arrangement at the moment when the crank stands 130 degrees round from the datum direction, turned anticlockwise.", "answer_schema": {"points": ["E", "O", "P", "R", "S", "T"], "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": "\"\"\"Quick-return drive with a slotted lever\n\nOracle formalization for mech_t3_mech_quick_return_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(31)\n\nO, E, P, S, T, R = scene.add.points([\"O\", \"E\", \"P\", \"S\", \"T\", \"R\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so the crank angle can be measured\")\nscene.constraint.eq(O.distance(P), 11.8, description=\"the crank OP is 11.8 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 130, description=\"the crank stands 130 degrees round from the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the crank is turned anticlockwise from the datum direction\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 20.1, description=\"the lever pivot S is 20.1 below the crank centre\")\nscene.constraint.eq(S.distance(T), 34.6, description=\"the slotted lever is 34.6 long from its pivot S to its tip T\")\nscene.relate.point_lies_on(P, scene.add.line_segment(S, T))\nscene.constraint.eq(T.distance(R), 18.2, description=\"the connecting rod from the lever tip to the block is 18.2 long\")\nscene.constraint.eq(R.y, 16.8, description=\"the block runs on a straight rail at height 16.8\")\nscene.constraint.gt(R.x - T.x, 0.05, description=\"the block lies to the right of the lever tip\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [-7.5848937943011645, 9.03932442880394], "S": [0.0, -20.1], "T": [-8.71586226146478, 13.384231289357295], "R": [9.1607292513319, 16.8]}, "circles": {}}, "implicit_facts": ["the crank pin position follows from the crank length and the stated angle", "'slides in the slot' means S, P and T are collinear, with P between S and T", "so the lever's direction is set by the line from S through P, and T is that far along it", "the block is where a circle of the rod length centred at T cuts the rail line"], "branch_facts": ["the block is to the right of the tip, picking one of the two crossings"]} {"unique_id": "mech_t3_mech_quick_return_0084", "problem_id": "mech_t3_mech_quick_return_0084", "domain": "PyGeoX-Mech", "template": "mech_quick_return", "tier": 3, "nl_description": "A quick-return drive works like this. A crank 9.1 mm long turns about a fixed centre O; set up coordinates with O at the origin and mark the datum direction with a construction point E, 10 mm from O and level with it. The crank pin at its far end is P. A long lever is pivoted to the bench at S, directly below O and 14.9 mm down from it. The lever has a straight slot cut along it, and the crank pin P slides in that slot, so P always sits somewhere along the lever between the pivot S and the lever's tip T. The lever measures 34.2 mm from S to T. A connecting rod 12 mm long runs from the tip T to a block whose centre is R, sliding on a straight rail at height 22.3; the block lies to the right of the lever tip. Show the arrangement at the moment when the crank stands 130 degrees round from the datum direction, turned anticlockwise.", "answer_schema": {"points": ["E", "O", "P", "R", "S", "T"], "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": "\"\"\"Quick-return drive with a slotted lever\n\nOracle formalization for mech_t3_mech_quick_return_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(34)\n\nO, E, P, S, T, R = scene.add.points([\"O\", \"E\", \"P\", \"S\", \"T\", \"R\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so the crank angle can be measured\")\nscene.constraint.eq(O.distance(P), 9.1, description=\"the crank OP is 9.1 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 130, description=\"the crank stands 130 degrees round from the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the crank is turned anticlockwise from the datum direction\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 14.9, description=\"the lever pivot S is 14.9 below the crank centre\")\nscene.constraint.eq(S.distance(T), 34.2, description=\"the slotted lever is 34.2 long from its pivot S to its tip T\")\nscene.relate.point_lies_on(P, scene.add.line_segment(S, T))\nscene.constraint.eq(T.distance(R), 12, description=\"the connecting rod from the lever tip to the block is 12 long\")\nscene.constraint.eq(R.y, 22.3, description=\"the block runs on a straight rail at height 22.3\")\nscene.constraint.gt(R.x - T.x, 0.05, description=\"the block lies to the right of the lever tip\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [-5.849367248147508, 6.9710044323827], "S": [0.0, -14.9], "T": [-8.836176110109102, 18.138795252719753], "R": [2.4192392554477067, 22.3]}, "circles": {}}, "implicit_facts": ["the crank pin position follows from the crank length and the stated angle", "'slides in the slot' means S, P and T are collinear, with P between S and T", "so the lever's direction is set by the line from S through P, and T is that far along it", "the block is where a circle of the rod length centred at T cuts the rail line"], "branch_facts": ["the block is to the right of the tip, picking one of the two crossings"]} {"unique_id": "mech_t3_mech_quick_return_0085", "problem_id": "mech_t3_mech_quick_return_0085", "domain": "PyGeoX-Mech", "template": "mech_quick_return", "tier": 3, "nl_description": "A quick-return drive works like this. A crank 10.2 mm long turns about a fixed centre O; set up coordinates with O at the origin and mark the datum direction with a construction point E, 10 mm from O and level with it. The crank pin at its far end is P. A long lever is pivoted to the bench at S, directly below O and 16.9 mm down from it. The lever has a straight slot cut along it, and the crank pin P slides in that slot, so P always sits somewhere along the lever between the pivot S and the lever's tip T. The lever measures 31.6 mm from S to T. A connecting rod 11 mm long runs from the tip T to a block whose centre is R, sliding on a straight rail at height 19.1; the block lies to the right of the lever tip. Show the arrangement at the moment when the crank stands 115 degrees round from the datum direction, turned anticlockwise.", "answer_schema": {"points": ["E", "O", "P", "R", "S", "T"], "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": "\"\"\"Quick-return drive with a slotted lever\n\nOracle formalization for mech_t3_mech_quick_return_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(30)\n\nO, E, P, S, T, R = scene.add.points([\"O\", \"E\", \"P\", \"S\", \"T\", \"R\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so the crank angle can be measured\")\nscene.constraint.eq(O.distance(P), 10.2, description=\"the crank OP is 10.2 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 115, description=\"the crank stands 115 degrees round from the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the crank is turned anticlockwise from the datum direction\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 16.9, description=\"the lever pivot S is 16.9 below the crank centre\")\nscene.constraint.eq(S.distance(T), 31.6, description=\"the slotted lever is 31.6 long from its pivot S to its tip T\")\nscene.relate.point_lies_on(P, scene.add.line_segment(S, T))\nscene.constraint.eq(T.distance(R), 11, description=\"the connecting rod from the lever tip to the block is 11 long\")\nscene.constraint.eq(R.y, 19.1, description=\"the block runs on a straight rail at height 19.1\")\nscene.constraint.gt(R.x - T.x, 0.05, description=\"the block lies to the right of the lever tip\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [-4.310706269755133, 9.24433942777383], "S": [0.0, -16.9], "T": [-5.140831211926284, 14.279029081266867], "R": [4.746445430341262, 19.1]}, "circles": {}}, "implicit_facts": ["the crank pin position follows from the crank length and the stated angle", "'slides in the slot' means S, P and T are collinear, with P between S and T", "so the lever's direction is set by the line from S through P, and T is that far along it", "the block is where a circle of the rod length centred at T cuts the rail line"], "branch_facts": ["the block is to the right of the tip, picking one of the two crossings"]} {"unique_id": "mech_t3_mech_quick_return_0086", "problem_id": "mech_t3_mech_quick_return_0086", "domain": "PyGeoX-Mech", "template": "mech_quick_return", "tier": 3, "nl_description": "A quick-return drive works like this. A crank 8.4 mm long turns about a fixed centre O; set up coordinates with O at the origin and mark the datum direction with a construction point E, 10 mm from O and level with it. The crank pin at its far end is P. A long lever is pivoted to the bench at S, directly below O and 19.6 mm down from it. The lever has a straight slot cut along it, and the crank pin P slides in that slot, so P always sits somewhere along the lever between the pivot S and the lever's tip T. The lever measures 41.3 mm from S to T. A connecting rod 14.6 mm long runs from the tip T to a block whose centre is R, sliding on a straight rail at height 26.4; the block lies to the right of the lever tip. Show the arrangement at the moment when the crank stands 100 degrees round from the datum direction, turned anticlockwise.", "answer_schema": {"points": ["E", "O", "P", "R", "S", "T"], "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": "\"\"\"Quick-return drive with a slotted lever\n\nOracle formalization for mech_t3_mech_quick_return_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(39)\n\nO, E, P, S, T, R = scene.add.points([\"O\", \"E\", \"P\", \"S\", \"T\", \"R\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(E.y, O.y, description=\"construction point E is level with O\")\nscene.constraint.eq(E.x - O.x, 10, description=\"construction point E is 10 mm from O in the datum direction, so the crank angle can be measured\")\nscene.constraint.eq(O.distance(P), 8.4, description=\"the crank OP is 8.4 long\")\nscene.constraint.eq(scene.add.angle(P, O, E).value, 100, description=\"the crank stands 100 degrees round from the datum direction\")\nscene.constraint.gt(cross(O, E, P), 0.05, description=\"the crank is turned anticlockwise from the datum direction\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 19.6, description=\"the lever pivot S is 19.6 below the crank centre\")\nscene.constraint.eq(S.distance(T), 41.3, description=\"the slotted lever is 41.3 long from its pivot S to its tip T\")\nscene.relate.point_lies_on(P, scene.add.line_segment(S, T))\nscene.constraint.eq(T.distance(R), 14.6, description=\"the connecting rod from the lever tip to the block is 14.6 long\")\nscene.constraint.eq(R.y, 26.4, description=\"the block runs on a straight rail at height 26.4\")\nscene.constraint.gt(R.x - T.x, 0.05, description=\"the block lies to the right of the lever tip\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "E": [10.0, 0.0], "P": [-1.4586446924022145, 8.272385125302547], "S": [0.0, -19.6], "T": [-2.1583980334409008, 21.6435609268797], "R": [11.645090197571378, 26.4]}, "circles": {}}, "implicit_facts": ["the crank pin position follows from the crank length and the stated angle", "'slides in the slot' means S, P and T are collinear, with P between S and T", "so the lever's direction is set by the line from S through P, and T is that far along it", "the block is where a circle of the rod length centred at T cuts the rail line"], "branch_facts": ["the block is to the right of the tip, picking one of the two crossings"]} {"unique_id": "mech_t3_mech_toggle_positions_0094", "problem_id": "mech_t3_mech_toggle_positions_0094", "domain": "PyGeoX-Mech", "template": "mech_toggle_positions", "tier": 3, "nl_description": "A four-bar mechanism has two fixed pins, O1 and O2, sitting at the same height 20.5 mm apart. Set up coordinates with O1 at the origin and O2 to its right. A crank 8.2 mm long turns about O1; a connecting bar 23 mm long runs from the end of the crank to the end of an output bar; the output bar is 19.3 mm long and turns about O2. The mechanism is to be shown in its two dead positions - the two positions at which the crank and the connecting bar lie in one straight line, so that turning the output bar can no longer drive the crank round. In the first of these the crank and connecting bar are stretched out end to end, with the crank end pin A1 lying between O1 and the output pin B1. In the second they are folded back on each other, with the fixed pin O1 lying between the crank end pin A2 and the output pin B2. In both positions the output pin is above the line joining the two fixed pins.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O1", "O2"], "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": "\"\"\"Four-bar at its two dead positions\n\nOracle formalization for mech_t3_mech_toggle_positions_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(37)\n\nO1, O2, A1, B1, A2, B2 = scene.add.points([\"O1\", \"O2\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O1.x, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (x)\")\nscene.constraint.eq(O1.y, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (y)\")\nscene.constraint.eq(O2.y, O1.y, description=\"the two fixed pins sit at the same height\")\nscene.constraint.gt(O2.x - O1.x, 1.0, description=\"datum: O2 is to the right of O1\")\nscene.constraint.eq(O1.distance(O2), 20.5, description=\"the two fixed pins are 20.5 apart\")\nscene.constraint.eq(O1.distance(A1), 8.2, description=\"the crank is 8.2 long (first position)\")\nscene.constraint.eq(A1.distance(B1), 23, description=\"the connecting bar is 23 long (first position)\")\nscene.constraint.eq(O2.distance(B1), 19.3, description=\"the output bar is 19.3 long (first position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O1, B1))\nscene.constraint.eq(O1.distance(A2), 8.2, description=\"the crank is 8.2 long (second position)\")\nscene.constraint.eq(A2.distance(B2), 23, description=\"the connecting bar is 23 long (second position)\")\nscene.constraint.eq(O2.distance(B2), 19.3, description=\"the output bar is 19.3 long (second position)\")\nscene.relate.point_lies_on(O1, scene.add.line_segment(A2, B2))\nscene.constraint.gt(cross(O1, O2, B1), 0.05, description=\"in the first position the output pin is above the line of fixed pins\")\nscene.constraint.gt(cross(O1, O2, B2), 0.05, description=\"in the second position the output pin is above the line of fixed pins\")\n", "possible_solution": {"points": {"O1": [0.0, 0.0], "O2": [20.5, 0.0], "A1": [6.546153846153846, 4.938407620122624], "B1": [24.90731707317073, 18.790038749734865], "A2": [-3.605405405405405, -7.364852467135608], "B2": [6.507317073170732, 13.292660550439882]}, "circles": {}}, "implicit_facts": ["a dead position is where the crank and the connecting bar become collinear", "stretched out, the distance from O1 to the output pin is crank + connecting bar", "folded back, that distance is connecting bar - crank", "in each case the output pin is then where a circle about O1 of that radius cuts a circle about O2 of the output-bar length", "the crank end pin then sits on the same straight line, at the crank length from O1"], "branch_facts": ["in both positions the output pin is above the line of fixed pins, which picks one of the two crossings"]} {"unique_id": "mech_t3_mech_toggle_positions_0095", "problem_id": "mech_t3_mech_toggle_positions_0095", "domain": "PyGeoX-Mech", "template": "mech_toggle_positions", "tier": 3, "nl_description": "A four-bar mechanism has two fixed pins, O1 and O2, sitting at the same height 19.4 mm apart. Set up coordinates with O1 at the origin and O2 to its right. A crank 6.6 mm long turns about O1; a connecting bar 19.1 mm long runs from the end of the crank to the end of an output bar; the output bar is 21.1 mm long and turns about O2. The mechanism is to be shown in its two dead positions - the two positions at which the crank and the connecting bar lie in one straight line, so that turning the output bar can no longer drive the crank round. In the first of these the crank and connecting bar are stretched out end to end, with the crank end pin A1 lying between O1 and the output pin B1. In the second they are folded back on each other, with the fixed pin O1 lying between the crank end pin A2 and the output pin B2. In both positions the output pin is above the line joining the two fixed pins.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O1", "O2"], "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": "\"\"\"Four-bar at its two dead positions\n\nOracle formalization for mech_t3_mech_toggle_positions_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(32)\n\nO1, O2, A1, B1, A2, B2 = scene.add.points([\"O1\", \"O2\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O1.x, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (x)\")\nscene.constraint.eq(O1.y, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (y)\")\nscene.constraint.eq(O2.y, O1.y, description=\"the two fixed pins sit at the same height\")\nscene.constraint.gt(O2.x - O1.x, 1.0, description=\"datum: O2 is to the right of O1\")\nscene.constraint.eq(O1.distance(O2), 19.4, description=\"the two fixed pins are 19.4 apart\")\nscene.constraint.eq(O1.distance(A1), 6.6, description=\"the crank is 6.6 long (first position)\")\nscene.constraint.eq(A1.distance(B1), 19.1, description=\"the connecting bar is 19.1 long (first position)\")\nscene.constraint.eq(O2.distance(B1), 21.1, description=\"the output bar is 21.1 long (first position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O1, B1))\nscene.constraint.eq(O1.distance(A2), 6.6, description=\"the crank is 6.6 long (second position)\")\nscene.constraint.eq(A2.distance(B2), 19.1, description=\"the connecting bar is 19.1 long (second position)\")\nscene.constraint.eq(O2.distance(B2), 21.1, description=\"the output bar is 21.1 long (second position)\")\nscene.relate.point_lies_on(O1, scene.add.line_segment(A2, B2))\nscene.constraint.gt(cross(O1, O2, B1), 0.05, description=\"in the first position the output pin is above the line of fixed pins\")\nscene.constraint.gt(cross(O1, O2, B2), 0.05, description=\"in the second position the output pin is above the line of fixed pins\")\n", "possible_solution": {"points": {"O1": [0.0, 0.0], "O2": [19.4, 0.0], "A1": [3.9159452846082883, 5.312755643538869], "B1": [15.248453608247427, 20.687548490749844], "A2": [-1.1893608247422676, -6.491950464118499], "B2": [2.2525773195876284, 12.29536072749716]}, "circles": {}}, "implicit_facts": ["a dead position is where the crank and the connecting bar become collinear", "stretched out, the distance from O1 to the output pin is crank + connecting bar", "folded back, that distance is connecting bar - crank", "in each case the output pin is then where a circle about O1 of that radius cuts a circle about O2 of the output-bar length", "the crank end pin then sits on the same straight line, at the crank length from O1"], "branch_facts": ["in both positions the output pin is above the line of fixed pins, which picks one of the two crossings"]} {"unique_id": "mech_t3_mech_toggle_positions_0096", "problem_id": "mech_t3_mech_toggle_positions_0096", "domain": "PyGeoX-Mech", "template": "mech_toggle_positions", "tier": 3, "nl_description": "A four-bar mechanism has two fixed pins, O1 and O2, sitting at the same height 22.1 mm apart. Set up coordinates with O1 at the origin and O2 to its right. A crank 7.5 mm long turns about O1; a connecting bar 21.2 mm long runs from the end of the crank to the end of an output bar; the output bar is 20.6 mm long and turns about O2. The mechanism is to be shown in its two dead positions - the two positions at which the crank and the connecting bar lie in one straight line, so that turning the output bar can no longer drive the crank round. In the first of these the crank and connecting bar are stretched out end to end, with the crank end pin A1 lying between O1 and the output pin B1. In the second they are folded back on each other, with the fixed pin O1 lying between the crank end pin A2 and the output pin B2. In both positions the output pin is above the line joining the two fixed pins.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O1", "O2"], "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": "\"\"\"Four-bar at its two dead positions\n\nOracle formalization for mech_t3_mech_toggle_positions_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(34)\n\nO1, O2, A1, B1, A2, B2 = scene.add.points([\"O1\", \"O2\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O1.x, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (x)\")\nscene.constraint.eq(O1.y, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (y)\")\nscene.constraint.eq(O2.y, O1.y, description=\"the two fixed pins sit at the same height\")\nscene.constraint.gt(O2.x - O1.x, 1.0, description=\"datum: O2 is to the right of O1\")\nscene.constraint.eq(O1.distance(O2), 22.1, description=\"the two fixed pins are 22.1 apart\")\nscene.constraint.eq(O1.distance(A1), 7.5, description=\"the crank is 7.5 long (first position)\")\nscene.constraint.eq(A1.distance(B1), 21.2, description=\"the connecting bar is 21.2 long (first position)\")\nscene.constraint.eq(O2.distance(B1), 20.6, description=\"the output bar is 20.6 long (first position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O1, B1))\nscene.constraint.eq(O1.distance(A2), 7.5, description=\"the crank is 7.5 long (second position)\")\nscene.constraint.eq(A2.distance(B2), 21.2, description=\"the connecting bar is 21.2 long (second position)\")\nscene.constraint.eq(O2.distance(B2), 20.6, description=\"the output bar is 20.6 long (second position)\")\nscene.relate.point_lies_on(O1, scene.add.line_segment(A2, B2))\nscene.constraint.gt(cross(O1, O2, B1), 0.05, description=\"in the first position the output pin is above the line of fixed pins\")\nscene.constraint.gt(cross(O1, O2, B2), 0.05, description=\"in the second position the output pin is above the line of fixed pins\")\n", "possible_solution": {"points": {"O1": [0.0, 0.0], "O2": [22.1, 0.0], "A1": [5.24859287054409, 5.357450221819494], "B1": [20.084615384615383, 20.5011761821626], "A2": [-3.1179608283515536, -6.82116707557184], "B2": [5.695475113122171, 12.459998524711228]}, "circles": {}}, "implicit_facts": ["a dead position is where the crank and the connecting bar become collinear", "stretched out, the distance from O1 to the output pin is crank + connecting bar", "folded back, that distance is connecting bar - crank", "in each case the output pin is then where a circle about O1 of that radius cuts a circle about O2 of the output-bar length", "the crank end pin then sits on the same straight line, at the crank length from O1"], "branch_facts": ["in both positions the output pin is above the line of fixed pins, which picks one of the two crossings"]} {"unique_id": "mech_t3_mech_toggle_positions_0097", "problem_id": "mech_t3_mech_toggle_positions_0097", "domain": "PyGeoX-Mech", "template": "mech_toggle_positions", "tier": 3, "nl_description": "A four-bar mechanism has two fixed pins, O1 and O2, sitting at the same height 23.2 mm apart. Set up coordinates with O1 at the origin and O2 to its right. A crank 5.2 mm long turns about O1; a connecting bar 19.6 mm long runs from the end of the crank to the end of an output bar; the output bar is 12.6 mm long and turns about O2. The mechanism is to be shown in its two dead positions - the two positions at which the crank and the connecting bar lie in one straight line, so that turning the output bar can no longer drive the crank round. In the first of these the crank and connecting bar are stretched out end to end, with the crank end pin A1 lying between O1 and the output pin B1. In the second they are folded back on each other, with the fixed pin O1 lying between the crank end pin A2 and the output pin B2. In both positions the output pin is above the line joining the two fixed pins.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O1", "O2"], "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": "\"\"\"Four-bar at its two dead positions\n\nOracle formalization for mech_t3_mech_toggle_positions_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(35)\n\nO1, O2, A1, B1, A2, B2 = scene.add.points([\"O1\", \"O2\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O1.x, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (x)\")\nscene.constraint.eq(O1.y, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (y)\")\nscene.constraint.eq(O2.y, O1.y, description=\"the two fixed pins sit at the same height\")\nscene.constraint.gt(O2.x - O1.x, 1.0, description=\"datum: O2 is to the right of O1\")\nscene.constraint.eq(O1.distance(O2), 23.2, description=\"the two fixed pins are 23.2 apart\")\nscene.constraint.eq(O1.distance(A1), 5.2, description=\"the crank is 5.2 long (first position)\")\nscene.constraint.eq(A1.distance(B1), 19.6, description=\"the connecting bar is 19.6 long (first position)\")\nscene.constraint.eq(O2.distance(B1), 12.6, description=\"the output bar is 12.6 long (first position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O1, B1))\nscene.constraint.eq(O1.distance(A2), 5.2, description=\"the crank is 5.2 long (second position)\")\nscene.constraint.eq(A2.distance(B2), 19.6, description=\"the connecting bar is 19.6 long (second position)\")\nscene.constraint.eq(O2.distance(B2), 12.6, description=\"the output bar is 12.6 long (second position)\")\nscene.relate.point_lies_on(O1, scene.add.line_segment(A2, B2))\nscene.constraint.gt(cross(O1, O2, B1), 0.05, description=\"in the first position the output pin is above the line of fixed pins\")\nscene.constraint.gt(cross(O1, O2, B2), 0.05, description=\"in the second position the output pin is above the line of fixed pins\")\n", "possible_solution": {"points": {"O1": [0.0, 0.0], "O2": [23.2, 0.0], "A1": [4.494146273637376, 2.6158458041618773], "B1": [21.433620689655175, 12.475572296772029], "A2": [-4.5671216475095795, -2.486242115492655], "B2": [12.647413793103452, 6.8849781659796605]}, "circles": {}}, "implicit_facts": ["a dead position is where the crank and the connecting bar become collinear", "stretched out, the distance from O1 to the output pin is crank + connecting bar", "folded back, that distance is connecting bar - crank", "in each case the output pin is then where a circle about O1 of that radius cuts a circle about O2 of the output-bar length", "the crank end pin then sits on the same straight line, at the crank length from O1"], "branch_facts": ["in both positions the output pin is above the line of fixed pins, which picks one of the two crossings"]} {"unique_id": "mech_t3_mech_toggle_positions_0098", "problem_id": "mech_t3_mech_toggle_positions_0098", "domain": "PyGeoX-Mech", "template": "mech_toggle_positions", "tier": 3, "nl_description": "A four-bar mechanism has two fixed pins, O1 and O2, sitting at the same height 17.8 mm apart. Set up coordinates with O1 at the origin and O2 to its right. A crank 5.6 mm long turns about O1; a connecting bar 16.1 mm long runs from the end of the crank to the end of an output bar; the output bar is 19.7 mm long and turns about O2. The mechanism is to be shown in its two dead positions - the two positions at which the crank and the connecting bar lie in one straight line, so that turning the output bar can no longer drive the crank round. In the first of these the crank and connecting bar are stretched out end to end, with the crank end pin A1 lying between O1 and the output pin B1. In the second they are folded back on each other, with the fixed pin O1 lying between the crank end pin A2 and the output pin B2. In both positions the output pin is above the line joining the two fixed pins.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O1", "O2"], "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": "\"\"\"Four-bar at its two dead positions\n\nOracle formalization for mech_t3_mech_toggle_positions_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(29)\n\nO1, O2, A1, B1, A2, B2 = scene.add.points([\"O1\", \"O2\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O1.x, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (x)\")\nscene.constraint.eq(O1.y, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (y)\")\nscene.constraint.eq(O2.y, O1.y, description=\"the two fixed pins sit at the same height\")\nscene.constraint.gt(O2.x - O1.x, 1.0, description=\"datum: O2 is to the right of O1\")\nscene.constraint.eq(O1.distance(O2), 17.8, description=\"the two fixed pins are 17.8 apart\")\nscene.constraint.eq(O1.distance(A1), 5.6, description=\"the crank is 5.6 long (first position)\")\nscene.constraint.eq(A1.distance(B1), 16.1, description=\"the connecting bar is 16.1 long (first position)\")\nscene.constraint.eq(O2.distance(B1), 19.7, description=\"the output bar is 19.7 long (first position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O1, B1))\nscene.constraint.eq(O1.distance(A2), 5.6, description=\"the crank is 5.6 long (second position)\")\nscene.constraint.eq(A2.distance(B2), 16.1, description=\"the connecting bar is 16.1 long (second position)\")\nscene.constraint.eq(O2.distance(B2), 19.7, description=\"the output bar is 19.7 long (second position)\")\nscene.relate.point_lies_on(O1, scene.add.line_segment(A2, B2))\nscene.constraint.gt(cross(O1, O2, B1), 0.05, description=\"in the first position the output pin is above the line of fixed pins\")\nscene.constraint.gt(cross(O1, O2, B2), 0.05, description=\"in the second position the output pin is above the line of fixed pins\")\n", "possible_solution": {"points": {"O1": [0.0, 0.0], "O2": [17.8, 0.0], "A1": [2.896991663646249, 4.792435633450296], "B1": [11.225842696629217, 18.5706880796199], "A2": [-0.58426966292135, -5.569437041657781], "B2": [1.0955056179775313, 10.442694453108341]}, "circles": {}}, "implicit_facts": ["a dead position is where the crank and the connecting bar become collinear", "stretched out, the distance from O1 to the output pin is crank + connecting bar", "folded back, that distance is connecting bar - crank", "in each case the output pin is then where a circle about O1 of that radius cuts a circle about O2 of the output-bar length", "the crank end pin then sits on the same straight line, at the crank length from O1"], "branch_facts": ["in both positions the output pin is above the line of fixed pins, which picks one of the two crossings"]} {"unique_id": "mech_t3_mech_toggle_positions_0099", "problem_id": "mech_t3_mech_toggle_positions_0099", "domain": "PyGeoX-Mech", "template": "mech_toggle_positions", "tier": 3, "nl_description": "A four-bar mechanism has two fixed pins, O1 and O2, sitting at the same height 23.3 mm apart. Set up coordinates with O1 at the origin and O2 to its right. A crank 7.1 mm long turns about O1; a connecting bar 18.2 mm long runs from the end of the crank to the end of an output bar; the output bar is 13.5 mm long and turns about O2. The mechanism is to be shown in its two dead positions - the two positions at which the crank and the connecting bar lie in one straight line, so that turning the output bar can no longer drive the crank round. In the first of these the crank and connecting bar are stretched out end to end, with the crank end pin A1 lying between O1 and the output pin B1. In the second they are folded back on each other, with the fixed pin O1 lying between the crank end pin A2 and the output pin B2. In both positions the output pin is above the line joining the two fixed pins.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O1", "O2"], "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": "\"\"\"Four-bar at its two dead positions\n\nOracle formalization for mech_t3_mech_toggle_positions_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(35)\n\nO1, O2, A1, B1, A2, B2 = scene.add.points([\"O1\", \"O2\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O1.x, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (x)\")\nscene.constraint.eq(O1.y, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (y)\")\nscene.constraint.eq(O2.y, O1.y, description=\"the two fixed pins sit at the same height\")\nscene.constraint.gt(O2.x - O1.x, 1.0, description=\"datum: O2 is to the right of O1\")\nscene.constraint.eq(O1.distance(O2), 23.3, description=\"the two fixed pins are 23.3 apart\")\nscene.constraint.eq(O1.distance(A1), 7.1, description=\"the crank is 7.1 long (first position)\")\nscene.constraint.eq(A1.distance(B1), 18.2, description=\"the connecting bar is 18.2 long (first position)\")\nscene.constraint.eq(O2.distance(B1), 13.5, description=\"the output bar is 13.5 long (first position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O1, B1))\nscene.constraint.eq(O1.distance(A2), 7.1, description=\"the crank is 7.1 long (second position)\")\nscene.constraint.eq(A2.distance(B2), 18.2, description=\"the connecting bar is 18.2 long (second position)\")\nscene.constraint.eq(O2.distance(B2), 13.5, description=\"the output bar is 13.5 long (second position)\")\nscene.relate.point_lies_on(O1, scene.add.line_segment(A2, B2))\nscene.constraint.gt(cross(O1, O2, B1), 0.05, description=\"in the first position the output pin is above the line of fixed pins\")\nscene.constraint.gt(cross(O1, O2, B2), 0.05, description=\"in the second position the output pin is above the line of fixed pins\")\n", "possible_solution": {"points": {"O1": [0.0, 0.0], "O2": [23.3, 0.0], "A1": [6.026550916894263, 3.7537559918141863], "B1": [21.474892703862654, 13.37606008350689], "A2": [-6.641408575957932, -2.5103171367762345], "B2": [10.383047210300429, 3.924580312424817]}, "circles": {}}, "implicit_facts": ["a dead position is where the crank and the connecting bar become collinear", "stretched out, the distance from O1 to the output pin is crank + connecting bar", "folded back, that distance is connecting bar - crank", "in each case the output pin is then where a circle about O1 of that radius cuts a circle about O2 of the output-bar length", "the crank end pin then sits on the same straight line, at the crank length from O1"], "branch_facts": ["in both positions the output pin is above the line of fixed pins, which picks one of the two crossings"]} {"unique_id": "mech_t3_mech_toggle_positions_0100", "problem_id": "mech_t3_mech_toggle_positions_0100", "domain": "PyGeoX-Mech", "template": "mech_toggle_positions", "tier": 3, "nl_description": "A four-bar mechanism has two fixed pins, O1 and O2, sitting at the same height 22.8 mm apart. Set up coordinates with O1 at the origin and O2 to its right. A crank 8.5 mm long turns about O1; a connecting bar 25.2 mm long runs from the end of the crank to the end of an output bar; the output bar is 15.6 mm long and turns about O2. The mechanism is to be shown in its two dead positions - the two positions at which the crank and the connecting bar lie in one straight line, so that turning the output bar can no longer drive the crank round. In the first of these the crank and connecting bar are stretched out end to end, with the crank end pin A1 lying between O1 and the output pin B1. In the second they are folded back on each other, with the fixed pin O1 lying between the crank end pin A2 and the output pin B2. In both positions the output pin is above the line joining the two fixed pins.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O1", "O2"], "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": "\"\"\"Four-bar at its two dead positions\n\nOracle formalization for mech_t3_mech_toggle_positions_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(45)\n\nO1, O2, A1, B1, A2, B2 = scene.add.points([\"O1\", \"O2\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O1.x, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (x)\")\nscene.constraint.eq(O1.y, 0, description=\"datum: O1 is the origin, the fixed pin the crank turns about (y)\")\nscene.constraint.eq(O2.y, O1.y, description=\"the two fixed pins sit at the same height\")\nscene.constraint.gt(O2.x - O1.x, 1.0, description=\"datum: O2 is to the right of O1\")\nscene.constraint.eq(O1.distance(O2), 22.8, description=\"the two fixed pins are 22.8 apart\")\nscene.constraint.eq(O1.distance(A1), 8.5, description=\"the crank is 8.5 long (first position)\")\nscene.constraint.eq(A1.distance(B1), 25.2, description=\"the connecting bar is 25.2 long (first position)\")\nscene.constraint.eq(O2.distance(B1), 15.6, description=\"the output bar is 15.6 long (first position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O1, B1))\nscene.constraint.eq(O1.distance(A2), 8.5, description=\"the crank is 8.5 long (second position)\")\nscene.constraint.eq(A2.distance(B2), 25.2, description=\"the connecting bar is 25.2 long (second position)\")\nscene.constraint.eq(O2.distance(B2), 15.6, description=\"the output bar is 15.6 long (second position)\")\nscene.relate.point_lies_on(O1, scene.add.line_segment(A2, B2))\nscene.constraint.gt(cross(O1, O2, B1), 0.05, description=\"in the first position the output pin is above the line of fixed pins\")\nscene.constraint.gt(cross(O1, O2, B2), 0.05, description=\"in the second position the output pin is above the line of fixed pins\")\n", "possible_solution": {"points": {"O1": [0.0, 0.0], "O2": [22.8, 0.0], "A1": [7.811081394138166, 3.35216459234158], "B1": [30.968640350877198, 13.290346677871911], "A2": [-6.198977045908184, -5.815727261856718], "B2": [12.179166666666667, 11.426193561530258]}, "circles": {}}, "implicit_facts": ["a dead position is where the crank and the connecting bar become collinear", "stretched out, the distance from O1 to the output pin is crank + connecting bar", "folded back, that distance is connecting bar - crank", "in each case the output pin is then where a circle about O1 of that radius cuts a circle about O2 of the output-bar length", "the crank end pin then sits on the same straight line, at the crank length from O1"], "branch_facts": ["in both positions the output pin is above the line of fixed pins, which picks one of the two crossings"]} {"unique_id": "mechh_t4_mechh_bell_crank_derived_0061", "problem_id": "mechh_t4_mechh_bell_crank_derived_0061", "domain": "PyGeoX-Mech", "template": "mechh_bell_crank_derived", "tier": 4, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 9.7 mm from F out to a pin H, the other runs 17.7 mm from F out to a pin G, and the two arms are set at a right angle to each other at F. A straight pull-rod connects pin G to a block whose centre is K; the block slides along a straight rail 16.3 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block has been driven to sit -6.5 mm relative to the level of F, measured up the rail. The pull-rod has not been cut to length yet - work out where every pin sits, and how long that rod has to be.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with the pull-rod length to be found\n\nOracle formalization for mechh_t4_mechh_bell_crank_derived_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(28)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(H), 9.7, description=\"the arm FH is 9.7 long\")\nscene.constraint.eq(F.distance(G), 17.7, description=\"the arm FG is 17.7 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 16.3, description=\"the block runs on a straight rail 16.3 to the right of F\")\nscene.constraint.eq(K.y - F.y, -6.5, description=\"in this position the block sits -6.5 relative to the level of F\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 17.7], "H": [9.7, 0.0], "K": [16.3, -6.5]}, "circles": {}}, "implicit_facts": ["the rod length is NOT given - it is whatever is needed to span from G to the stated block position", "'H is level with F' means H has the same height as F, so arm FH lies along the datum direction", "the right angle at F then forces arm FG to stand straight up", "with both G and K fully pinned, the rod length is simply the distance between them"], "branch_facts": ["arm FG stands up above F, not below it"]} {"unique_id": "mechh_t4_mechh_bell_crank_derived_0062", "problem_id": "mechh_t4_mechh_bell_crank_derived_0062", "domain": "PyGeoX-Mech", "template": "mechh_bell_crank_derived", "tier": 4, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 12.5 mm from F out to a pin H, the other runs 17.1 mm from F out to a pin G, and the two arms are set at a right angle to each other at F. A straight pull-rod connects pin G to a block whose centre is K; the block slides along a straight rail 22.3 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block has been driven to sit -5.1 mm relative to the level of F, measured up the rail. The pull-rod has not been cut to length yet - work out where every pin sits, and how long that rod has to be.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with the pull-rod length to be found\n\nOracle formalization for mechh_t4_mechh_bell_crank_derived_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(34)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(H), 12.5, description=\"the arm FH is 12.5 long\")\nscene.constraint.eq(F.distance(G), 17.1, description=\"the arm FG is 17.1 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 22.3, description=\"the block runs on a straight rail 22.3 to the right of F\")\nscene.constraint.eq(K.y - F.y, -5.1, description=\"in this position the block sits -5.1 relative to the level of F\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 17.1], "H": [12.5, 0.0], "K": [22.3, -5.1]}, "circles": {}}, "implicit_facts": ["the rod length is NOT given - it is whatever is needed to span from G to the stated block position", "'H is level with F' means H has the same height as F, so arm FH lies along the datum direction", "the right angle at F then forces arm FG to stand straight up", "with both G and K fully pinned, the rod length is simply the distance between them"], "branch_facts": ["arm FG stands up above F, not below it"]} {"unique_id": "mechh_t4_mechh_bell_crank_derived_0063", "problem_id": "mechh_t4_mechh_bell_crank_derived_0063", "domain": "PyGeoX-Mech", "template": "mechh_bell_crank_derived", "tier": 4, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 12.1 mm from F out to a pin H, the other runs 15.4 mm from F out to a pin G, and the two arms are set at a right angle to each other at F. A straight pull-rod connects pin G to a block whose centre is K; the block slides along a straight rail 16.7 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block has been driven to sit -5.5 mm relative to the level of F, measured up the rail. The pull-rod has not been cut to length yet - work out where every pin sits, and how long that rod has to be.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with the pull-rod length to be found\n\nOracle formalization for mechh_t4_mechh_bell_crank_derived_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(27)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(H), 12.1, description=\"the arm FH is 12.1 long\")\nscene.constraint.eq(F.distance(G), 15.4, description=\"the arm FG is 15.4 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 16.7, description=\"the block runs on a straight rail 16.7 to the right of F\")\nscene.constraint.eq(K.y - F.y, -5.5, description=\"in this position the block sits -5.5 relative to the level of F\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 15.4], "H": [12.1, 0.0], "K": [16.7, -5.5]}, "circles": {}}, "implicit_facts": ["the rod length is NOT given - it is whatever is needed to span from G to the stated block position", "'H is level with F' means H has the same height as F, so arm FH lies along the datum direction", "the right angle at F then forces arm FG to stand straight up", "with both G and K fully pinned, the rod length is simply the distance between them"], "branch_facts": ["arm FG stands up above F, not below it"]} {"unique_id": "mechh_t4_mechh_bell_crank_derived_0064", "problem_id": "mechh_t4_mechh_bell_crank_derived_0064", "domain": "PyGeoX-Mech", "template": "mechh_bell_crank_derived", "tier": 4, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 13.5 mm from F out to a pin H, the other runs 14.9 mm from F out to a pin G, and the two arms are set at a right angle to each other at F. A straight pull-rod connects pin G to a block whose centre is K; the block slides along a straight rail 22.5 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block has been driven to sit -7.3 mm relative to the level of F, measured up the rail. The pull-rod has not been cut to length yet - work out where every pin sits, and how long that rod has to be.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with the pull-rod length to be found\n\nOracle formalization for mechh_t4_mechh_bell_crank_derived_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(34)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(H), 13.5, description=\"the arm FH is 13.5 long\")\nscene.constraint.eq(F.distance(G), 14.9, description=\"the arm FG is 14.9 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 22.5, description=\"the block runs on a straight rail 22.5 to the right of F\")\nscene.constraint.eq(K.y - F.y, -7.3, description=\"in this position the block sits -7.3 relative to the level of F\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 14.9], "H": [13.5, 0.0], "K": [22.5, -7.3]}, "circles": {}}, "implicit_facts": ["the rod length is NOT given - it is whatever is needed to span from G to the stated block position", "'H is level with F' means H has the same height as F, so arm FH lies along the datum direction", "the right angle at F then forces arm FG to stand straight up", "with both G and K fully pinned, the rod length is simply the distance between them"], "branch_facts": ["arm FG stands up above F, not below it"]} {"unique_id": "mechh_t4_mechh_bell_crank_derived_0065", "problem_id": "mechh_t4_mechh_bell_crank_derived_0065", "domain": "PyGeoX-Mech", "template": "mechh_bell_crank_derived", "tier": 4, "nl_description": "A bent lever turns about a fixed pin F. One arm runs 13.2 mm from F out to a pin H, the other runs 16.9 mm from F out to a pin G, and the two arms are set at a right angle to each other at F. A straight pull-rod connects pin G to a block whose centre is K; the block slides along a straight rail 20.2 mm to the right of F. Set up coordinates with F at the origin. Place the lever so that pin H is level with F, with the arm FH reaching out to the right and the arm FG standing up above F. In this position the block has been driven to sit -5.4 mm relative to the level of F, measured up the rail. The pull-rod has not been cut to length yet - work out where every pin sits, and how long that rod has to be.", "answer_schema": {"points": ["F", "G", "H", "K"], "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": "\"\"\"Bell crank with the pull-rod length to be found\n\nOracle formalization for mechh_t4_mechh_bell_crank_derived_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(31)\n\nF, G, H, K = scene.add.points([\"F\", \"G\", \"H\", \"K\"])\n\nscene.constraint.eq(F.x, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (x)\")\nscene.constraint.eq(F.y, 0, description=\"datum: F is the origin, the fixed pin of the bent lever (y)\")\nscene.constraint.eq(H.y, F.y, description=\"pin H is level with the fixed pin F\")\nscene.constraint.gt(H.x - F.x, 1.0, description=\"the arm FH reaches out in the +x direction\")\nscene.constraint.eq(F.distance(H), 13.2, description=\"the arm FH is 13.2 long\")\nscene.constraint.eq(F.distance(G), 16.9, description=\"the arm FG is 16.9 long\")\nscene.relate.right_angle(G, F, H)\nscene.constraint.gt(G.y - F.y, 0.05, description=\"the arm FG stands up above the fixed pin\")\nscene.constraint.eq(K.x - F.x, 20.2, description=\"the block runs on a straight rail 20.2 to the right of F\")\nscene.constraint.eq(K.y - F.y, -5.4, description=\"in this position the block sits -5.4 relative to the level of F\")\n", "possible_solution": {"points": {"F": [0.0, 0.0], "G": [0.0, 16.9], "H": [13.2, 0.0], "K": [20.2, -5.4]}, "circles": {}}, "implicit_facts": ["the rod length is NOT given - it is whatever is needed to span from G to the stated block position", "'H is level with F' means H has the same height as F, so arm FH lies along the datum direction", "the right angle at F then forces arm FG to stand straight up", "with both G and K fully pinned, the rod length is simply the distance between them"], "branch_facts": ["arm FG stands up above F, not below it"]} {"unique_id": "mechh_t4_mechh_fourbar_both_pivots_0001", "problem_id": "mechh_t4_mechh_fourbar_both_pivots_0001", "domain": "PyGeoX-Mech", "template": "mechh_fourbar_both_pivots", "tier": 4, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid and the two fixed pins never move. Neither fixed pin has been located yet - that is what has to be worked out. As the mechanism is worked through its travel, the two moving pins are required to pass through three matching pairs of positions, measured on the bench grid: C reaches (18.9855, 16.3744) while D is at (39.7057, 11.204); then C reaches (13.5046, 18.8038) while D is at (35.6394, 12.9406); then C reaches (7.5127, 19.0017) while D is at (31.2188, 13.0296). Write down where each of the two fixed pins must be placed.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar with both fixed pivots unknown\n\nOracle formalization for mechh_t4_mechh_fourbar_both_pivots_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(57)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 18.9855, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 16.3744, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 13.5046, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 18.8038, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 7.5127, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 19.0017, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(D1.x, 39.7057, description=\"matching position 1 of the moving pin D (x)\")\nscene.constraint.eq(D1.y, 11.204, description=\"matching position 1 of the moving pin D (y)\")\nscene.constraint.eq(D2.x, 35.6394, description=\"matching position 2 of the moving pin D (x)\")\nscene.constraint.eq(D2.y, 12.9406, description=\"matching position 2 of the moving pin D (y)\")\nscene.constraint.eq(D3.x, 31.2188, description=\"matching position 3 of the moving pin D (x)\")\nscene.constraint.eq(D3.y, 13.0296, description=\"matching position 3 of the moving pin D (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.distance(D1), B.distance(D2), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 2\")\nscene.constraint.eq(B.distance(D1), B.distance(D3), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 3\")\n", "possible_solution": {"points": {"A": [10.0, 3.5], "B": [33.2, 1.6], "C1": [18.985547062253403, 16.374391014414204], "C2": [13.504638235282108, 18.803839741705303], "C3": [7.512691436938617, 19.001719134087082], "D1": [39.705692905102424, 11.203955425995064], "D2": [35.63944310693812, 12.940595986455566], "D3": [31.21876507050815, 13.029554153778765]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the fixed pin A, so all three positions of C are the SAME distance from A", "so A is the circumcentre of the three C positions", "the same argument applies independently to B and the three D positions", "both pivots must be found this way - neither is given, and neither can be read off the other"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_fourbar_both_pivots_0002", "problem_id": "mechh_t4_mechh_fourbar_both_pivots_0002", "domain": "PyGeoX-Mech", "template": "mechh_fourbar_both_pivots", "tier": 4, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid and the two fixed pins never move. Neither fixed pin has been located yet - that is what has to be worked out. As the mechanism is worked through its travel, the two moving pins are required to pass through three matching pairs of positions, measured on the bench grid: C reaches (15.7869, 19.0127) while D is at (32.4387, 10.6711); then C reaches (10.6065, 20.7411) while D is at (28.7136, 12.0931); then C reaches (5.1486, 20.5528) while D is at (24.7342, 11.8431). Write down where each of the two fixed pins must be placed.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar with both fixed pivots unknown\n\nOracle formalization for mechh_t4_mechh_fourbar_both_pivots_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\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 15.7869, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 19.0127, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 10.6065, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 20.7411, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 5.1486, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 20.5528, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(D1.x, 32.4387, description=\"matching position 1 of the moving pin D (x)\")\nscene.constraint.eq(D1.y, 10.6711, description=\"matching position 1 of the moving pin D (y)\")\nscene.constraint.eq(D2.x, 28.7136, description=\"matching position 2 of the moving pin D (x)\")\nscene.constraint.eq(D2.y, 12.0931, description=\"matching position 2 of the moving pin D (y)\")\nscene.constraint.eq(D3.x, 24.7342, description=\"matching position 3 of the moving pin D (x)\")\nscene.constraint.eq(D3.y, 11.8431, description=\"matching position 3 of the moving pin D (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.distance(D1), B.distance(D2), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 2\")\nscene.constraint.eq(B.distance(D1), B.distance(D3), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 3\")\n", "possible_solution": {"points": {"A": [8.4, 5.5], "B": [27.3, 2.8], "C1": [15.78689904224884, 19.012724467686862], "C2": [10.606465390896968, 20.741112507910106], "C3": [5.148556174939747, 20.552844018738703], "D1": [32.438691840778105, 10.671076556959699], "D2": [28.71363856669628, 12.093095609254696], "D3": [24.734236580206467, 11.843055792908135]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the fixed pin A, so all three positions of C are the SAME distance from A", "so A is the circumcentre of the three C positions", "the same argument applies independently to B and the three D positions", "both pivots must be found this way - neither is given, and neither can be read off the other"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_fourbar_both_pivots_0003", "problem_id": "mechh_t4_mechh_fourbar_both_pivots_0003", "domain": "PyGeoX-Mech", "template": "mechh_fourbar_both_pivots", "tier": 4, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid and the two fixed pins never move. Neither fixed pin has been located yet - that is what has to be worked out. As the mechanism is worked through its travel, the two moving pins are required to pass through three matching pairs of positions, measured on the bench grid: C reaches (18.8261, 14.9418) while D is at (35.2283, 15.7443); then C reaches (12.5716, 18.3278) while D is at (29.1967, 16.8908); then C reaches (5.4624, 18.1219) while D is at (23.2649, 15.3077). Write down where each of the two fixed pins must be placed.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar with both fixed pivots unknown\n\nOracle formalization for mechh_t4_mechh_fourbar_both_pivots_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(51)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 18.8261, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 14.9418, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 12.5716, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 18.3278, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 5.4624, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 18.1219, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(D1.x, 35.2283, description=\"matching position 1 of the moving pin D (x)\")\nscene.constraint.eq(D1.y, 15.7443, description=\"matching position 1 of the moving pin D (y)\")\nscene.constraint.eq(D2.x, 29.1967, description=\"matching position 2 of the moving pin D (x)\")\nscene.constraint.eq(D2.y, 16.8908, description=\"matching position 2 of the moving pin D (y)\")\nscene.constraint.eq(D3.x, 23.2649, description=\"matching position 3 of the moving pin D (x)\")\nscene.constraint.eq(D3.y, 15.3077, description=\"matching position 3 of the moving pin D (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.distance(D1), B.distance(D2), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 2\")\nscene.constraint.eq(B.distance(D1), B.distance(D3), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 3\")\n", "possible_solution": {"points": {"A": [9.4, 5.0], "B": [29.7, 3.1], "C1": [18.826081040221844, 14.941780334687056], "C2": [12.571629905197577, 18.327819166857584], "C3": [5.462408238171294, 18.121942353065673], "D1": [35.228255349241735, 15.744302780049999], "D2": [29.196749148787312, 16.890820808811696], "D3": [23.264850811743997, 15.307737502292886]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the fixed pin A, so all three positions of C are the SAME distance from A", "so A is the circumcentre of the three C positions", "the same argument applies independently to B and the three D positions", "both pivots must be found this way - neither is given, and neither can be read off the other"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_fourbar_both_pivots_0004", "problem_id": "mechh_t4_mechh_fourbar_both_pivots_0004", "domain": "PyGeoX-Mech", "template": "mechh_fourbar_both_pivots", "tier": 4, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid and the two fixed pins never move. Neither fixed pin has been located yet - that is what has to be worked out. As the mechanism is worked through its travel, the two moving pins are required to pass through three matching pairs of positions, measured on the bench grid: C reaches (21.3491, 14.9782) while D is at (34.995, 18.6944); then C reaches (14.2335, 19.1298) while D is at (30.0432, 18.325); then C reaches (5.9956, 19.1908) while D is at (25.6566, 15.9979). Write down where each of the two fixed pins must be placed.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar with both fixed pivots unknown\n\nOracle formalization for mechh_t4_mechh_fourbar_both_pivots_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(50)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 21.3491, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 14.9782, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 14.2335, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 19.1298, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 5.9956, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 19.1908, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(D1.x, 34.995, description=\"matching position 1 of the moving pin D (x)\")\nscene.constraint.eq(D1.y, 18.6944, description=\"matching position 1 of the moving pin D (y)\")\nscene.constraint.eq(D2.x, 30.0432, description=\"matching position 2 of the moving pin D (x)\")\nscene.constraint.eq(D2.y, 18.325, description=\"matching position 2 of the moving pin D (y)\")\nscene.constraint.eq(D3.x, 25.6566, description=\"matching position 3 of the moving pin D (x)\")\nscene.constraint.eq(D3.y, 15.9979, description=\"matching position 3 of the moving pin D (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.distance(D1), B.distance(D2), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 2\")\nscene.constraint.eq(B.distance(D1), B.distance(D3), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 3\")\n", "possible_solution": {"points": {"A": [10.0, 3.7], "B": [33.4, 6.7], "C1": [21.3490979033856, 14.97820804823925], "C2": [14.233525900202743, 19.129752378191704], "C3": [5.9956282337685085, 19.190803941623173], "D1": [34.99503098971496, 18.694410204001233], "D2": [30.043157961325864, 18.325042431208157], "D3": [25.65662379102094, 15.997855929525748]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the fixed pin A, so all three positions of C are the SAME distance from A", "so A is the circumcentre of the three C positions", "the same argument applies independently to B and the three D positions", "both pivots must be found this way - neither is given, and neither can be read off the other"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_fourbar_both_pivots_0005", "problem_id": "mechh_t4_mechh_fourbar_both_pivots_0005", "domain": "PyGeoX-Mech", "template": "mechh_fourbar_both_pivots", "tier": 4, "nl_description": "A mechanism is built from three rigid bars pinned together in the plane. Bar 1 turns about a fixed pin A on the bench and carries a moving pin C at its far end. Bar 2 turns about a second fixed pin B and carries a moving pin D at its far end. Bar 3 joins pin C to pin D. The bars are rigid and the two fixed pins never move. Neither fixed pin has been located yet - that is what has to be worked out. As the mechanism is worked through its travel, the two moving pins are required to pass through three matching pairs of positions, measured on the bench grid: C reaches (17.8448, 16.9825) while D is at (36.1835, 15.0712); then C reaches (11.8502, 18.8603) while D is at (31.6025, 16.8345); then C reaches (5.641, 17.9072) while D is at (26.6973, 16.6497). Write down where each of the two fixed pins must be placed.", "answer_schema": {"points": ["A", "B", "C1", "C2", "C3", "D1", "D2", "D3"], "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": "\"\"\"Four-bar with both fixed pivots unknown\n\nOracle formalization for mechh_t4_mechh_fourbar_both_pivots_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(52)\n\nA, B, C1, C2, C3, D1, D2, D3 = scene.add.points([\"A\", \"B\", \"C1\", \"C2\", \"C3\", \"D1\", \"D2\", \"D3\"])\n\nscene.constraint.eq(C1.x, 17.8448, description=\"required position 1 of the moving pin C (x)\")\nscene.constraint.eq(C1.y, 16.9825, description=\"required position 1 of the moving pin C (y)\")\nscene.constraint.eq(C2.x, 11.8502, description=\"required position 2 of the moving pin C (x)\")\nscene.constraint.eq(C2.y, 18.8603, description=\"required position 2 of the moving pin C (y)\")\nscene.constraint.eq(C3.x, 5.641, description=\"required position 3 of the moving pin C (x)\")\nscene.constraint.eq(C3.y, 17.9072, description=\"required position 3 of the moving pin C (y)\")\nscene.constraint.eq(D1.x, 36.1835, description=\"matching position 1 of the moving pin D (x)\")\nscene.constraint.eq(D1.y, 15.0712, description=\"matching position 1 of the moving pin D (y)\")\nscene.constraint.eq(D2.x, 31.6025, description=\"matching position 2 of the moving pin D (x)\")\nscene.constraint.eq(D2.y, 16.8345, description=\"matching position 2 of the moving pin D (y)\")\nscene.constraint.eq(D3.x, 26.6973, description=\"matching position 3 of the moving pin D (x)\")\nscene.constraint.eq(D3.y, 16.6497, description=\"matching position 3 of the moving pin D (y)\")\nscene.constraint.eq(A.distance(C1), A.distance(C2), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 2\")\nscene.constraint.eq(A.distance(C1), A.distance(C3), description=\"bar AC is rigid, so C is the same distance from A in positions 1 and 3\")\nscene.constraint.eq(B.distance(D1), B.distance(D2), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 2\")\nscene.constraint.eq(B.distance(D1), B.distance(D3), description=\"bar BD is rigid, so D is the same distance from B in positions 1 and 3\")\n", "possible_solution": {"points": {"A": [10.8, 5.0], "B": [29.6, 4.8], "C1": [17.84484233283951, 16.982495420630563], "C2": [11.850164232963252, 18.86027254724108], "C3": [5.640995143607335, 17.907155724314975], "D1": [36.183549428136814, 15.07116726216059], "D2": [31.602531504347176, 16.834528140899295], "D3": [26.697331031289778, 16.64966298500032]}, "circles": {}}, "implicit_facts": ["bar 1 turns about the fixed pin A, so all three positions of C are the SAME distance from A", "so A is the circumcentre of the three C positions", "the same argument applies independently to B and the three D positions", "both pivots must be found this way - neither is given, and neither can be read off the other"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_quick_return_extremes_0071", "problem_id": "mechh_t4_mechh_quick_return_extremes_0071", "domain": "PyGeoX-Mech", "template": "mechh_quick_return_extremes", "tier": 4, "nl_description": "A quick-return drive has a crank 8.7 mm long turning about a fixed centre O, and a long slotted lever pivoted to the bench at S, directly below O and 20.7 mm down from it. Set up coordinates with O at the origin. The crank pin slides in the lever's slot, so it always sits somewhere along the lever between the pivot S and the lever's tip; the lever measures 31.1 mm from S to its tip. As the crank goes round, the lever rocks back and forth between two extreme positions, and it reverses at exactly the moments when the crank has swung as far to one side as it can take the lever - which is when the crank itself lies square to the lever. Show the mechanism at both of those turning points: the crank pin at P1 with the lever tip at T1 for the extreme where the tip has swung to the right, and the crank pin at P2 with the lever tip at T2 for the extreme where it has swung to the left.", "answer_schema": {"points": ["O", "P1", "P2", "S", "T1", "T2"], "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": "\"\"\"Quick-return drive at both reversal points\n\nOracle formalization for mechh_t4_mechh_quick_return_extremes_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(32)\n\nO, S, P1, T1, P2, T2 = scene.add.points([\"O\", \"S\", \"P1\", \"T1\", \"P2\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 20.7, description=\"the lever pivot S is 20.7 below the crank centre\")\nscene.constraint.eq(O.distance(P1), 8.7, description=\"the crank is 8.7 long (first extreme)\")\nscene.constraint.eq(O.distance(P2), 8.7, description=\"the crank is 8.7 long (second extreme)\")\nscene.constraint.eq(S.distance(T1), 31.1, description=\"the lever measures 31.1 from its pivot to its tip (first extreme)\")\nscene.constraint.eq(S.distance(T2), 31.1, description=\"the lever measures 31.1 from its pivot to its tip (second extreme)\")\nscene.relate.point_lies_on(P1, scene.add.line_segment(S, T1))\nscene.relate.point_lies_on(P2, scene.add.line_segment(S, T2))\nscene.relate.perpendicular(scene.add.line_segment(O, P1), scene.add.line_segment(S, P1))\nscene.relate.perpendicular(scene.add.line_segment(O, P2), scene.add.line_segment(S, P2))\nscene.constraint.gt(T1.x - S.x, 0.05, description=\"the first extreme is the one where the lever tip has swung to the right\")\nscene.constraint.lt(T2.x - S.x, -0.05, description=\"the second extreme is the one where the lever tip has swung to the left\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -20.7], "P1": [7.894292164042734, -3.6565217391304343], "T1": [13.071014492753623, 7.519826011693002], "P2": [-7.894292164042734, -3.6565217391304343], "T2": [-13.071014492753623, 7.519826011693002]}, "circles": {}}, "implicit_facts": ["the lever reverses exactly when the crank is square to the lever, which is the geometric condition behind 'as far as it can take it'", "so at each extreme, the crank OP is perpendicular to the lever line SP", "that makes O, P and S a right-angled triangle with the right angle at P, fixing the crank pin", "the lever tip then lies on the ray from S through P at the lever's full length"], "branch_facts": ["P1/T1 is the right-hand extreme and P2/T2 the left-hand one"]} {"unique_id": "mechh_t4_mechh_quick_return_extremes_0072", "problem_id": "mechh_t4_mechh_quick_return_extremes_0072", "domain": "PyGeoX-Mech", "template": "mechh_quick_return_extremes", "tier": 4, "nl_description": "A quick-return drive has a crank 9.6 mm long turning about a fixed centre O, and a long slotted lever pivoted to the bench at S, directly below O and 18.3 mm down from it. Set up coordinates with O at the origin. The crank pin slides in the lever's slot, so it always sits somewhere along the lever between the pivot S and the lever's tip; the lever measures 30.3 mm from S to its tip. As the crank goes round, the lever rocks back and forth between two extreme positions, and it reverses at exactly the moments when the crank has swung as far to one side as it can take the lever - which is when the crank itself lies square to the lever. Show the mechanism at both of those turning points: the crank pin at P1 with the lever tip at T1 for the extreme where the tip has swung to the right, and the crank pin at P2 with the lever tip at T2 for the extreme where it has swung to the left.", "answer_schema": {"points": ["O", "P1", "P2", "S", "T1", "T2"], "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": "\"\"\"Quick-return drive at both reversal points\n\nOracle formalization for mechh_t4_mechh_quick_return_extremes_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(29)\n\nO, S, P1, T1, P2, T2 = scene.add.points([\"O\", \"S\", \"P1\", \"T1\", \"P2\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 18.3, description=\"the lever pivot S is 18.3 below the crank centre\")\nscene.constraint.eq(O.distance(P1), 9.6, description=\"the crank is 9.6 long (first extreme)\")\nscene.constraint.eq(O.distance(P2), 9.6, description=\"the crank is 9.6 long (second extreme)\")\nscene.constraint.eq(S.distance(T1), 30.3, description=\"the lever measures 30.3 from its pivot to its tip (first extreme)\")\nscene.constraint.eq(S.distance(T2), 30.3, description=\"the lever measures 30.3 from its pivot to its tip (second extreme)\")\nscene.relate.point_lies_on(P1, scene.add.line_segment(S, T1))\nscene.relate.point_lies_on(P2, scene.add.line_segment(S, T2))\nscene.relate.perpendicular(scene.add.line_segment(O, P1), scene.add.line_segment(S, P1))\nscene.relate.perpendicular(scene.add.line_segment(O, P2), scene.add.line_segment(S, P2))\nscene.constraint.gt(T1.x - S.x, 0.05, description=\"the first extreme is the one where the lever tip has swung to the right\")\nscene.constraint.lt(T2.x - S.x, -0.05, description=\"the second extreme is the one where the lever tip has swung to the left\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -18.3], "P1": [8.173007007013005, -5.036065573770491], "T1": [15.895081967213113, 7.496053365884798], "P2": [-8.173007007013005, -5.036065573770491], "T2": [-15.895081967213113, 7.496053365884798]}, "circles": {}}, "implicit_facts": ["the lever reverses exactly when the crank is square to the lever, which is the geometric condition behind 'as far as it can take it'", "so at each extreme, the crank OP is perpendicular to the lever line SP", "that makes O, P and S a right-angled triangle with the right angle at P, fixing the crank pin", "the lever tip then lies on the ray from S through P at the lever's full length"], "branch_facts": ["P1/T1 is the right-hand extreme and P2/T2 the left-hand one"]} {"unique_id": "mechh_t4_mechh_quick_return_extremes_0073", "problem_id": "mechh_t4_mechh_quick_return_extremes_0073", "domain": "PyGeoX-Mech", "template": "mechh_quick_return_extremes", "tier": 4, "nl_description": "A quick-return drive has a crank 10 mm long turning about a fixed centre O, and a long slotted lever pivoted to the bench at S, directly below O and 19.1 mm down from it. Set up coordinates with O at the origin. The crank pin slides in the lever's slot, so it always sits somewhere along the lever between the pivot S and the lever's tip; the lever measures 29.8 mm from S to its tip. As the crank goes round, the lever rocks back and forth between two extreme positions, and it reverses at exactly the moments when the crank has swung as far to one side as it can take the lever - which is when the crank itself lies square to the lever. Show the mechanism at both of those turning points: the crank pin at P1 with the lever tip at T1 for the extreme where the tip has swung to the right, and the crank pin at P2 with the lever tip at T2 for the extreme where it has swung to the left.", "answer_schema": {"points": ["O", "P1", "P2", "S", "T1", "T2"], "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": "\"\"\"Quick-return drive at both reversal points\n\nOracle formalization for mechh_t4_mechh_quick_return_extremes_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(30)\n\nO, S, P1, T1, P2, T2 = scene.add.points([\"O\", \"S\", \"P1\", \"T1\", \"P2\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 19.1, description=\"the lever pivot S is 19.1 below the crank centre\")\nscene.constraint.eq(O.distance(P1), 10, description=\"the crank is 10 long (first extreme)\")\nscene.constraint.eq(O.distance(P2), 10, description=\"the crank is 10 long (second extreme)\")\nscene.constraint.eq(S.distance(T1), 29.8, description=\"the lever measures 29.8 from its pivot to its tip (first extreme)\")\nscene.constraint.eq(S.distance(T2), 29.8, description=\"the lever measures 29.8 from its pivot to its tip (second extreme)\")\nscene.relate.point_lies_on(P1, scene.add.line_segment(S, T1))\nscene.relate.point_lies_on(P2, scene.add.line_segment(S, T2))\nscene.relate.perpendicular(scene.add.line_segment(O, P1), scene.add.line_segment(S, P1))\nscene.relate.perpendicular(scene.add.line_segment(O, P2), scene.add.line_segment(S, P2))\nscene.constraint.gt(T1.x - S.x, 0.05, description=\"the first extreme is the one where the lever tip has swung to the right\")\nscene.constraint.lt(T2.x - S.x, -0.05, description=\"the second extreme is the one where the lever tip has swung to the left\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -19.1], "P1": [8.519886778049397, -5.2356020942408374], "T1": [15.602094240837694, 6.289262598587207], "P2": [-8.519886778049397, -5.2356020942408374], "T2": [-15.602094240837694, 6.289262598587207]}, "circles": {}}, "implicit_facts": ["the lever reverses exactly when the crank is square to the lever, which is the geometric condition behind 'as far as it can take it'", "so at each extreme, the crank OP is perpendicular to the lever line SP", "that makes O, P and S a right-angled triangle with the right angle at P, fixing the crank pin", "the lever tip then lies on the ray from S through P at the lever's full length"], "branch_facts": ["P1/T1 is the right-hand extreme and P2/T2 the left-hand one"]} {"unique_id": "mechh_t4_mechh_quick_return_extremes_0074", "problem_id": "mechh_t4_mechh_quick_return_extremes_0074", "domain": "PyGeoX-Mech", "template": "mechh_quick_return_extremes", "tier": 4, "nl_description": "A quick-return drive has a crank 10.3 mm long turning about a fixed centre O, and a long slotted lever pivoted to the bench at S, directly below O and 21.2 mm down from it. Set up coordinates with O at the origin. The crank pin slides in the lever's slot, so it always sits somewhere along the lever between the pivot S and the lever's tip; the lever measures 27.3 mm from S to its tip. As the crank goes round, the lever rocks back and forth between two extreme positions, and it reverses at exactly the moments when the crank has swung as far to one side as it can take the lever - which is when the crank itself lies square to the lever. Show the mechanism at both of those turning points: the crank pin at P1 with the lever tip at T1 for the extreme where the tip has swung to the right, and the crank pin at P2 with the lever tip at T2 for the extreme where it has swung to the left.", "answer_schema": {"points": ["O", "P1", "P2", "S", "T1", "T2"], "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": "\"\"\"Quick-return drive at both reversal points\n\nOracle formalization for mechh_t4_mechh_quick_return_extremes_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(33)\n\nO, S, P1, T1, P2, T2 = scene.add.points([\"O\", \"S\", \"P1\", \"T1\", \"P2\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 21.2, description=\"the lever pivot S is 21.2 below the crank centre\")\nscene.constraint.eq(O.distance(P1), 10.3, description=\"the crank is 10.3 long (first extreme)\")\nscene.constraint.eq(O.distance(P2), 10.3, description=\"the crank is 10.3 long (second extreme)\")\nscene.constraint.eq(S.distance(T1), 27.3, description=\"the lever measures 27.3 from its pivot to its tip (first extreme)\")\nscene.constraint.eq(S.distance(T2), 27.3, description=\"the lever measures 27.3 from its pivot to its tip (second extreme)\")\nscene.relate.point_lies_on(P1, scene.add.line_segment(S, T1))\nscene.relate.point_lies_on(P2, scene.add.line_segment(S, T2))\nscene.relate.perpendicular(scene.add.line_segment(O, P1), scene.add.line_segment(S, P1))\nscene.relate.perpendicular(scene.add.line_segment(O, P2), scene.add.line_segment(S, P2))\nscene.constraint.gt(T1.x - S.x, 0.05, description=\"the first extreme is the one where the lever tip has swung to the right\")\nscene.constraint.lt(T2.x - S.x, -0.05, description=\"the second extreme is the one where the lever tip has swung to the left\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -21.2], "P1": [9.002640120952487, -5.004245283018869], "T1": [13.263679245283019, 2.6613665341750377], "P2": [-9.002640120952487, -5.004245283018869], "T2": [-13.263679245283019, 2.6613665341750377]}, "circles": {}}, "implicit_facts": ["the lever reverses exactly when the crank is square to the lever, which is the geometric condition behind 'as far as it can take it'", "so at each extreme, the crank OP is perpendicular to the lever line SP", "that makes O, P and S a right-angled triangle with the right angle at P, fixing the crank pin", "the lever tip then lies on the ray from S through P at the lever's full length"], "branch_facts": ["P1/T1 is the right-hand extreme and P2/T2 the left-hand one"]} {"unique_id": "mechh_t4_mechh_quick_return_extremes_0075", "problem_id": "mechh_t4_mechh_quick_return_extremes_0075", "domain": "PyGeoX-Mech", "template": "mechh_quick_return_extremes", "tier": 4, "nl_description": "A quick-return drive has a crank 7.1 mm long turning about a fixed centre O, and a long slotted lever pivoted to the bench at S, directly below O and 23.8 mm down from it. Set up coordinates with O at the origin. The crank pin slides in the lever's slot, so it always sits somewhere along the lever between the pivot S and the lever's tip; the lever measures 31.4 mm from S to its tip. As the crank goes round, the lever rocks back and forth between two extreme positions, and it reverses at exactly the moments when the crank has swung as far to one side as it can take the lever - which is when the crank itself lies square to the lever. Show the mechanism at both of those turning points: the crank pin at P1 with the lever tip at T1 for the extreme where the tip has swung to the right, and the crank pin at P2 with the lever tip at T2 for the extreme where it has swung to the left.", "answer_schema": {"points": ["O", "P1", "P2", "S", "T1", "T2"], "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": "\"\"\"Quick-return drive at both reversal points\n\nOracle formalization for mechh_t4_mechh_quick_return_extremes_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(36)\n\nO, S, P1, T1, P2, T2 = scene.add.points([\"O\", \"S\", \"P1\", \"T1\", \"P2\", \"T2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the crank centre (y)\")\nscene.constraint.eq(S.x, O.x, description=\"the lever pivot S is directly below the crank centre\")\nscene.constraint.eq(O.y - S.y, 23.8, description=\"the lever pivot S is 23.8 below the crank centre\")\nscene.constraint.eq(O.distance(P1), 7.1, description=\"the crank is 7.1 long (first extreme)\")\nscene.constraint.eq(O.distance(P2), 7.1, description=\"the crank is 7.1 long (second extreme)\")\nscene.constraint.eq(S.distance(T1), 31.4, description=\"the lever measures 31.4 from its pivot to its tip (first extreme)\")\nscene.constraint.eq(S.distance(T2), 31.4, description=\"the lever measures 31.4 from its pivot to its tip (second extreme)\")\nscene.relate.point_lies_on(P1, scene.add.line_segment(S, T1))\nscene.relate.point_lies_on(P2, scene.add.line_segment(S, T2))\nscene.relate.perpendicular(scene.add.line_segment(O, P1), scene.add.line_segment(S, P1))\nscene.relate.perpendicular(scene.add.line_segment(O, P2), scene.add.line_segment(S, P2))\nscene.constraint.gt(T1.x - S.x, 0.05, description=\"the first extreme is the one where the lever tip has swung to the right\")\nscene.constraint.lt(T2.x - S.x, -0.05, description=\"the second extreme is the one where the lever tip has swung to the left\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "S": [0.0, -23.8], "P1": [6.7767094686412, -2.118067226890756], "T1": [9.3672268907563, 6.170236241596296], "P2": [-6.7767094686412, -2.118067226890756], "T2": [-9.3672268907563, 6.170236241596296]}, "circles": {}}, "implicit_facts": ["the lever reverses exactly when the crank is square to the lever, which is the geometric condition behind 'as far as it can take it'", "so at each extreme, the crank OP is perpendicular to the lever line SP", "that makes O, P and S a right-angled triangle with the right angle at P, fixing the crank pin", "the lever tip then lies on the ray from S through P at the lever's full length"], "branch_facts": ["P1/T1 is the right-hand extreme and P2/T2 the left-hand one"]} {"unique_id": "mechh_t4_mechh_rigid_body_guidance_0011", "problem_id": "mechh_t4_mechh_rigid_body_guidance_0011", "domain": "PyGeoX-Mech", "template": "mechh_rigid_body_guidance", "tier": 4, "nl_description": "A rigid carrier plate has two pins driven into it, P and Q. The plate has to be swung between two working poses by hanging it on two bars, one pinned to the bench at A and carrying P, the other pinned to the bench at B and carrying Q. Both bench pins must be mounted somewhere on the bench base line, which is the line at height 0 on the bench grid. In the first pose, P sits at (13, 11.4) and Q sits at (25.7775, 18.8683). In the second pose, P has moved to (14.7237, 8.6862) and Q to (22.8066, 21.0841). Both bars are rigid, so neither changes length as the plate swings. Work out where the two bench pins A and B have to go.", "answer_schema": {"points": ["A", "B", "P1", "P2", "Q1", "Q2"], "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": "\"\"\"Rigid-body guidance through two poses\n\nOracle formalization for mechh_t4_mechh_rigid_body_guidance_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(39)\n\nA, B, P1, Q1, P2, Q2 = scene.add.points([\"A\", \"B\", \"P1\", \"Q1\", \"P2\", \"Q2\"])\n\nscene.constraint.eq(P1.x, 13, description=\"first pose, pin P (x)\")\nscene.constraint.eq(P1.y, 11.4, description=\"first pose, pin P (y)\")\nscene.constraint.eq(Q1.x, 25.7775, description=\"first pose, pin Q (x)\")\nscene.constraint.eq(Q1.y, 18.8683, description=\"first pose, pin Q (y)\")\nscene.constraint.eq(P2.x, 14.7237, description=\"second pose, pin P (x)\")\nscene.constraint.eq(P2.y, 8.6862, description=\"second pose, pin P (y)\")\nscene.constraint.eq(Q2.x, 22.8066, description=\"second pose, pin Q (x)\")\nscene.constraint.eq(Q2.y, 21.0841, description=\"second pose, pin Q (y)\")\nscene.constraint.eq(A.y, 0, description=\"the fixed pin A is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(B.y, 0, description=\"the fixed pin B is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(A.distance(P1), A.distance(P2), description=\"the bar from A to P is rigid, so P is the same distance from A in both poses\")\nscene.constraint.eq(B.distance(Q1), B.distance(Q2), description=\"the bar from B to Q is rigid, so Q is the same distance from B in both poses\")\n", "possible_solution": {"points": {"A": [-1.9500043243135146, 0.0], "B": [9.393360181183553, 0.0], "P1": [13.0, 11.4], "Q1": [25.777494894521325, 18.86830798912856], "P2": [14.72369423683444, 8.686219292014531], "Q2": [22.806567391625233, 21.08408852682913]}, "circles": {}}, "implicit_facts": ["the bar from A to P is rigid and A is fixed, so P is the same distance from A in both poses", "that puts A on the perpendicular bisector of the segment joining P's two positions", "combined with 'A is on the base line', that pins A completely", "the identical argument, applied separately to Q, pins B - the two pivots are found independently"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_rigid_body_guidance_0012", "problem_id": "mechh_t4_mechh_rigid_body_guidance_0012", "domain": "PyGeoX-Mech", "template": "mechh_rigid_body_guidance", "tier": 4, "nl_description": "A rigid carrier plate has two pins driven into it, P and Q. The plate has to be swung between two working poses by hanging it on two bars, one pinned to the bench at A and carrying P, the other pinned to the bench at B and carrying Q. Both bench pins must be mounted somewhere on the bench base line, which is the line at height 0 on the bench grid. In the first pose, P sits at (10.7, 12.8) and Q sits at (24.8014, 15.7174). In the second pose, P has moved to (12.6307, 8.9079) and Q to (22.407, 19.4806). Both bars are rigid, so neither changes length as the plate swings. Work out where the two bench pins A and B have to go.", "answer_schema": {"points": ["A", "B", "P1", "P2", "Q1", "Q2"], "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": "\"\"\"Rigid-body guidance through two poses\n\nOracle formalization for mechh_t4_mechh_rigid_body_guidance_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(37)\n\nA, B, P1, Q1, P2, Q2 = scene.add.points([\"A\", \"B\", \"P1\", \"Q1\", \"P2\", \"Q2\"])\n\nscene.constraint.eq(P1.x, 10.7, description=\"first pose, pin P (x)\")\nscene.constraint.eq(P1.y, 12.8, description=\"first pose, pin P (y)\")\nscene.constraint.eq(Q1.x, 24.8014, description=\"first pose, pin Q (x)\")\nscene.constraint.eq(Q1.y, 15.7174, description=\"first pose, pin Q (y)\")\nscene.constraint.eq(P2.x, 12.6307, description=\"second pose, pin P (x)\")\nscene.constraint.eq(P2.y, 8.9079, description=\"second pose, pin P (y)\")\nscene.constraint.eq(Q2.x, 22.407, description=\"second pose, pin Q (x)\")\nscene.constraint.eq(Q2.y, 19.4806, description=\"second pose, pin Q (y)\")\nscene.constraint.eq(A.y, 0, description=\"the fixed pin A is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(B.y, 0, description=\"the fixed pin B is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(A.distance(P1), A.distance(P2), description=\"the bar from A to P is rigid, so P is the same distance from A in both poses\")\nscene.constraint.eq(B.distance(Q1), B.distance(Q2), description=\"the bar from B to Q is rigid, so Q is the same distance from B in both poses\")\n", "possible_solution": {"points": {"A": [-10.215505798061443, 0.0], "B": [-4.056205257998403, 0.0], "P1": [10.7, 12.8], "Q1": [24.80136777331723, 15.717434990132634], "P2": [12.630680773482757, 8.907869662659827], "Q2": [22.407030433406543, 19.480617050491816]}, "circles": {}}, "implicit_facts": ["the bar from A to P is rigid and A is fixed, so P is the same distance from A in both poses", "that puts A on the perpendicular bisector of the segment joining P's two positions", "combined with 'A is on the base line', that pins A completely", "the identical argument, applied separately to Q, pins B - the two pivots are found independently"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_rigid_body_guidance_0013", "problem_id": "mechh_t4_mechh_rigid_body_guidance_0013", "domain": "PyGeoX-Mech", "template": "mechh_rigid_body_guidance", "tier": 4, "nl_description": "A rigid carrier plate has two pins driven into it, P and Q. The plate has to be swung between two working poses by hanging it on two bars, one pinned to the bench at A and carrying P, the other pinned to the bench at B and carrying Q. Both bench pins must be mounted somewhere on the bench base line, which is the line at height 0 on the bench grid. In the first pose, P sits at (11, 12.3) and Q sits at (19.1215, 17.2287). In the second pose, P has moved to (13.0629, 8.8146) and Q to (16.501, 17.6707). Both bars are rigid, so neither changes length as the plate swings. Work out where the two bench pins A and B have to go.", "answer_schema": {"points": ["A", "B", "P1", "P2", "Q1", "Q2"], "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": "\"\"\"Rigid-body guidance through two poses\n\nOracle formalization for mechh_t4_mechh_rigid_body_guidance_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(30)\n\nA, B, P1, Q1, P2, Q2 = scene.add.points([\"A\", \"B\", \"P1\", \"Q1\", \"P2\", \"Q2\"])\n\nscene.constraint.eq(P1.x, 11, description=\"first pose, pin P (x)\")\nscene.constraint.eq(P1.y, 12.3, description=\"first pose, pin P (y)\")\nscene.constraint.eq(Q1.x, 19.1215, description=\"first pose, pin Q (x)\")\nscene.constraint.eq(Q1.y, 17.2287, description=\"first pose, pin Q (y)\")\nscene.constraint.eq(P2.x, 13.0629, description=\"second pose, pin P (x)\")\nscene.constraint.eq(P2.y, 8.8146, description=\"second pose, pin P (y)\")\nscene.constraint.eq(Q2.x, 16.501, description=\"second pose, pin Q (x)\")\nscene.constraint.eq(Q2.y, 17.6707, description=\"second pose, pin Q (y)\")\nscene.constraint.eq(A.y, 0, description=\"the fixed pin A is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(B.y, 0, description=\"the fixed pin B is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(A.distance(P1), A.distance(P2), description=\"the bar from A to P is rigid, so P is the same distance from A in both poses\")\nscene.constraint.eq(B.distance(Q1), B.distance(Q2), description=\"the bar from B to Q is rigid, so Q is the same distance from B in both poses\")\n", "possible_solution": {"points": {"A": [-5.80551576485127, 0.0], "B": [14.868102301205974, 0.0], "P1": [11.0, 12.3], "Q1": [19.12145779348456, 17.22868372982574], "P2": [13.062932168789128, 8.814592032429292], "Q2": [16.500952591400967, 17.670664275455767]}, "circles": {}}, "implicit_facts": ["the bar from A to P is rigid and A is fixed, so P is the same distance from A in both poses", "that puts A on the perpendicular bisector of the segment joining P's two positions", "combined with 'A is on the base line', that pins A completely", "the identical argument, applied separately to Q, pins B - the two pivots are found independently"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_rigid_body_guidance_0014", "problem_id": "mechh_t4_mechh_rigid_body_guidance_0014", "domain": "PyGeoX-Mech", "template": "mechh_rigid_body_guidance", "tier": 4, "nl_description": "A rigid carrier plate has two pins driven into it, P and Q. The plate has to be swung between two working poses by hanging it on two bars, one pinned to the bench at A and carrying P, the other pinned to the bench at B and carrying Q. Both bench pins must be mounted somewhere on the bench base line, which is the line at height 0 on the bench grid. In the first pose, P sits at (9.1, 10.1) and Q sits at (22.0426, 14.2771). In the second pose, P has moved to (11.2469, 8.0938) and Q to (20.1216, 18.3992). Both bars are rigid, so neither changes length as the plate swings. Work out where the two bench pins A and B have to go.", "answer_schema": {"points": ["A", "B", "P1", "P2", "Q1", "Q2"], "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": "\"\"\"Rigid-body guidance through two poses\n\nOracle formalization for mechh_t4_mechh_rigid_body_guidance_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(34)\n\nA, B, P1, Q1, P2, Q2 = scene.add.points([\"A\", \"B\", \"P1\", \"Q1\", \"P2\", \"Q2\"])\n\nscene.constraint.eq(P1.x, 9.1, description=\"first pose, pin P (x)\")\nscene.constraint.eq(P1.y, 10.1, description=\"first pose, pin P (y)\")\nscene.constraint.eq(Q1.x, 22.0426, description=\"first pose, pin Q (x)\")\nscene.constraint.eq(Q1.y, 14.2771, description=\"first pose, pin Q (y)\")\nscene.constraint.eq(P2.x, 11.2469, description=\"second pose, pin P (x)\")\nscene.constraint.eq(P2.y, 8.0938, description=\"second pose, pin P (y)\")\nscene.constraint.eq(Q2.x, 20.1216, description=\"second pose, pin Q (x)\")\nscene.constraint.eq(Q2.y, 18.3992, description=\"second pose, pin Q (y)\")\nscene.constraint.eq(A.y, 0, description=\"the fixed pin A is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(B.y, 0, description=\"the fixed pin B is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(A.distance(P1), A.distance(P2), description=\"the bar from A to P is rigid, so P is the same distance from A in both poses\")\nscene.constraint.eq(B.distance(Q1), B.distance(Q2), description=\"the bar from B to Q is rigid, so Q is the same distance from B in both poses\")\n", "possible_solution": {"points": {"A": [1.6727260613064576, 0.0], "B": [-13.974783367652307, 0.0], "P1": [9.1, 10.1], "Q1": [22.04263191547887, 14.277113728452655], "P2": [11.246875713197849, 8.093828303317549], "Q2": [20.121566867417734, 18.39915978353745]}, "circles": {}}, "implicit_facts": ["the bar from A to P is rigid and A is fixed, so P is the same distance from A in both poses", "that puts A on the perpendicular bisector of the segment joining P's two positions", "combined with 'A is on the base line', that pins A completely", "the identical argument, applied separately to Q, pins B - the two pivots are found independently"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_rigid_body_guidance_0015", "problem_id": "mechh_t4_mechh_rigid_body_guidance_0015", "domain": "PyGeoX-Mech", "template": "mechh_rigid_body_guidance", "tier": 4, "nl_description": "A rigid carrier plate has two pins driven into it, P and Q. The plate has to be swung between two working poses by hanging it on two bars, one pinned to the bench at A and carrying P, the other pinned to the bench at B and carrying Q. Both bench pins must be mounted somewhere on the bench base line, which is the line at height 0 on the bench grid. In the first pose, P sits at (9.4, 11.7) and Q sits at (18.362, 16.9765). In the second pose, P has moved to (11.3119, 9.3569) and Q to (15.4184, 18.9118). Both bars are rigid, so neither changes length as the plate swings. Work out where the two bench pins A and B have to go.", "answer_schema": {"points": ["A", "B", "P1", "P2", "Q1", "Q2"], "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": "\"\"\"Rigid-body guidance through two poses\n\nOracle formalization for mechh_t4_mechh_rigid_body_guidance_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(30)\n\nA, B, P1, Q1, P2, Q2 = scene.add.points([\"A\", \"B\", \"P1\", \"Q1\", \"P2\", \"Q2\"])\n\nscene.constraint.eq(P1.x, 9.4, description=\"first pose, pin P (x)\")\nscene.constraint.eq(P1.y, 11.7, description=\"first pose, pin P (y)\")\nscene.constraint.eq(Q1.x, 18.362, description=\"first pose, pin Q (x)\")\nscene.constraint.eq(Q1.y, 16.9765, description=\"first pose, pin Q (y)\")\nscene.constraint.eq(P2.x, 11.3119, description=\"second pose, pin P (x)\")\nscene.constraint.eq(P2.y, 9.3569, description=\"second pose, pin P (y)\")\nscene.constraint.eq(Q2.x, 15.4184, description=\"second pose, pin Q (x)\")\nscene.constraint.eq(Q2.y, 18.9118, description=\"second pose, pin Q (y)\")\nscene.constraint.eq(A.y, 0, description=\"the fixed pin A is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(B.y, 0, description=\"the fixed pin B is mounted on the bench base line, the line y = 0\")\nscene.constraint.eq(A.distance(P1), A.distance(P2), description=\"the bar from A to P is rigid, so P is the same distance from A in both poses\")\nscene.constraint.eq(B.distance(Q1), B.distance(Q2), description=\"the bar from B to Q is rigid, so Q is the same distance from B in both poses\")\n", "possible_solution": {"points": {"A": [-2.547369927641453, 0.0], "B": [5.093076421663792, 0.0], "P1": [9.4, 11.7], "Q1": [18.362045549418237, 16.97652722632536], "P2": [11.311876297919335, 9.356866048372734], "Q2": [15.418428534099021, 18.911772051653333]}, "circles": {}}, "implicit_facts": ["the bar from A to P is rigid and A is fixed, so P is the same distance from A in both poses", "that puts A on the perpendicular bisector of the segment joining P's two positions", "combined with 'A is on the base line', that pins A completely", "the identical argument, applied separately to Q, pins B - the two pivots are found independently"], "branch_facts": []} {"unique_id": "mechh_t4_mechh_scissor_unequal_0031", "problem_id": "mechh_t4_mechh_scissor_unequal_0031", "domain": "PyGeoX-Mech", "template": "mechh_scissor_unequal", "tier": 4, "nl_description": "Two straight bars of different lengths are crossed over each other and pinned together at a single point X. The first bar is 36.3 mm long and the pin sits 17.424 mm along it from its lower end; the second bar is 28.2 mm long and the pin sits 16.638 mm along that one from its lower end - so the pin is NOT at the middle of either bar. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart. The upper ends are U1 on the first bar and U2 on the second, and both bars run upward from the floor through the crossing pin. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 8.7 mm above the floor, with F2 to the right of F1.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair with unequal bars pinned off-centre\n\nOracle formalization for mechh_t4_mechh_scissor_unequal_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(46)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 3.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.constraint.eq(F1.distance(U1), 36.3, description=\"the first bar is 36.3 long\")\nscene.constraint.eq(F2.distance(U2), 28.2, description=\"the second bar is 28.2 long\")\nscene.relate.point_lies_on(X, scene.add.line_segment(F1, U1))\nscene.relate.point_lies_on(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(X), 17.424, description=\"the crossing pin is 17.424 along the first bar from its lower end\")\nscene.constraint.eq(F2.distance(X), 16.638, description=\"the crossing pin is 16.638 along the second bar from its lower end\")\nscene.constraint.eq(X.y - F1.y, 8.7, description=\"the crossing pin sits 8.7 above the floor\")\nscene.constraint.gt(U1.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\nscene.constraint.gt(U2.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [29.27868668238522, 0.0], "U1": [31.45114266604633, 18.125], "U2": [5.241164304956442, 14.745762711864405], "X": [15.09654847970224, 8.7]}, "circles": {}}, "implicit_facts": ["the pin is NOT the midpoint here, so the usual 'upper end is twice the pin height' shortcut does not apply", "each lower end is sqrt(its own pin distance squared minus the pin height squared) horizontally from X", "each upper end lies on the same straight line out through X, at its own bar length from its lower end", "the two bars have different lengths AND different pin distances, so the two halves are not mirror images"], "branch_facts": ["F2 lies to the right of F1, which picks which side the second bar leans"]} {"unique_id": "mechh_t4_mechh_scissor_unequal_0032", "problem_id": "mechh_t4_mechh_scissor_unequal_0032", "domain": "PyGeoX-Mech", "template": "mechh_scissor_unequal", "tier": 4, "nl_description": "Two straight bars of different lengths are crossed over each other and pinned together at a single point X. The first bar is 28.1 mm long and the pin sits 15.455 mm along it from its lower end; the second bar is 35.6 mm long and the pin sits 16.732 mm along that one from its lower end - so the pin is NOT at the middle of either bar. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart. The upper ends are U1 on the first bar and U2 on the second, and both bars run upward from the floor through the crossing pin. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 8.9 mm above the floor, with F2 to the right of F1.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair with unequal bars pinned off-centre\n\nOracle formalization for mechh_t4_mechh_scissor_unequal_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(40)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 3.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.constraint.eq(F1.distance(U1), 28.1, description=\"the first bar is 28.1 long\")\nscene.constraint.eq(F2.distance(U2), 35.6, description=\"the second bar is 35.6 long\")\nscene.relate.point_lies_on(X, scene.add.line_segment(F1, U1))\nscene.relate.point_lies_on(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(X), 15.455, description=\"the crossing pin is 15.455 along the first bar from its lower end\")\nscene.constraint.eq(F2.distance(X), 16.732, description=\"the crossing pin is 16.732 along the second bar from its lower end\")\nscene.constraint.eq(X.y - F1.y, 8.9, description=\"the crossing pin sits 8.9 above the floor\")\nscene.constraint.gt(U1.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\nscene.constraint.gt(U2.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [26.803771479443274, 0.0], "U1": [22.973000681900015, 16.181818181818183], "U2": [-3.3422308703402734, 18.93617021276596], "X": [12.635150375045008, 8.9]}, "circles": {}}, "implicit_facts": ["the pin is NOT the midpoint here, so the usual 'upper end is twice the pin height' shortcut does not apply", "each lower end is sqrt(its own pin distance squared minus the pin height squared) horizontally from X", "each upper end lies on the same straight line out through X, at its own bar length from its lower end", "the two bars have different lengths AND different pin distances, so the two halves are not mirror images"], "branch_facts": ["F2 lies to the right of F1, which picks which side the second bar leans"]} {"unique_id": "mechh_t4_mechh_scissor_unequal_0033", "problem_id": "mechh_t4_mechh_scissor_unequal_0033", "domain": "PyGeoX-Mech", "template": "mechh_scissor_unequal", "tier": 4, "nl_description": "Two straight bars of different lengths are crossed over each other and pinned together at a single point X. The first bar is 31 mm long and the pin sits 13.02 mm along it from its lower end; the second bar is 28.8 mm long and the pin sits 13.248 mm along that one from its lower end - so the pin is NOT at the middle of either bar. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart. The upper ends are U1 on the first bar and U2 on the second, and both bars run upward from the floor through the crossing pin. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 8.3 mm above the floor, with F2 to the right of F1.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair with unequal bars pinned off-centre\n\nOracle formalization for mechh_t4_mechh_scissor_unequal_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(36)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 3.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.constraint.eq(F1.distance(U1), 31, description=\"the first bar is 31 long\")\nscene.constraint.eq(F2.distance(U2), 28.8, description=\"the second bar is 28.8 long\")\nscene.relate.point_lies_on(X, scene.add.line_segment(F1, U1))\nscene.relate.point_lies_on(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(X), 13.02, description=\"the crossing pin is 13.02 along the first bar from its lower end\")\nscene.constraint.eq(F2.distance(X), 13.248, description=\"the crossing pin is 13.248 along the second bar from its lower end\")\nscene.constraint.eq(X.y - F1.y, 8.3, description=\"the crossing pin sits 8.3 above the floor\")\nscene.constraint.gt(U1.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\nscene.constraint.gt(U2.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [20.357142565106578, 0.0], "U1": [23.884453524864366, 19.761904761904763], "U2": [-2.0899706624228678, 18.04347826086957], "X": [10.031470480443033, 8.3]}, "circles": {}}, "implicit_facts": ["the pin is NOT the midpoint here, so the usual 'upper end is twice the pin height' shortcut does not apply", "each lower end is sqrt(its own pin distance squared minus the pin height squared) horizontally from X", "each upper end lies on the same straight line out through X, at its own bar length from its lower end", "the two bars have different lengths AND different pin distances, so the two halves are not mirror images"], "branch_facts": ["F2 lies to the right of F1, which picks which side the second bar leans"]} {"unique_id": "mechh_t4_mechh_scissor_unequal_0034", "problem_id": "mechh_t4_mechh_scissor_unequal_0034", "domain": "PyGeoX-Mech", "template": "mechh_scissor_unequal", "tier": 4, "nl_description": "Two straight bars of different lengths are crossed over each other and pinned together at a single point X. The first bar is 35.4 mm long and the pin sits 17.7 mm along it from its lower end; the second bar is 37.2 mm long and the pin sits 16.368 mm along that one from its lower end - so the pin is NOT at the middle of either bar. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart. The upper ends are U1 on the first bar and U2 on the second, and both bars run upward from the floor through the crossing pin. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 12.6 mm above the floor, with F2 to the right of F1.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair with unequal bars pinned off-centre\n\nOracle formalization for mechh_t4_mechh_scissor_unequal_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(42)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 3.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.constraint.eq(F1.distance(U1), 35.4, description=\"the first bar is 35.4 long\")\nscene.constraint.eq(F2.distance(U2), 37.2, description=\"the second bar is 37.2 long\")\nscene.relate.point_lies_on(X, scene.add.line_segment(F1, U1))\nscene.relate.point_lies_on(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(X), 17.7, description=\"the crossing pin is 17.7 along the first bar from its lower end\")\nscene.constraint.eq(F2.distance(X), 16.368, description=\"the crossing pin is 16.368 along the second bar from its lower end\")\nscene.constraint.eq(X.y - F1.y, 12.6, description=\"the crossing pin sits 12.6 above the floor\")\nscene.constraint.gt(U1.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\nscene.constraint.gt(U2.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [22.878565499486655, 0.0], "U1": [24.862019226120793, 25.2], "U2": [-0.8658796969366627, 28.63636363636364], "X": [12.431009613060397, 12.6]}, "circles": {}}, "implicit_facts": ["the pin is NOT the midpoint here, so the usual 'upper end is twice the pin height' shortcut does not apply", "each lower end is sqrt(its own pin distance squared minus the pin height squared) horizontally from X", "each upper end lies on the same straight line out through X, at its own bar length from its lower end", "the two bars have different lengths AND different pin distances, so the two halves are not mirror images"], "branch_facts": ["F2 lies to the right of F1, which picks which side the second bar leans"]} {"unique_id": "mechh_t4_mechh_scissor_unequal_0035", "problem_id": "mechh_t4_mechh_scissor_unequal_0035", "domain": "PyGeoX-Mech", "template": "mechh_scissor_unequal", "tier": 4, "nl_description": "Two straight bars of different lengths are crossed over each other and pinned together at a single point X. The first bar is 31.5 mm long and the pin sits 15.12 mm along it from its lower end; the second bar is 28.4 mm long and the pin sits 15.336 mm along that one from its lower end - so the pin is NOT at the middle of either bar. The lower end of the first bar is F1 and the lower end of the second is F2; both rest on the same flat floor, a distance apart. The upper ends are U1 on the first bar and U2 on the second, and both bars run upward from the floor through the crossing pin. Set up coordinates with F1 at the origin and the floor as the line through it. Arrange the pair so that the crossing pin X sits 11.4 mm above the floor, with F2 to the right of F1.", "answer_schema": {"points": ["F1", "F2", "U1", "U2", "X"], "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": "\"\"\"Scissor pair with unequal bars pinned off-centre\n\nOracle formalization for mechh_t4_mechh_scissor_unequal_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(36)\n\nF1, F2, U1, U2, X = scene.add.points([\"F1\", \"F2\", \"U1\", \"U2\", \"X\"])\n\nscene.constraint.eq(F1.x, 0, description=\"datum: F1 is the origin, the lower end of the first bar (x)\")\nscene.constraint.eq(F1.y, 0, description=\"datum: F1 is the origin, the lower end of the first bar (y)\")\nscene.constraint.eq(F2.y, F1.y, description=\"both lower ends rest on the same flat floor\")\nscene.constraint.gt(F2.x - F1.x, 3.0, description=\"the two lower ends rest apart from each other on the floor\")\nscene.constraint.eq(F1.distance(U1), 31.5, description=\"the first bar is 31.5 long\")\nscene.constraint.eq(F2.distance(U2), 28.4, description=\"the second bar is 28.4 long\")\nscene.relate.point_lies_on(X, scene.add.line_segment(F1, U1))\nscene.relate.point_lies_on(X, scene.add.line_segment(F2, U2))\nscene.constraint.eq(F1.distance(X), 15.12, description=\"the crossing pin is 15.12 along the first bar from its lower end\")\nscene.constraint.eq(F2.distance(X), 15.336, description=\"the crossing pin is 15.336 along the second bar from its lower end\")\nscene.constraint.eq(X.y - F1.y, 11.4, description=\"the crossing pin sits 11.4 above the floor\")\nscene.constraint.gt(U1.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\nscene.constraint.gt(U2.y - X.y, 0.05, description=\"both bars run upward from the floor through the crossing pin\")\n", "possible_solution": {"points": {"F1": [0.0, 0.0], "F2": [20.19080076692246, 0.0], "U1": [20.692691946675282, 23.75], "U2": [1.1939329289255607, 21.11111111111111], "X": [9.932492134404134, 11.4]}, "circles": {}}, "implicit_facts": ["the pin is NOT the midpoint here, so the usual 'upper end is twice the pin height' shortcut does not apply", "each lower end is sqrt(its own pin distance squared minus the pin height squared) horizontally from X", "each upper end lies on the same straight line out through X, at its own bar length from its lower end", "the two bars have different lengths AND different pin distances, so the two halves are not mirror images"], "branch_facts": ["F2 lies to the right of F1, which picks which side the second bar leans"]} {"unique_id": "mechh_t4_mechh_slider_dead_centres_0021", "problem_id": "mechh_t4_mechh_slider_dead_centres_0021", "domain": "PyGeoX-Mech", "template": "mechh_slider_dead_centres", "tier": 4, "nl_description": "A crank 9.7 mm long turns about a fixed centre O and drives a block along a straight rail through a connecting rod 30.6 mm long. Set up coordinates with O at the origin; the rail runs 2.6 mm above the level of O. The mechanism is to be shown at the two ends of the block's travel - the outermost and innermost positions the block can reach, which are exactly the two positions where the crank and the connecting rod fall into one straight line. At the outer position the crank pin A1 lies between O and the block centre B1, with the crank and rod stretched out end to end. At the inner position the crank has come round so that O itself lies between the crank pin A2 and the block centre B2, with the crank folded back along the rod. Both extreme positions are on the positive x side of the crank centre.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O"], "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": "\"\"\"Slider-crank at both ends of its travel\n\nOracle formalization for mechh_t4_mechh_slider_dead_centres_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(57)\n\nO, A1, B1, A2, B2 = scene.add.points([\"O\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank centre (y)\")\nscene.constraint.eq(O.distance(A1), 9.7, description=\"the crank is 9.7 long (outer position)\")\nscene.constraint.eq(A1.distance(B1), 30.6, description=\"the connecting rod is 30.6 long (outer position)\")\nscene.constraint.eq(B1.y, 2.6, description=\"the block runs on a straight rail 2.6 above the level of O (outer position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O, B1))\nscene.constraint.eq(O.distance(A2), 9.7, description=\"the crank is 9.7 long (inner position)\")\nscene.constraint.eq(A2.distance(B2), 30.6, description=\"the connecting rod is 30.6 long (inner position)\")\nscene.constraint.eq(B2.y, 2.6, description=\"the block runs on the same rail 2.6 above the level of O (inner position)\")\nscene.relate.point_lies_on(O, scene.add.line_segment(A2, B2))\nscene.constraint.gt(B1.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\nscene.constraint.gt(B2.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A1": [9.679791644716309, 0.6258064516129033], "B1": [40.21604157547085, 2.6], "A2": [-9.624649529941776, -1.2066985645933013], "B2": [20.73764692533847, 2.6]}, "circles": {}}, "implicit_facts": ["the ends of travel are exactly where the crank and the connecting rod become collinear", "stretched out, the distance from O to the block is crank + rod", "folded back, that distance is rod - crank", "the block is then where a circle of that radius about O cuts the rail line, and the crank pin sits on the same straight line at the crank length from O"], "branch_facts": ["both extreme positions are on the +x side of the crank centre"]} {"unique_id": "mechh_t4_mechh_slider_dead_centres_0022", "problem_id": "mechh_t4_mechh_slider_dead_centres_0022", "domain": "PyGeoX-Mech", "template": "mechh_slider_dead_centres", "tier": 4, "nl_description": "A crank 7.5 mm long turns about a fixed centre O and drives a block along a straight rail through a connecting rod 24.5 mm long. Set up coordinates with O at the origin; the rail runs 5.6 mm above the level of O. The mechanism is to be shown at the two ends of the block's travel - the outermost and innermost positions the block can reach, which are exactly the two positions where the crank and the connecting rod fall into one straight line. At the outer position the crank pin A1 lies between O and the block centre B1, with the crank and rod stretched out end to end. At the inner position the crank has come round so that O itself lies between the crank pin A2 and the block centre B2, with the crank folded back along the rod. Both extreme positions are on the positive x side of the crank centre.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O"], "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": "\"\"\"Slider-crank at both ends of its travel\n\nOracle formalization for mechh_t4_mechh_slider_dead_centres_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(46)\n\nO, A1, B1, A2, B2 = scene.add.points([\"O\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank centre (y)\")\nscene.constraint.eq(O.distance(A1), 7.5, description=\"the crank is 7.5 long (outer position)\")\nscene.constraint.eq(A1.distance(B1), 24.5, description=\"the connecting rod is 24.5 long (outer position)\")\nscene.constraint.eq(B1.y, 5.6, description=\"the block runs on a straight rail 5.6 above the level of O (outer position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O, B1))\nscene.constraint.eq(O.distance(A2), 7.5, description=\"the crank is 7.5 long (inner position)\")\nscene.constraint.eq(A2.distance(B2), 24.5, description=\"the connecting rod is 24.5 long (inner position)\")\nscene.constraint.eq(B2.y, 5.6, description=\"the block runs on the same rail 5.6 above the level of O (inner position)\")\nscene.relate.point_lies_on(O, scene.add.line_segment(A2, B2))\nscene.constraint.gt(B1.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\nscene.constraint.gt(B2.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A1": [7.384263250318206, 1.3125], "B1": [31.506189868024347, 5.6], "A2": [-7.081397727258813, -2.4705882352941178], "B2": [16.051168181786643, 5.6]}, "circles": {}}, "implicit_facts": ["the ends of travel are exactly where the crank and the connecting rod become collinear", "stretched out, the distance from O to the block is crank + rod", "folded back, that distance is rod - crank", "the block is then where a circle of that radius about O cuts the rail line, and the crank pin sits on the same straight line at the crank length from O"], "branch_facts": ["both extreme positions are on the +x side of the crank centre"]} {"unique_id": "mechh_t4_mechh_slider_dead_centres_0023", "problem_id": "mechh_t4_mechh_slider_dead_centres_0023", "domain": "PyGeoX-Mech", "template": "mechh_slider_dead_centres", "tier": 4, "nl_description": "A crank 7.5 mm long turns about a fixed centre O and drives a block along a straight rail through a connecting rod 27.2 mm long. Set up coordinates with O at the origin; the rail runs 6.4 mm above the level of O. The mechanism is to be shown at the two ends of the block's travel - the outermost and innermost positions the block can reach, which are exactly the two positions where the crank and the connecting rod fall into one straight line. At the outer position the crank pin A1 lies between O and the block centre B1, with the crank and rod stretched out end to end. At the inner position the crank has come round so that O itself lies between the crank pin A2 and the block centre B2, with the crank folded back along the rod. Both extreme positions are on the positive x side of the crank centre.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O"], "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": "\"\"\"Slider-crank at both ends of its travel\n\nOracle formalization for mechh_t4_mechh_slider_dead_centres_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(49)\n\nO, A1, B1, A2, B2 = scene.add.points([\"O\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank centre (y)\")\nscene.constraint.eq(O.distance(A1), 7.5, description=\"the crank is 7.5 long (outer position)\")\nscene.constraint.eq(A1.distance(B1), 27.2, description=\"the connecting rod is 27.2 long (outer position)\")\nscene.constraint.eq(B1.y, 6.4, description=\"the block runs on a straight rail 6.4 above the level of O (outer position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O, B1))\nscene.constraint.eq(O.distance(A2), 7.5, description=\"the crank is 7.5 long (inner position)\")\nscene.constraint.eq(A2.distance(B2), 27.2, description=\"the connecting rod is 27.2 long (inner position)\")\nscene.constraint.eq(B2.y, 6.4, description=\"the block runs on the same rail 6.4 above the level of O (inner position)\")\nscene.relate.point_lies_on(O, scene.add.line_segment(A2, B2))\nscene.constraint.gt(B1.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\nscene.constraint.gt(B2.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A1": [7.371331071904747, 1.3832853025936598], "B1": [34.10469175934596, 6.4], "A2": [-7.093182131828333, -2.436548223350254], "B2": [18.631425066269085, 6.4]}, "circles": {}}, "implicit_facts": ["the ends of travel are exactly where the crank and the connecting rod become collinear", "stretched out, the distance from O to the block is crank + rod", "folded back, that distance is rod - crank", "the block is then where a circle of that radius about O cuts the rail line, and the crank pin sits on the same straight line at the crank length from O"], "branch_facts": ["both extreme positions are on the +x side of the crank centre"]} {"unique_id": "mechh_t4_mechh_slider_dead_centres_0024", "problem_id": "mechh_t4_mechh_slider_dead_centres_0024", "domain": "PyGeoX-Mech", "template": "mechh_slider_dead_centres", "tier": 4, "nl_description": "A crank 8.2 mm long turns about a fixed centre O and drives a block along a straight rail through a connecting rod 21.6 mm long. Set up coordinates with O at the origin; the rail runs 1.2 mm above the level of O. The mechanism is to be shown at the two ends of the block's travel - the outermost and innermost positions the block can reach, which are exactly the two positions where the crank and the connecting rod fall into one straight line. At the outer position the crank pin A1 lies between O and the block centre B1, with the crank and rod stretched out end to end. At the inner position the crank has come round so that O itself lies between the crank pin A2 and the block centre B2, with the crank folded back along the rod. Both extreme positions are on the positive x side of the crank centre.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O"], "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": "\"\"\"Slider-crank at both ends of its travel\n\nOracle formalization for mechh_t4_mechh_slider_dead_centres_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(44)\n\nO, A1, B1, A2, B2 = scene.add.points([\"O\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank centre (y)\")\nscene.constraint.eq(O.distance(A1), 8.2, description=\"the crank is 8.2 long (outer position)\")\nscene.constraint.eq(A1.distance(B1), 21.6, description=\"the connecting rod is 21.6 long (outer position)\")\nscene.constraint.eq(B1.y, 1.2, description=\"the block runs on a straight rail 1.2 above the level of O (outer position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O, B1))\nscene.constraint.eq(O.distance(A2), 8.2, description=\"the crank is 8.2 long (inner position)\")\nscene.constraint.eq(A2.distance(B2), 21.6, description=\"the connecting rod is 21.6 long (inner position)\")\nscene.constraint.eq(B2.y, 1.2, description=\"the block runs on the same rail 1.2 above the level of O (inner position)\")\nscene.relate.point_lies_on(O, scene.add.line_segment(A2, B2))\nscene.constraint.gt(B1.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\nscene.constraint.gt(B2.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A1": [8.193348953483872, 0.33020134228187914], "B1": [29.77582912363651, 1.2], "A2": [-8.167053438194838, -0.7343283582089549], "B2": [13.3461604965623, 1.2]}, "circles": {}}, "implicit_facts": ["the ends of travel are exactly where the crank and the connecting rod become collinear", "stretched out, the distance from O to the block is crank + rod", "folded back, that distance is rod - crank", "the block is then where a circle of that radius about O cuts the rail line, and the crank pin sits on the same straight line at the crank length from O"], "branch_facts": ["both extreme positions are on the +x side of the crank centre"]} {"unique_id": "mechh_t4_mechh_slider_dead_centres_0025", "problem_id": "mechh_t4_mechh_slider_dead_centres_0025", "domain": "PyGeoX-Mech", "template": "mechh_slider_dead_centres", "tier": 4, "nl_description": "A crank 8.2 mm long turns about a fixed centre O and drives a block along a straight rail through a connecting rod 27 mm long. Set up coordinates with O at the origin; the rail runs -0.9 mm above the level of O. The mechanism is to be shown at the two ends of the block's travel - the outermost and innermost positions the block can reach, which are exactly the two positions where the crank and the connecting rod fall into one straight line. At the outer position the crank pin A1 lies between O and the block centre B1, with the crank and rod stretched out end to end. At the inner position the crank has come round so that O itself lies between the crank pin A2 and the block centre B2, with the crank folded back along the rod. Both extreme positions are on the positive x side of the crank centre.", "answer_schema": {"points": ["A1", "A2", "B1", "B2", "O"], "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": "\"\"\"Slider-crank at both ends of its travel\n\nOracle formalization for mechh_t4_mechh_slider_dead_centres_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(51)\n\nO, A1, B1, A2, B2 = scene.add.points([\"O\", \"A1\", \"B1\", \"A2\", \"B2\"])\n\nscene.constraint.eq(O.x, 0, description=\"datum: O is the origin, the fixed crank centre (x)\")\nscene.constraint.eq(O.y, 0, description=\"datum: O is the origin, the fixed crank centre (y)\")\nscene.constraint.eq(O.distance(A1), 8.2, description=\"the crank is 8.2 long (outer position)\")\nscene.constraint.eq(A1.distance(B1), 27, description=\"the connecting rod is 27 long (outer position)\")\nscene.constraint.eq(B1.y, -0.9, description=\"the block runs on a straight rail -0.9 above the level of O (outer position)\")\nscene.relate.point_lies_on(A1, scene.add.line_segment(O, B1))\nscene.constraint.eq(O.distance(A2), 8.2, description=\"the crank is 8.2 long (inner position)\")\nscene.constraint.eq(A2.distance(B2), 27, description=\"the connecting rod is 27 long (inner position)\")\nscene.constraint.eq(B2.y, -0.9, description=\"the block runs on the same rail -0.9 above the level of O (inner position)\")\nscene.relate.point_lies_on(O, scene.add.line_segment(A2, B2))\nscene.constraint.gt(B1.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\nscene.constraint.gt(B2.x, 0.05, description=\"both extreme positions are on the +x side of the crank centre\")\n", "possible_solution": {"points": {"O": [0.0, 0.0], "A1": [8.197319260929097, -0.20965909090909088], "B1": [35.188492437159056, -0.9], "A2": [-8.190598390340691, 0.3925531914893617], "B2": [18.77844509004939, -0.9]}, "circles": {}}, "implicit_facts": ["the ends of travel are exactly where the crank and the connecting rod become collinear", "stretched out, the distance from O to the block is crank + rod", "folded back, that distance is rod - crank", "the block is then where a circle of that radius about O cuts the rail line, and the crank pin sits on the same straight line at the crank length from O"], "branch_facts": ["both extreme positions are on the +x side of the crank centre"]} {"unique_id": "mechh_t4_mechh_toggle_derived_bar_0041", "problem_id": "mechh_t4_mechh_toggle_derived_bar_0041", "domain": "PyGeoX-Mech", "template": "mechh_toggle_derived_bar", "tier": 4, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 13.1 mm long, ending at a pin K. The second runs from K out to a clamping pad at L. Set up coordinates with J at the origin. The pad has to reach the point (26.9, -9.2) on the workpiece, and in that clamped position the angle at pin K, measured between the two bars, must come out at 160 degrees. The length of the second bar has not been decided yet - it has to be chosen to make that happen. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp with the second bar length to be found\n\nOracle formalization for mechh_t4_mechh_toggle_derived_bar_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(40)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.x, 26.9, description=\"the clamping pad has to reach this point on the workpiece (x)\")\nscene.constraint.eq(L.y, -9.2, description=\"the clamping pad has to reach this point on the workpiece (y)\")\nscene.constraint.eq(J.distance(K), 13.1, description=\"the first bar JK is 13.1 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 160, description=\"in the clamped position the angle at pin K, between the two bars, is 160 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [12.974143303182098, -1.8115202313237044], "L": [26.9, -9.2]}, "circles": {}}, "implicit_facts": ["the second bar's length is NOT given - it is what makes the stated angle come out right", "the distance from J to the pad follows straight from the pad's stated position", "the cosine rule on the angle at K then gives a QUADRATIC in the unknown bar length, not a direct formula", "K itself is where a circle of the first bar's length about J meets a circle of the derived length about the pad"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two possible K positions"]} {"unique_id": "mechh_t4_mechh_toggle_derived_bar_0042", "problem_id": "mechh_t4_mechh_toggle_derived_bar_0042", "domain": "PyGeoX-Mech", "template": "mechh_toggle_derived_bar", "tier": 4, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 15.8 mm long, ending at a pin K. The second runs from K out to a clamping pad at L. Set up coordinates with J at the origin. The pad has to reach the point (23.5, -15.7) on the workpiece, and in that clamped position the angle at pin K, measured between the two bars, must come out at 150 degrees. The length of the second bar has not been decided yet - it has to be chosen to make that happen. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp with the second bar length to be found\n\nOracle formalization for mechh_t4_mechh_toggle_derived_bar_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(36)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.x, 23.5, description=\"the clamping pad has to reach this point on the workpiece (x)\")\nscene.constraint.eq(L.y, -15.7, description=\"the clamping pad has to reach this point on the workpiece (y)\")\nscene.constraint.eq(J.distance(K), 15.8, description=\"the first bar JK is 15.8 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 150, description=\"in the clamped position the angle at pin K, between the two bars, is 150 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [14.849189470412545, -5.398293440689305], "L": [23.5, -15.7]}, "circles": {}}, "implicit_facts": ["the second bar's length is NOT given - it is what makes the stated angle come out right", "the distance from J to the pad follows straight from the pad's stated position", "the cosine rule on the angle at K then gives a QUADRATIC in the unknown bar length, not a direct formula", "K itself is where a circle of the first bar's length about J meets a circle of the derived length about the pad"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two possible K positions"]} {"unique_id": "mechh_t4_mechh_toggle_derived_bar_0043", "problem_id": "mechh_t4_mechh_toggle_derived_bar_0043", "domain": "PyGeoX-Mech", "template": "mechh_toggle_derived_bar", "tier": 4, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 17 mm long, ending at a pin K. The second runs from K out to a clamping pad at L. Set up coordinates with J at the origin. The pad has to reach the point (22.3, -12.4) on the workpiece, and in that clamped position the angle at pin K, measured between the two bars, must come out at 155 degrees. The length of the second bar has not been decided yet - it has to be chosen to make that happen. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp with the second bar length to be found\n\nOracle formalization for mechh_t4_mechh_toggle_derived_bar_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(34)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.x, 22.3, description=\"the clamping pad has to reach this point on the workpiece (x)\")\nscene.constraint.eq(L.y, -12.4, description=\"the clamping pad has to reach this point on the workpiece (y)\")\nscene.constraint.eq(J.distance(K), 17, description=\"the first bar JK is 17 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 155, description=\"in the clamped position the angle at pin K, between the two bars, is 155 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [15.93063875625335, -5.934201615867014], "L": [22.3, -12.4]}, "circles": {}}, "implicit_facts": ["the second bar's length is NOT given - it is what makes the stated angle come out right", "the distance from J to the pad follows straight from the pad's stated position", "the cosine rule on the angle at K then gives a QUADRATIC in the unknown bar length, not a direct formula", "K itself is where a circle of the first bar's length about J meets a circle of the derived length about the pad"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two possible K positions"]} {"unique_id": "mechh_t4_mechh_toggle_derived_bar_0044", "problem_id": "mechh_t4_mechh_toggle_derived_bar_0044", "domain": "PyGeoX-Mech", "template": "mechh_toggle_derived_bar", "tier": 4, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 17.3 mm long, ending at a pin K. The second runs from K out to a clamping pad at L. Set up coordinates with J at the origin. The pad has to reach the point (31.3, -8.3) on the workpiece, and in that clamped position the angle at pin K, measured between the two bars, must come out at 165 degrees. The length of the second bar has not been decided yet - it has to be chosen to make that happen. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp with the second bar length to be found\n\nOracle formalization for mechh_t4_mechh_toggle_derived_bar_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(46)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.x, 31.3, description=\"the clamping pad has to reach this point on the workpiece (x)\")\nscene.constraint.eq(L.y, -8.3, description=\"the clamping pad has to reach this point on the workpiece (y)\")\nscene.constraint.eq(J.distance(K), 17.3, description=\"the first bar JK is 17.3 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 165, description=\"in the clamped position the angle at pin K, between the two bars, is 165 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [17.139952590460986, -2.347770260640891], "L": [31.3, -8.3]}, "circles": {}}, "implicit_facts": ["the second bar's length is NOT given - it is what makes the stated angle come out right", "the distance from J to the pad follows straight from the pad's stated position", "the cosine rule on the angle at K then gives a QUADRATIC in the unknown bar length, not a direct formula", "K itself is where a circle of the first bar's length about J meets a circle of the derived length about the pad"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two possible K positions"]} {"unique_id": "mechh_t4_mechh_toggle_derived_bar_0045", "problem_id": "mechh_t4_mechh_toggle_derived_bar_0045", "domain": "PyGeoX-Mech", "template": "mechh_toggle_derived_bar", "tier": 4, "nl_description": "A toggle clamp is made of two bars. The first is pinned to the bench at a fixed pin J and is 13.8 mm long, ending at a pin K. The second runs from K out to a clamping pad at L. Set up coordinates with J at the origin. The pad has to reach the point (29.6, -12.6) on the workpiece, and in that clamped position the angle at pin K, measured between the two bars, must come out at 150 degrees. The length of the second bar has not been decided yet - it has to be chosen to make that happen. The elbow at K stands above the straight line from J to the pad.", "answer_schema": {"points": ["J", "K", "L"], "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": "\"\"\"Toggle clamp with the second bar length to be found\n\nOracle formalization for mechh_t4_mechh_toggle_derived_bar_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(43)\n\nJ, K, L = scene.add.points([\"J\", \"K\", \"L\"])\n\nscene.constraint.eq(J.x, 0, description=\"datum: J is the origin, the fixed pin of the clamp (x)\")\nscene.constraint.eq(J.y, 0, description=\"datum: J is the origin, the fixed pin of the clamp (y)\")\nscene.constraint.eq(L.x, 29.6, description=\"the clamping pad has to reach this point on the workpiece (x)\")\nscene.constraint.eq(L.y, -12.6, description=\"the clamping pad has to reach this point on the workpiece (y)\")\nscene.constraint.eq(J.distance(K), 13.8, description=\"the first bar JK is 13.8 long\")\nscene.constraint.eq(scene.add.angle(J, K, L).value, 150, description=\"in the clamped position the angle at pin K, between the two bars, is 150 degrees\")\nscene.constraint.gt(cross(J, L, K), 0.05, description=\"the elbow at K stands above the straight line from J to the pad\")\n", "possible_solution": {"points": {"J": [0.0, 0.0], "K": [13.737763919158775, -1.3091380757809503], "L": [29.6, -12.6]}, "circles": {}}, "implicit_facts": ["the second bar's length is NOT given - it is what makes the stated angle come out right", "the distance from J to the pad follows straight from the pad's stated position", "the cosine rule on the angle at K then gives a QUADRATIC in the unknown bar length, not a direct formula", "K itself is where a circle of the first bar's length about J meets a circle of the derived length about the pad"], "branch_facts": ["the elbow stands above the line JL, which picks one of the two possible K positions"]} {"unique_id": "mechh_t4_mechh_trammel_two_poses_0051", "problem_id": "mechh_t4_mechh_trammel_two_poses_0051", "domain": "PyGeoX-Mech", "template": "mechh_trammel_two_poses", "tier": 4, "nl_description": "A straight bar has a small block pinned at each end: one block slides along a horizontal rail, the other along a vertical rail, and the two rails cross at Z, the origin of the coordinates. A scriber is clamped to the bar somewhere along its length and stays put there as the bar moves. Neither the bar's length nor where the scriber is clamped has been recorded - both have to be recovered. The mechanism is caught in two poses. In pose 1 the blocks are Bh1 on the horizontal rail and Bv1 on the vertical rail, and the scriber is at (4.1479, 19.0814). In pose 2 the blocks are Bh2 and Bv2, and the scriber has moved to (9.2898, 10.2716). In both poses the vertical-rail block is on the upper half of its rail, and between pose 1 and pose 2 the horizontal-rail block has moved further out along its own rail.", "answer_schema": {"points": ["Bh1", "Bh2", "Bv1", "Bv2", "T1", "T2", "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": "\"\"\"Elliptic trammel with unknown bar length and scriber position\n\nOracle formalization for mechh_t4_mechh_trammel_two_poses_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\nZ, Bh1, Bv1, T1, Bh2, Bv2, T2 = scene.add.points([\"Z\", \"Bh1\", \"Bv1\", \"T1\", \"Bh2\", \"Bv2\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, where the two rails cross (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, where the two rails cross (y)\")\nscene.constraint.eq(Bh1.y, Z.y, description=\"in pose 1 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bh2.y, Z.y, description=\"in pose 2 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bv1.x, Z.x, description=\"in pose 1 the second block sits on the vertical rail\")\nscene.constraint.eq(Bv2.x, Z.x, description=\"in pose 2 the second block sits on the vertical rail\")\nscene.constraint.eq(T1.x, 4.1479, description=\"the scriber passes through this point in pose 1 (x)\")\nscene.constraint.eq(T1.y, 19.0814, description=\"the scriber passes through this point in pose 1 (y)\")\nscene.constraint.eq(T2.x, 9.2898, description=\"the scriber passes through this point in pose 2 (x)\")\nscene.constraint.eq(T2.y, 10.2716, description=\"the scriber passes through this point in pose 2 (y)\")\nscene.relate.point_lies_on(T1, scene.add.line_segment(Bh1, Bv1))\nscene.relate.point_lies_on(T2, scene.add.line_segment(Bh2, Bv2))\nscene.constraint.eq(Bh1.distance(Bv1), Bh2.distance(Bv2), description=\"the bar is rigid, so it is the same length in both poses\")\nscene.constraint.eq(Bh1.distance(T1), Bh2.distance(T2), description=\"the scriber is clamped to the bar, so it sits the same distance along it in both poses\")\nscene.constraint.gt(Bv1.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bv2.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bh2.x - Bh1.x, 0.05, description=\"the first block has moved outward along its rail from pose 1 to pose 2\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh1": [12.172244722423919, 0.0], "Bv1": [0.0, 28.94471382510843], "T1": [4.147866832163565, 19.081387776424982], "Bh2": [27.261518878443365, 0.0], "Bv2": [0.0, 15.581065061165617], "T2": [9.289753248386752, 10.271593846055042]}, "circles": {}}, "implicit_facts": ["two quantities are unknown at once - the bar length and the scriber's distance along it - so they cannot be solved one at a time", "each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so its length is the same in both poses; the scriber is clamped, so its distance along the bar is too", "the scriber lies ON the bar between the blocks in each pose, which ties the two poses together"], "branch_facts": ["the vertical block is on the upper half of its rail in both poses, and the horizontal block moves outward between poses"]} {"unique_id": "mechh_t4_mechh_trammel_two_poses_0052", "problem_id": "mechh_t4_mechh_trammel_two_poses_0052", "domain": "PyGeoX-Mech", "template": "mechh_trammel_two_poses", "tier": 4, "nl_description": "A straight bar has a small block pinned at each end: one block slides along a horizontal rail, the other along a vertical rail, and the two rails cross at Z, the origin of the coordinates. A scriber is clamped to the bar somewhere along its length and stays put there as the bar moves. Neither the bar's length nor where the scriber is clamped has been recorded - both have to be recovered. The mechanism is caught in two poses. In pose 1 the blocks are Bh1 on the horizontal rail and Bv1 on the vertical rail, and the scriber is at (2.8887, 16.8653). In pose 2 the blocks are Bh2 and Bv2, and the scriber has moved to (5.33, 10.4624). In both poses the vertical-rail block is on the upper half of its rail, and between pose 1 and pose 2 the horizontal-rail block has moved further out along its own rail.", "answer_schema": {"points": ["Bh1", "Bh2", "Bv1", "Bv2", "T1", "T2", "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": "\"\"\"Elliptic trammel with unknown bar length and scriber position\n\nOracle formalization for mechh_t4_mechh_trammel_two_poses_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(34)\n\nZ, Bh1, Bv1, T1, Bh2, Bv2, T2 = scene.add.points([\"Z\", \"Bh1\", \"Bv1\", \"T1\", \"Bh2\", \"Bv2\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, where the two rails cross (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, where the two rails cross (y)\")\nscene.constraint.eq(Bh1.y, Z.y, description=\"in pose 1 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bh2.y, Z.y, description=\"in pose 2 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bv1.x, Z.x, description=\"in pose 1 the second block sits on the vertical rail\")\nscene.constraint.eq(Bv2.x, Z.x, description=\"in pose 2 the second block sits on the vertical rail\")\nscene.constraint.eq(T1.x, 2.8887, description=\"the scriber passes through this point in pose 1 (x)\")\nscene.constraint.eq(T1.y, 16.8653, description=\"the scriber passes through this point in pose 1 (y)\")\nscene.constraint.eq(T2.x, 5.33, description=\"the scriber passes through this point in pose 2 (x)\")\nscene.constraint.eq(T2.y, 10.4624, description=\"the scriber passes through this point in pose 2 (y)\")\nscene.relate.point_lies_on(T1, scene.add.line_segment(Bh1, Bv1))\nscene.relate.point_lies_on(T2, scene.add.line_segment(Bh2, Bv2))\nscene.constraint.eq(Bh1.distance(Bv1), Bh2.distance(Bv2), description=\"the bar is rigid, so it is the same length in both poses\")\nscene.constraint.eq(Bh1.distance(T1), Bh2.distance(T2), description=\"the scriber is clamped to the bar, so it sits the same distance along it in both poses\")\nscene.constraint.gt(Bv1.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bv2.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bh2.x - Bh1.x, 0.05, description=\"the first block has moved outward along its rail from pose 1 to pose 2\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh1": [11.419402406302133, 0.0], "Bv1": [0.0, 22.576254088819542], "T1": [2.8887025849934265, 16.86526491220116], "Bh2": [21.06999179564216, 0.0], "Bv2": [0.0, 14.005193527101726], "T2": [5.329958398897622, 10.462377773210381]}, "circles": {}}, "implicit_facts": ["two quantities are unknown at once - the bar length and the scriber's distance along it - so they cannot be solved one at a time", "each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so its length is the same in both poses; the scriber is clamped, so its distance along the bar is too", "the scriber lies ON the bar between the blocks in each pose, which ties the two poses together"], "branch_facts": ["the vertical block is on the upper half of its rail in both poses, and the horizontal block moves outward between poses"]} {"unique_id": "mechh_t4_mechh_trammel_two_poses_0053", "problem_id": "mechh_t4_mechh_trammel_two_poses_0053", "domain": "PyGeoX-Mech", "template": "mechh_trammel_two_poses", "tier": 4, "nl_description": "A straight bar has a small block pinned at each end: one block slides along a horizontal rail, the other along a vertical rail, and the two rails cross at Z, the origin of the coordinates. A scriber is clamped to the bar somewhere along its length and stays put there as the bar moves. Neither the bar's length nor where the scriber is clamped has been recorded - both have to be recovered. The mechanism is caught in two poses. In pose 1 the blocks are Bh1 on the horizontal rail and Bv1 on the vertical rail, and the scriber is at (2.9819, 21.6225). In pose 2 the blocks are Bh2 and Bv2, and the scriber has moved to (5.9929, 14.9777). In both poses the vertical-rail block is on the upper half of its rail, and between pose 1 and pose 2 the horizontal-rail block has moved further out along its own rail.", "answer_schema": {"points": ["Bh1", "Bh2", "Bv1", "Bv2", "T1", "T2", "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": "\"\"\"Elliptic trammel with unknown bar length and scriber position\n\nOracle formalization for mechh_t4_mechh_trammel_two_poses_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(42)\n\nZ, Bh1, Bv1, T1, Bh2, Bv2, T2 = scene.add.points([\"Z\", \"Bh1\", \"Bv1\", \"T1\", \"Bh2\", \"Bv2\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, where the two rails cross (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, where the two rails cross (y)\")\nscene.constraint.eq(Bh1.y, Z.y, description=\"in pose 1 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bh2.y, Z.y, description=\"in pose 2 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bv1.x, Z.x, description=\"in pose 1 the second block sits on the vertical rail\")\nscene.constraint.eq(Bv2.x, Z.x, description=\"in pose 2 the second block sits on the vertical rail\")\nscene.constraint.eq(T1.x, 2.9819, description=\"the scriber passes through this point in pose 1 (x)\")\nscene.constraint.eq(T1.y, 21.6225, description=\"the scriber passes through this point in pose 1 (y)\")\nscene.constraint.eq(T2.x, 5.9929, description=\"the scriber passes through this point in pose 2 (x)\")\nscene.constraint.eq(T2.y, 14.9777, description=\"the scriber passes through this point in pose 2 (y)\")\nscene.relate.point_lies_on(T1, scene.add.line_segment(Bh1, Bv1))\nscene.relate.point_lies_on(T2, scene.add.line_segment(Bh2, Bv2))\nscene.constraint.eq(Bh1.distance(Bv1), Bh2.distance(Bv2), description=\"the bar is rigid, so it is the same length in both poses\")\nscene.constraint.eq(Bh1.distance(T1), Bh2.distance(T2), description=\"the scriber is clamped to the bar, so it sits the same distance along it in both poses\")\nscene.constraint.gt(Bv1.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bv2.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bh2.x - Bh1.x, 0.05, description=\"the first block has moved outward along its rail from pose 1 to pose 2\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh1": [11.927649861576116, 0.0], "Bv1": [0.0, 28.83003934752159], "T1": [2.9819124653940285, 21.622529510641193], "Bh2": [23.97145450130007, 0.0], "Bv2": [0.0, 19.970212044244814], "T2": [5.992863625325018, 14.97765903318361]}, "circles": {}}, "implicit_facts": ["two quantities are unknown at once - the bar length and the scriber's distance along it - so they cannot be solved one at a time", "each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so its length is the same in both poses; the scriber is clamped, so its distance along the bar is too", "the scriber lies ON the bar between the blocks in each pose, which ties the two poses together"], "branch_facts": ["the vertical block is on the upper half of its rail in both poses, and the horizontal block moves outward between poses"]} {"unique_id": "mechh_t4_mechh_trammel_two_poses_0054", "problem_id": "mechh_t4_mechh_trammel_two_poses_0054", "domain": "PyGeoX-Mech", "template": "mechh_trammel_two_poses", "tier": 4, "nl_description": "A straight bar has a small block pinned at each end: one block slides along a horizontal rail, the other along a vertical rail, and the two rails cross at Z, the origin of the coordinates. A scriber is clamped to the bar somewhere along its length and stays put there as the bar moves. Neither the bar's length nor where the scriber is clamped has been recorded - both have to be recovered. The mechanism is caught in two poses. In pose 1 the blocks are Bh1 on the horizontal rail and Bv1 on the vertical rail, and the scriber is at (2.8278, 15.525). In pose 2 the blocks are Bh2 and Bv2, and the scriber has moved to (5.988, 9.8708). In both poses the vertical-rail block is on the upper half of its rail, and between pose 1 and pose 2 the horizontal-rail block has moved further out along its own rail.", "answer_schema": {"points": ["Bh1", "Bh2", "Bv1", "Bv2", "T1", "T2", "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": "\"\"\"Elliptic trammel with unknown bar length and scriber position\n\nOracle formalization for mechh_t4_mechh_trammel_two_poses_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(34)\n\nZ, Bh1, Bv1, T1, Bh2, Bv2, T2 = scene.add.points([\"Z\", \"Bh1\", \"Bv1\", \"T1\", \"Bh2\", \"Bv2\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, where the two rails cross (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, where the two rails cross (y)\")\nscene.constraint.eq(Bh1.y, Z.y, description=\"in pose 1 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bh2.y, Z.y, description=\"in pose 2 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bv1.x, Z.x, description=\"in pose 1 the second block sits on the vertical rail\")\nscene.constraint.eq(Bv2.x, Z.x, description=\"in pose 2 the second block sits on the vertical rail\")\nscene.constraint.eq(T1.x, 2.8278, description=\"the scriber passes through this point in pose 1 (x)\")\nscene.constraint.eq(T1.y, 15.525, description=\"the scriber passes through this point in pose 1 (y)\")\nscene.constraint.eq(T2.x, 5.988, description=\"the scriber passes through this point in pose 2 (x)\")\nscene.constraint.eq(T2.y, 9.8708, description=\"the scriber passes through this point in pose 2 (y)\")\nscene.relate.point_lies_on(T1, scene.add.line_segment(Bh1, Bv1))\nscene.relate.point_lies_on(T2, scene.add.line_segment(Bh2, Bv2))\nscene.constraint.eq(Bh1.distance(Bv1), Bh2.distance(Bv2), description=\"the bar is rigid, so it is the same length in both poses\")\nscene.constraint.eq(Bh1.distance(T1), Bh2.distance(T2), description=\"the scriber is clamped to the bar, so it sits the same distance along it in both poses\")\nscene.constraint.gt(Bv1.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bv2.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bh2.x - Bh1.x, 0.05, description=\"the first block has moved outward along its rail from pose 1 to pose 2\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh1": [9.247760758735604, 0.0], "Bv1": [0.0, 22.363338770165555], "T1": [2.82782767002659, 15.524962452015759], "Bh2": [19.58238312285071, 0.0], "Bv2": [0.0, 14.218659262739635], "T2": [5.988001450789058, 9.870804777439087]}, "circles": {}}, "implicit_facts": ["two quantities are unknown at once - the bar length and the scriber's distance along it - so they cannot be solved one at a time", "each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so its length is the same in both poses; the scriber is clamped, so its distance along the bar is too", "the scriber lies ON the bar between the blocks in each pose, which ties the two poses together"], "branch_facts": ["the vertical block is on the upper half of its rail in both poses, and the horizontal block moves outward between poses"]} {"unique_id": "mechh_t4_mechh_trammel_two_poses_0055", "problem_id": "mechh_t4_mechh_trammel_two_poses_0055", "domain": "PyGeoX-Mech", "template": "mechh_trammel_two_poses", "tier": 4, "nl_description": "A straight bar has a small block pinned at each end: one block slides along a horizontal rail, the other along a vertical rail, and the two rails cross at Z, the origin of the coordinates. A scriber is clamped to the bar somewhere along its length and stays put there as the bar moves. Neither the bar's length nor where the scriber is clamped has been recorded - both have to be recovered. The mechanism is caught in two poses. In pose 1 the blocks are Bh1 on the horizontal rail and Bv1 on the vertical rail, and the scriber is at (5.2045, 11.5775). In pose 2 the blocks are Bh2 and Bv2, and the scriber has moved to (10.0739, 6.7175). In both poses the vertical-rail block is on the upper half of its rail, and between pose 1 and pose 2 the horizontal-rail block has moved further out along its own rail.", "answer_schema": {"points": ["Bh1", "Bh2", "Bv1", "Bv2", "T1", "T2", "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": "\"\"\"Elliptic trammel with unknown bar length and scriber position\n\nOracle formalization for mechh_t4_mechh_trammel_two_poses_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(34)\n\nZ, Bh1, Bv1, T1, Bh2, Bv2, T2 = scene.add.points([\"Z\", \"Bh1\", \"Bv1\", \"T1\", \"Bh2\", \"Bv2\", \"T2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, where the two rails cross (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, where the two rails cross (y)\")\nscene.constraint.eq(Bh1.y, Z.y, description=\"in pose 1 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bh2.y, Z.y, description=\"in pose 2 the first block sits on the horizontal rail\")\nscene.constraint.eq(Bv1.x, Z.x, description=\"in pose 1 the second block sits on the vertical rail\")\nscene.constraint.eq(Bv2.x, Z.x, description=\"in pose 2 the second block sits on the vertical rail\")\nscene.constraint.eq(T1.x, 5.2045, description=\"the scriber passes through this point in pose 1 (x)\")\nscene.constraint.eq(T1.y, 11.5775, description=\"the scriber passes through this point in pose 1 (y)\")\nscene.constraint.eq(T2.x, 10.0739, description=\"the scriber passes through this point in pose 2 (x)\")\nscene.constraint.eq(T2.y, 6.7175, description=\"the scriber passes through this point in pose 2 (y)\")\nscene.relate.point_lies_on(T1, scene.add.line_segment(Bh1, Bv1))\nscene.relate.point_lies_on(T2, scene.add.line_segment(Bh2, Bv2))\nscene.constraint.eq(Bh1.distance(Bv1), Bh2.distance(Bv2), description=\"the bar is rigid, so it is the same length in both poses\")\nscene.constraint.eq(Bh1.distance(T1), Bh2.distance(T2), description=\"the scriber is clamped to the bar, so it sits the same distance along it in both poses\")\nscene.constraint.gt(Bv1.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bv2.y - Z.y, 0.05, description=\"both poses have the second block on the upper half of the vertical rail\")\nscene.constraint.gt(Bh2.x - Bh1.x, 0.05, description=\"the first block has moved outward along its rail from pose 1 to pose 2\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "Bh1": [10.894069533443028, 0.0], "Bv1": [0.0, 22.16775245712809], "T1": [5.20445427103756, 11.577490149674185], "Bh2": [21.086859345620923, 0.0], "Bv2": [0.0, 12.862129020422687], "T2": [10.07388422179461, 6.717468192852334]}, "circles": {}}, "implicit_facts": ["two quantities are unknown at once - the bar length and the scriber's distance along it - so they cannot be solved one at a time", "each block lies on its own rail, so one has height 0 and the other has abscissa 0", "the bar is rigid, so its length is the same in both poses; the scriber is clamped, so its distance along the bar is too", "the scriber lies ON the bar between the blocks in each pose, which ties the two poses together"], "branch_facts": ["the vertical block is on the upper half of its rail in both poses, and the horizontal block moves outward between poses"]} {"unique_id": "mechh_t4_mechh_two_cranks_derived_link_0091", "problem_id": "mechh_t4_mechh_two_cranks_derived_link_0091", "domain": "PyGeoX-Mech", "template": "mechh_two_cranks_derived_link", "tier": 4, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 25.3 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 10 mm long and ends at a pin W; the bar at V is 13.3 mm long and ends at a pin X. A third bar joins W to X, but it has not been cut to length yet. In the position of interest both moving pins have been raised to exactly 6 mm above the line joining the two fixed pins, and the two bars are splayed apart - W has swung out to the left of U and X has swung out to the right of V. Work out where W and X sit, and how long the joining bar has to be.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks at a stated level, joining link to be found\n\nOracle formalization for mechh_t4_mechh_two_cranks_derived_link_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(53)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 25.3, description=\"the two fixed pins are 25.3 apart\")\nscene.constraint.eq(U.distance(W), 10, description=\"the bar at U is 10 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 13.3, description=\"the bar at V is 13.3 long, ending at pin X\")\nscene.constraint.eq(W.y - U.y, 6, description=\"both moving pins are raised 6 above the line of the fixed pins\")\nscene.constraint.eq(X.y - U.y, 6, description=\"both moving pins are raised 6 above the line of the fixed pins\")\nscene.constraint.lt(W.x - U.x, -0.05, description=\"the two bars are splayed apart: W swings out to the left of U\")\nscene.constraint.gt(X.x - V.x, 0.05, description=\"the two bars are splayed apart: X swings out to the right of V\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [25.3, 0.0], "W": [-8.0, 6.0], "X": [37.169709347747315, 6.0]}, "circles": {}}, "implicit_facts": ["the joining bar's length is NOT given - it follows once both moving pins are placed", "each moving pin is at its own bar length from its own fixed pin AND at the stated height", "so each is found from a right-angled triangle, and each has a left/right choice to resolve", "the joining length is then just the distance between the two pins once both are placed"], "branch_facts": ["the bars are splayed apart - W to the left of U, X to the right of V"]} {"unique_id": "mechh_t4_mechh_two_cranks_derived_link_0092", "problem_id": "mechh_t4_mechh_two_cranks_derived_link_0092", "domain": "PyGeoX-Mech", "template": "mechh_two_cranks_derived_link", "tier": 4, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 33.7 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 11.5 mm long and ends at a pin W; the bar at V is 13.9 mm long and ends at a pin X. A third bar joins W to X, but it has not been cut to length yet. In the position of interest both moving pins have been raised to exactly 9.5 mm above the line joining the two fixed pins, and the two bars are splayed apart - W has swung out to the left of U and X has swung out to the right of V. Work out where W and X sit, and how long the joining bar has to be.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks at a stated level, joining link to be found\n\nOracle formalization for mechh_t4_mechh_two_cranks_derived_link_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(62)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 33.7, description=\"the two fixed pins are 33.7 apart\")\nscene.constraint.eq(U.distance(W), 11.5, description=\"the bar at U is 11.5 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 13.9, description=\"the bar at V is 13.9 long, ending at pin X\")\nscene.constraint.eq(W.y - U.y, 9.5, description=\"both moving pins are raised 9.5 above the line of the fixed pins\")\nscene.constraint.eq(X.y - U.y, 9.5, description=\"both moving pins are raised 9.5 above the line of the fixed pins\")\nscene.constraint.lt(W.x - U.x, -0.05, description=\"the two bars are splayed apart: W swings out to the left of U\")\nscene.constraint.gt(X.x - V.x, 0.05, description=\"the two bars are splayed apart: X swings out to the right of V\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [33.7, 0.0], "W": [-6.48074069840786, 9.5], "X": [43.84692071517266, 9.5]}, "circles": {}}, "implicit_facts": ["the joining bar's length is NOT given - it follows once both moving pins are placed", "each moving pin is at its own bar length from its own fixed pin AND at the stated height", "so each is found from a right-angled triangle, and each has a left/right choice to resolve", "the joining length is then just the distance between the two pins once both are placed"], "branch_facts": ["the bars are splayed apart - W to the left of U, X to the right of V"]} {"unique_id": "mechh_t4_mechh_two_cranks_derived_link_0093", "problem_id": "mechh_t4_mechh_two_cranks_derived_link_0093", "domain": "PyGeoX-Mech", "template": "mechh_two_cranks_derived_link", "tier": 4, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 24.8 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 10.2 mm long and ends at a pin W; the bar at V is 16.6 mm long and ends at a pin X. A third bar joins W to X, but it has not been cut to length yet. In the position of interest both moving pins have been raised to exactly 5.7 mm above the line joining the two fixed pins, and the two bars are splayed apart - W has swung out to the left of U and X has swung out to the right of V. Work out where W and X sit, and how long the joining bar has to be.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks at a stated level, joining link to be found\n\nOracle formalization for mechh_t4_mechh_two_cranks_derived_link_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(58)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 24.8, description=\"the two fixed pins are 24.8 apart\")\nscene.constraint.eq(U.distance(W), 10.2, description=\"the bar at U is 10.2 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 16.6, description=\"the bar at V is 16.6 long, ending at pin X\")\nscene.constraint.eq(W.y - U.y, 5.7, description=\"both moving pins are raised 5.7 above the line of the fixed pins\")\nscene.constraint.eq(X.y - U.y, 5.7, description=\"both moving pins are raised 5.7 above the line of the fixed pins\")\nscene.constraint.lt(W.x - U.x, -0.05, description=\"the two bars are splayed apart: W swings out to the left of U\")\nscene.constraint.gt(X.x - V.x, 0.05, description=\"the two bars are splayed apart: X swings out to the right of V\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [24.8, 0.0], "W": [-8.458723307923009, 5.7], "X": [40.390702357495, 5.7]}, "circles": {}}, "implicit_facts": ["the joining bar's length is NOT given - it follows once both moving pins are placed", "each moving pin is at its own bar length from its own fixed pin AND at the stated height", "so each is found from a right-angled triangle, and each has a left/right choice to resolve", "the joining length is then just the distance between the two pins once both are placed"], "branch_facts": ["the bars are splayed apart - W to the left of U, X to the right of V"]} {"unique_id": "mechh_t4_mechh_two_cranks_derived_link_0094", "problem_id": "mechh_t4_mechh_two_cranks_derived_link_0094", "domain": "PyGeoX-Mech", "template": "mechh_two_cranks_derived_link", "tier": 4, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 24.2 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 11.3 mm long and ends at a pin W; the bar at V is 11.4 mm long and ends at a pin X. A third bar joins W to X, but it has not been cut to length yet. In the position of interest both moving pins have been raised to exactly 7.2 mm above the line joining the two fixed pins, and the two bars are splayed apart - W has swung out to the left of U and X has swung out to the right of V. Work out where W and X sit, and how long the joining bar has to be.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks at a stated level, joining link to be found\n\nOracle formalization for mechh_t4_mechh_two_cranks_derived_link_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(48)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 24.2, description=\"the two fixed pins are 24.2 apart\")\nscene.constraint.eq(U.distance(W), 11.3, description=\"the bar at U is 11.3 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 11.4, description=\"the bar at V is 11.4 long, ending at pin X\")\nscene.constraint.eq(W.y - U.y, 7.2, description=\"both moving pins are raised 7.2 above the line of the fixed pins\")\nscene.constraint.eq(X.y - U.y, 7.2, description=\"both moving pins are raised 7.2 above the line of the fixed pins\")\nscene.constraint.lt(W.x - U.x, -0.05, description=\"the two bars are splayed apart: W swings out to the left of U\")\nscene.constraint.gt(X.x - V.x, 0.05, description=\"the two bars are splayed apart: X swings out to the right of V\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [24.2, 0.0], "W": [-8.709190547921201, 7.2], "X": [33.03855191759374, 7.2]}, "circles": {}}, "implicit_facts": ["the joining bar's length is NOT given - it follows once both moving pins are placed", "each moving pin is at its own bar length from its own fixed pin AND at the stated height", "so each is found from a right-angled triangle, and each has a left/right choice to resolve", "the joining length is then just the distance between the two pins once both are placed"], "branch_facts": ["the bars are splayed apart - W to the left of U, X to the right of V"]} {"unique_id": "mechh_t4_mechh_two_cranks_derived_link_0095", "problem_id": "mechh_t4_mechh_two_cranks_derived_link_0095", "domain": "PyGeoX-Mech", "template": "mechh_two_cranks_derived_link", "tier": 4, "nl_description": "Two bars are pinned to the bench at fixed pins U and V, which sit at the same height and are 27 mm apart. Set up coordinates with U at the origin and V to its right. The bar at U is 12.6 mm long and ends at a pin W; the bar at V is 16 mm long and ends at a pin X. A third bar joins W to X, but it has not been cut to length yet. In the position of interest both moving pins have been raised to exactly 8.1 mm above the line joining the two fixed pins, and the two bars are splayed apart - W has swung out to the left of U and X has swung out to the right of V. Work out where W and X sit, and how long the joining bar has to be.", "answer_schema": {"points": ["U", "V", "W", "X"], "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": "\"\"\"Two cranks at a stated level, joining link to be found\n\nOracle formalization for mechh_t4_mechh_two_cranks_derived_link_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(58)\n\nU, V, W, X = scene.add.points([\"U\", \"V\", \"W\", \"X\"])\n\nscene.constraint.eq(U.x, 0, description=\"datum: U is the origin, the first fixed pin (x)\")\nscene.constraint.eq(U.y, 0, description=\"datum: U is the origin, the first fixed pin (y)\")\nscene.constraint.eq(V.y, U.y, description=\"the two fixed pins sit at the same height on the bench\")\nscene.constraint.gt(V.x - U.x, 1.0, description=\"datum: V is to the right of U\")\nscene.constraint.eq(U.distance(V), 27, description=\"the two fixed pins are 27 apart\")\nscene.constraint.eq(U.distance(W), 12.6, description=\"the bar at U is 12.6 long, ending at pin W\")\nscene.constraint.eq(V.distance(X), 16, description=\"the bar at V is 16 long, ending at pin X\")\nscene.constraint.eq(W.y - U.y, 8.1, description=\"both moving pins are raised 8.1 above the line of the fixed pins\")\nscene.constraint.eq(X.y - U.y, 8.1, description=\"both moving pins are raised 8.1 above the line of the fixed pins\")\nscene.constraint.lt(W.x - U.x, -0.05, description=\"the two bars are splayed apart: W swings out to the left of U\")\nscene.constraint.gt(X.x - V.x, 0.05, description=\"the two bars are splayed apart: X swings out to the right of V\")\n", "possible_solution": {"points": {"U": [0.0, 0.0], "V": [27.0, 0.0], "W": [-9.651424765287247, 8.1], "X": [40.79818828687303, 8.1]}, "circles": {}}, "implicit_facts": ["the joining bar's length is NOT given - it follows once both moving pins are placed", "each moving pin is at its own bar length from its own fixed pin AND at the stated height", "so each is found from a right-angled triangle, and each has a left/right choice to resolve", "the joining length is then just the distance between the two pins once both are placed"], "branch_facts": ["the bars are splayed apart - W to the left of U, X to the right of V"]} {"unique_id": "mechh_t4_mechh_walking_beam_two_rods_0081", "problem_id": "mechh_t4_mechh_walking_beam_two_rods_0081", "domain": "PyGeoX-Mech", "template": "mechh_walking_beam_two_rods", "tier": 4, "nl_description": "A straight beam 36.3 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 on the left and E2 on the right. Each end drives its own block down a rail of its own: E1 is connected by a rod 14.8 mm long to a block K1 sliding on a vertical rail 19 mm to the left of the pivot, and E2 is connected by a rod 17.9 mm long to a block K2 sliding on a vertical rail 23 mm to the right of the pivot. Each block hangs below the beam end it is connected to. Arrange the beam so that the right-hand end E2 sits 10.6 mm above the height of the pivot.", "answer_schema": {"points": ["E1", "E2", "K1", "K2", "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": "\"\"\"Walking beam driving two blocks on separate rails\n\nOracle formalization for mechh_t4_mechh_walking_beam_two_rods_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(37)\n\nZ, E1, E2, K1, K2 = scene.add.points([\"Z\", \"E1\", \"E2\", \"K1\", \"K2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 36.3, description=\"the beam is 36.3 long overall\")\nscene.constraint.eq(E2.y - Z.y, 10.6, description=\"the right-hand end E2 sits 10.6 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"E2 is the end on the right of the pivot\")\nscene.constraint.eq(K1.x - Z.x, -19, description=\"the left-hand block runs on a rail 19 to the left of the pivot\")\nscene.constraint.eq(K2.x - Z.x, 23, description=\"the right-hand block runs on a rail 23 to the right of the pivot\")\nscene.constraint.eq(E1.distance(K1), 14.8, description=\"the left-hand rod is 14.8 long\")\nscene.constraint.eq(E2.distance(K2), 17.9, description=\"the right-hand rod is 17.9 long\")\nscene.constraint.lt(K1.y - E1.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\nscene.constraint.lt(K2.y - E2.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-14.733041098157567, -10.6], "E2": [14.733041098157567, 10.6], "K1": [-19.0, -24.771558196965767], "K2": [23.0, -5.276630326213688]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length from it", "the two ends are diametrically opposite about the pivot, so if E2 is dy above it then E1 is dy BELOW it, with the opposite x-offset too", "each block is where a circle of its own rod length, centred on its own beam end, cuts its own rail line", "the two sides are worked out separately but both depend on first getting the beam's tilt right"], "branch_facts": ["E2 is on the right of the pivot, and each block hangs below its beam end"]} {"unique_id": "mechh_t4_mechh_walking_beam_two_rods_0082", "problem_id": "mechh_t4_mechh_walking_beam_two_rods_0082", "domain": "PyGeoX-Mech", "template": "mechh_walking_beam_two_rods", "tier": 4, "nl_description": "A straight beam 31.9 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 on the left and E2 on the right. Each end drives its own block down a rail of its own: E1 is connected by a rod 13.1 mm long to a block K1 sliding on a vertical rail 17.7 mm to the left of the pivot, and E2 is connected by a rod 16.1 mm long to a block K2 sliding on a vertical rail 19.3 mm to the right of the pivot. Each block hangs below the beam end it is connected to. Arrange the beam so that the right-hand end E2 sits 11.8 mm above the height of the pivot.", "answer_schema": {"points": ["E1", "E2", "K1", "K2", "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": "\"\"\"Walking beam driving two blocks on separate rails\n\nOracle formalization for mechh_t4_mechh_walking_beam_two_rods_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(35)\n\nZ, E1, E2, K1, K2 = scene.add.points([\"Z\", \"E1\", \"E2\", \"K1\", \"K2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 31.9, description=\"the beam is 31.9 long overall\")\nscene.constraint.eq(E2.y - Z.y, 11.8, description=\"the right-hand end E2 sits 11.8 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"E2 is the end on the right of the pivot\")\nscene.constraint.eq(K1.x - Z.x, -17.7, description=\"the left-hand block runs on a rail 17.7 to the left of the pivot\")\nscene.constraint.eq(K2.x - Z.x, 19.3, description=\"the right-hand block runs on a rail 19.3 to the right of the pivot\")\nscene.constraint.eq(E1.distance(K1), 13.1, description=\"the left-hand rod is 13.1 long\")\nscene.constraint.eq(E2.distance(K2), 16.1, description=\"the right-hand rod is 16.1 long\")\nscene.constraint.lt(K1.y - E1.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\nscene.constraint.lt(K2.y - E2.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-10.731379221703051, -11.8], "E2": [10.731379221703051, 11.8], "K1": [-17.7, -22.892714926846722], "K2": [19.3, -1.8304342541878604]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length from it", "the two ends are diametrically opposite about the pivot, so if E2 is dy above it then E1 is dy BELOW it, with the opposite x-offset too", "each block is where a circle of its own rod length, centred on its own beam end, cuts its own rail line", "the two sides are worked out separately but both depend on first getting the beam's tilt right"], "branch_facts": ["E2 is on the right of the pivot, and each block hangs below its beam end"]} {"unique_id": "mechh_t4_mechh_walking_beam_two_rods_0083", "problem_id": "mechh_t4_mechh_walking_beam_two_rods_0083", "domain": "PyGeoX-Mech", "template": "mechh_walking_beam_two_rods", "tier": 4, "nl_description": "A straight beam 36.5 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 on the left and E2 on the right. Each end drives its own block down a rail of its own: E1 is connected by a rod 16.3 mm long to a block K1 sliding on a vertical rail 24.9 mm to the left of the pivot, and E2 is connected by a rod 17.7 mm long to a block K2 sliding on a vertical rail 22.2 mm to the right of the pivot. Each block hangs below the beam end it is connected to. Arrange the beam so that the right-hand end E2 sits 9.7 mm above the height of the pivot.", "answer_schema": {"points": ["E1", "E2", "K1", "K2", "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": "\"\"\"Walking beam driving two blocks on separate rails\n\nOracle formalization for mechh_t4_mechh_walking_beam_two_rods_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(37)\n\nZ, E1, E2, K1, K2 = scene.add.points([\"Z\", \"E1\", \"E2\", \"K1\", \"K2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 36.5, description=\"the beam is 36.5 long overall\")\nscene.constraint.eq(E2.y - Z.y, 9.7, description=\"the right-hand end E2 sits 9.7 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"E2 is the end on the right of the pivot\")\nscene.constraint.eq(K1.x - Z.x, -24.9, description=\"the left-hand block runs on a rail 24.9 to the left of the pivot\")\nscene.constraint.eq(K2.x - Z.x, 22.2, description=\"the right-hand block runs on a rail 22.2 to the right of the pivot\")\nscene.constraint.eq(E1.distance(K1), 16.3, description=\"the left-hand rod is 16.3 long\")\nscene.constraint.eq(E2.distance(K2), 17.7, description=\"the right-hand rod is 17.7 long\")\nscene.constraint.lt(K1.y - E1.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\nscene.constraint.lt(K2.y - E2.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-15.45873539459163, -9.7], "E2": [15.45873539459163, 9.7], "K1": [-24.9, -22.987306824585], "K2": [22.2, -6.6659815324308695]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length from it", "the two ends are diametrically opposite about the pivot, so if E2 is dy above it then E1 is dy BELOW it, with the opposite x-offset too", "each block is where a circle of its own rod length, centred on its own beam end, cuts its own rail line", "the two sides are worked out separately but both depend on first getting the beam's tilt right"], "branch_facts": ["E2 is on the right of the pivot, and each block hangs below its beam end"]} {"unique_id": "mechh_t4_mechh_walking_beam_two_rods_0084", "problem_id": "mechh_t4_mechh_walking_beam_two_rods_0084", "domain": "PyGeoX-Mech", "template": "mechh_walking_beam_two_rods", "tier": 4, "nl_description": "A straight beam 39.9 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 on the left and E2 on the right. Each end drives its own block down a rail of its own: E1 is connected by a rod 22.9 mm long to a block K1 sliding on a vertical rail 24.2 mm to the left of the pivot, and E2 is connected by a rod 23.2 mm long to a block K2 sliding on a vertical rail 19.2 mm to the right of the pivot. Each block hangs below the beam end it is connected to. Arrange the beam so that the right-hand end E2 sits 14.8 mm above the height of the pivot.", "answer_schema": {"points": ["E1", "E2", "K1", "K2", "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": "\"\"\"Walking beam driving two blocks on separate rails\n\nOracle formalization for mechh_t4_mechh_walking_beam_two_rods_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(50)\n\nZ, E1, E2, K1, K2 = scene.add.points([\"Z\", \"E1\", \"E2\", \"K1\", \"K2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 39.9, description=\"the beam is 39.9 long overall\")\nscene.constraint.eq(E2.y - Z.y, 14.8, description=\"the right-hand end E2 sits 14.8 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"E2 is the end on the right of the pivot\")\nscene.constraint.eq(K1.x - Z.x, -24.2, description=\"the left-hand block runs on a rail 24.2 to the left of the pivot\")\nscene.constraint.eq(K2.x - Z.x, 19.2, description=\"the right-hand block runs on a rail 19.2 to the right of the pivot\")\nscene.constraint.eq(E1.distance(K1), 22.9, description=\"the left-hand rod is 22.9 long\")\nscene.constraint.eq(E2.distance(K2), 23.2, description=\"the right-hand rod is 23.2 long\")\nscene.constraint.lt(K1.y - E1.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\nscene.constraint.lt(K2.y - E2.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-13.377686646053569, -14.8], "E2": [13.377686646053569, 14.8], "K1": [-24.2, -34.98136600106625], "K2": [19.2, -7.657530300735587]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length from it", "the two ends are diametrically opposite about the pivot, so if E2 is dy above it then E1 is dy BELOW it, with the opposite x-offset too", "each block is where a circle of its own rod length, centred on its own beam end, cuts its own rail line", "the two sides are worked out separately but both depend on first getting the beam's tilt right"], "branch_facts": ["E2 is on the right of the pivot, and each block hangs below its beam end"]} {"unique_id": "mechh_t4_mechh_walking_beam_two_rods_0085", "problem_id": "mechh_t4_mechh_walking_beam_two_rods_0085", "domain": "PyGeoX-Mech", "template": "mechh_walking_beam_two_rods", "tier": 4, "nl_description": "A straight beam 37.5 mm long rocks about a fixed pivot placed exactly at its middle. Set up coordinates with that pivot at the origin Z. The two ends of the beam are E1 on the left and E2 on the right. Each end drives its own block down a rail of its own: E1 is connected by a rod 13.7 mm long to a block K1 sliding on a vertical rail 16.4 mm to the left of the pivot, and E2 is connected by a rod 15.8 mm long to a block K2 sliding on a vertical rail 17.2 mm to the right of the pivot. Each block hangs below the beam end it is connected to. Arrange the beam so that the right-hand end E2 sits 14.2 mm above the height of the pivot.", "answer_schema": {"points": ["E1", "E2", "K1", "K2", "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": "\"\"\"Walking beam driving two blocks on separate rails\n\nOracle formalization for mechh_t4_mechh_walking_beam_two_rods_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(40)\n\nZ, E1, E2, K1, K2 = scene.add.points([\"Z\", \"E1\", \"E2\", \"K1\", \"K2\"])\n\nscene.constraint.eq(Z.x, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (x)\")\nscene.constraint.eq(Z.y, 0, description=\"datum: Z is the origin, the fixed pivot of the beam (y)\")\nscene.relate.is_midpoint(Z, scene.add.line_segment(E1, E2))\nscene.constraint.eq(E1.distance(E2), 37.5, description=\"the beam is 37.5 long overall\")\nscene.constraint.eq(E2.y - Z.y, 14.2, description=\"the right-hand end E2 sits 14.2 above the height of the pivot\")\nscene.constraint.gt(E2.x - Z.x, 0.05, description=\"E2 is the end on the right of the pivot\")\nscene.constraint.eq(K1.x - Z.x, -16.4, description=\"the left-hand block runs on a rail 16.4 to the left of the pivot\")\nscene.constraint.eq(K2.x - Z.x, 17.2, description=\"the right-hand block runs on a rail 17.2 to the right of the pivot\")\nscene.constraint.eq(E1.distance(K1), 13.7, description=\"the left-hand rod is 13.7 long\")\nscene.constraint.eq(E2.distance(K2), 15.8, description=\"the right-hand rod is 15.8 long\")\nscene.constraint.lt(K1.y - E1.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\nscene.constraint.lt(K2.y - E2.y, -0.05, description=\"each block hangs below the beam end it is connected to\")\n", "possible_solution": {"points": {"Z": [0.0, 0.0], "E1": [-12.244284380885638, -14.2], "E2": [12.244284380885638, 14.2], "K1": [-16.4, -27.25450220012425], "K2": [17.2, -0.8026958478290158]}, "circles": {}}, "implicit_facts": ["the pivot is at the middle, so each end is half the beam length from it", "the two ends are diametrically opposite about the pivot, so if E2 is dy above it then E1 is dy BELOW it, with the opposite x-offset too", "each block is where a circle of its own rod length, centred on its own beam end, cuts its own rail line", "the two sides are worked out separately but both depend on first getting the beam's tilt right"], "branch_facts": ["E2 is on the right of the pivot, and each block hangs below its beam end"]}