id int32 1 6.98k | annotation stringclasses 132
values | source stringlengths 7 17 | problem_level int32 0 28 | problem_text_cn stringlengths 20 201 | problem_text_en stringlengths 58 424 | problem_img stringlengths 5 8 | construction_cdl listlengths 1 28 | text_cdl listlengths 0 16 | image_cdl listlengths 0 16 | goal_cdl stringlengths 8 131 | problem_answer stringclasses 906
values | theorem_seqs listlengths 0 28 | theorem_seqs_dag_json stringlengths 13 3.3k | image imagewidth (px) 48 1.6k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
201 | XiaokaiZhang_2023-03-12 | Geometry3k-203 | 1 | 如图所示，∠EHC=35°，∠GCH=28°，∠HDF=25°，∠HFG=51°，CE⊥HE，FG⊥HG，HC⊥AC。求∠GHF的大小。 | As shown in the diagram, ∠EHC=35°, ∠GCH=28°, ∠HDF=25°, ∠HFG=51°, CE⊥HE, FG⊥HG, HC is perpendicular to AC. Find the measure of ∠GHF. | 201.png | [
"Shape(HD,DF,FH)",
"Shape(HF,FG,GH)",
"Shape(HG,GC,CH)",
"Shape(HC,CE,EH)",
"Shape(EC,CA,AE)",
"Collinear(DFGC)",
"Collinear(HEA)"
] | [
"Equal(MeasureOfAngle(EHC),35)",
"Equal(MeasureOfAngle(GCH),28)",
"Equal(MeasureOfAngle(HDF),25)",
"Equal(MeasureOfAngle(HFG),51)",
"PerpendicularBetweenLine(CE,HE)",
"PerpendicularBetweenLine(FG,HG)",
"PerpendicularBetweenLine(HC,AC)"
] | [
"Equal(MeasureOfAngle(EHC),35)",
"Equal(MeasureOfAngle(GCH),28)",
"Equal(MeasureOfAngle(HDF),25)",
"Equal(MeasureOfAngle(HFG),51)",
"PerpendicularBetweenLine(CE,HE)",
"PerpendicularBetweenLine(FG,HG)",
"PerpendicularBetweenLine(HC,AC)"
] | Value(MeasureOfAngle(GHF)) | 39 | [
"triangle_property_angle_sum(1,HFG)"
] | {"START": ["triangle_property_angle_sum(1,HFG)"]} | |
202 | XiaokaiZhang_2023-04-02 | Geometry3k-204 | 3 | 如图所示，∠ADE=43°，AD平行于BC，DC∥AB。求∠ABC的大小。 | As shown in the diagram, ∠ADE=43°, AD is parallel to BC, DC∥AB. Find the measure of ∠ABC. | 202.png | [
"Shape(AB,BC,CD,DA)",
"Shape(AD,DE)",
"Collinear(EDC)"
] | [
"Equal(MeasureOfAngle(ADE),43)",
"ParallelBetweenLine(AD,BC)",
"ParallelBetweenLine(DC,AB)"
] | [
"ParallelBetweenLine(AD,BC)",
"ParallelBetweenLine(DC,AB)"
] | Value(MeasureOfAngle(ABC)) | 137 | [
"adjacent_complementary_angle(1,CDA,ADE)",
"parallelogram_judgment_parallel_and_parallel(1,ABCD)",
"parallelogram_property_opposite_angle_equal(1,BCDA)"
] | {"START": ["adjacent_complementary_angle(1,CDA,ADE)", "parallelogram_judgment_parallel_and_parallel(1,ABCD)"], "parallelogram_judgment_parallel_and_parallel(1,ABCD)": ["parallelogram_property_opposite_angle_equal(1,BCDA)"]} | |
203 | XiaokaiZhang_2023-03-12 | Geometry3k-205 | 3 | 如图所示，∠BDE=35°，∠CAE=28°，∠EBD=75°。求∠BEA的大小。 | As shown in the diagram, ∠BDE=35°, ∠CAE=28°, ∠EBD=75°. Find the measure of ∠BEA. | 203.png | [
"Shape(BD,DE,EB)",
"Shape(EC,CA,AE)",
"Shape(BE,EA)",
"Shape(CE,ED)",
"Collinear(DEA)",
"Collinear(BEC)"
] | [
"Equal(MeasureOfAngle(BDE),35)",
"Equal(MeasureOfAngle(CAE),28)",
"Equal(MeasureOfAngle(EBD),75)"
] | [
"Equal(MeasureOfAngle(BDE),35)",
"Equal(MeasureOfAngle(CAE),28)",
"Equal(MeasureOfAngle(EBD),75)"
] | Value(MeasureOfAngle(BEA)) | 110 | [
"angle_addition(1,DEB,BEA)",
"triangle_property_angle_sum(1,BDE)",
"flat_angle(1,DEA)"
] | {"START": ["angle_addition(1,DEB,BEA)", "triangle_property_angle_sum(1,BDE)", "flat_angle(1,DEA)"]} | |
204 | XiaokaiZhang_2023-04-02 | Geometry3k-206 | 2 | 如图所示，∠ABE=32°。求∠EDC的大小。 | As shown in the diagram, ∠ABE=32°. Find the measure of ∠EDC. | 204.png | [
"Shape(HCA,AE,EC)",
"Shape(HAB,BA)",
"Shape(HBD,DE,EB)",
"Shape(HDC,CD)",
"Shape(EA,AB,BE)",
"Shape(ED,DC,CE)",
"Collinear(AED)",
"Collinear(CEB)",
"Cocircular(H,ABDC)"
] | [
"Equal(MeasureOfAngle(ABE),32)"
] | [
"Equal(MeasureOfAngle(ABE),32)"
] | Value(MeasureOfAngle(EDC)) | 32 | [
"arc_property_circumference_angle_external(1,HCA,D)",
"arc_property_circumference_angle_external(1,HCA,B)"
] | {"START": ["arc_property_circumference_angle_external(1,HCA,D)", "arc_property_circumference_angle_external(1,HCA,B)"]} | |
205 | XiaokaiZhang_2023-04-02 | Geometry3k-207 | 4 | 如图所示，AB=3，AC=5，AE=y，BE=x，CD=7/2，DE=3，EB平行于DC。求直线BE的长度。 | As shown in the diagram, AB=3, AC=5, AE=y, BE=x, CD=7/2, DE=3, EB∥DC. Find the length of line BE. | 205.png | [
"Shape(BA,AE,EB)",
"Shape(BE,ED,DC,CB)",
"Collinear(ABC)",
"Collinear(AED)"
] | [
"Equal(LengthOfLine(AB),3)",
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(AE),y)",
"Equal(LengthOfLine(BE),x)",
"Equal(LengthOfLine(CD),7/2)",
"Equal(LengthOfLine(DE),3)",
"ParallelBetweenLine(EB,DC)"
] | [
"Equal(LengthOfLine(AB),3)",
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(AE),y)",
"Equal(LengthOfLine(BE),x)",
"Equal(LengthOfLine(CD),7/2)",
"Equal(LengthOfLine(DE),3)",
"ParallelBetweenLine(EB,DC)"
] | Value(LengthOfLine(BE)) | 21/10 | [
"parallel_property_corresponding_angle(1,EB,DC,A)",
"similar_triangle_judgment_aa(1,BAE,CAD)",
"similar_triangle_property_line_ratio(1,EBA,DCA)",
"similar_triangle_property_line_ratio(1,AEB,ADC)"
] | {"START": ["parallel_property_corresponding_angle(1,EB,DC,A)"], "parallel_property_corresponding_angle(1,EB,DC,A)": ["similar_triangle_judgment_aa(1,BAE,CAD)"], "similar_triangle_judgment_aa(1,BAE,CAD)": ["similar_triangle_property_line_ratio(1,AEB,ADC)", "similar_triangle_property_line_ratio(1,EBA,DCA)"]} | |
206 | XiaokaiZhang_2023-04-02 | Geometry3k-208 | 3 | 如图所示，CE=ED，EC=6，EF=5，FE=EA，CFDA是菱形。求四边形CFDA的面积。 | As shown in the diagram, CE=ED, EC=6, EF=5, FE=EA, quadrilateral CFDA is a rhombus. Find the area of CFDA. | 206.png | [
"Shape(CF,FE,EC)",
"Shape(EF,FD,DE)",
"Shape(ED,DA,AE)",
"Shape(CE,EA,AC)",
"Collinear(FEA)",
"Collinear(CED)"
] | [
"Equal(LengthOfLine(CE),LengthOfLine(ED))",
"Equal(LengthOfLine(EC),6)",
"Equal(LengthOfLine(EF),5)",
"Equal(LengthOfLine(FE),LengthOfLine(EA))",
"Rhombus(CFDA)"
] | [
"Equal(LengthOfLine(CE),LengthOfLine(ED))",
"Equal(LengthOfLine(EC),6)",
"Equal(LengthOfLine(EF),5)",
"Equal(LengthOfLine(FE),LengthOfLine(EA))"
] | Value(AreaOfQuadrilateral(CFDA)) | 60 | [
"line_addition(1,FE,EA)",
"line_addition(1,CE,ED)",
"kite_area_formula_diagonal(1,CFDA)"
] | {"START": ["line_addition(1,FE,EA)", "line_addition(1,CE,ED)", "kite_area_formula_diagonal(1,CFDA)"]} | |
207 | XiaokaiZhang_2023-04-02 | Geometry3k-209 | 2 | 如图所示，AC=16，CD=23，∠BDE=60°，BA和DC是平行四边形BDCA的一组对边，DE⊥BE。求BDCA的面积。 | As shown in the diagram, AC=16, CD=23, ∠BDE=60°, quadrilateral BDCA is a parallelogram, DE is perpendicular to BE. Find the area of quadrilateral BDCA. | 207.png | [
"Shape(BD,DE,EB)",
"Shape(BE,EC,CA,AB)",
"Collinear(DEC)"
] | [
"Equal(LengthOfLine(AC),16)",
"Equal(LengthOfLine(CD),23)",
"Equal(MeasureOfAngle(BDE),60)",
"Parallelogram(BDCA)",
"PerpendicularBetweenLine(DE,BE)"
] | [
"Equal(LengthOfLine(AC),16)",
"Equal(LengthOfLine(CD),23)",
"Equal(MeasureOfAngle(BDE),60)",
"PerpendicularBetweenLine(DE,BE)"
] | Value(AreaOfQuadrilateral(BDCA)) | 184*sqrt(3) | [
"parallelogram_property_opposite_line_equal(1,BDCA)",
"parallelogram_area_formula_sine(1,BDCA)"
] | {"START": ["parallelogram_property_opposite_line_equal(1,BDCA)", "parallelogram_area_formula_sine(1,BDCA)"]} | |
208 | XiaokaiZhang_2023-04-02 | Geometry3k-210 | 4 | 如图所示，LM=MN，∠BNM=2*x-5°，B是⊙B的圆心，NL是⊙B的直径。求x的值。 | As shown in the diagram, LM=MN, ∠BNM=2*x-5°, the center of circle B is B, the diameter of ⊙B is NL. Find the value of x. | 208.png | [
"Shape(BML,LM)",
"Shape(BLN,NB,BL)",
"Shape(ML,LB,BN,NM)",
"Shape(BNM,MN)",
"Collinear(LBN)",
"Cocircular(B,LNM)"
] | [
"Equal(LengthOfLine(LM),LengthOfLine(MN))",
"Equal(MeasureOfAngle(BNM),2*x-5)",
"IsCentreOfCircle(B,B)",
"IsDiameterOfCircle(NL,B)"
] | [
"Equal(LengthOfLine(LM),LengthOfLine(MN))",
"Equal(MeasureOfAngle(BNM),2*x-5)",
"IsCentreOfCircle(B,B)",
"IsDiameterOfCircle(NL,B)"
] | Value(x) | 25 | [
"isosceles_triangle_judgment_line_equal(1,MLN)",
"isosceles_triangle_property_angle_equal(1,MLN)",
"diameter_of_circle_property_right_angle(1,NML,B)",
"triangle_property_angle_sum(1,MLN)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,MLN)", "diameter_of_circle_property_right_angle(1,NML,B)", "triangle_property_angle_sum(1,MLN)"], "isosceles_triangle_judgment_line_equal(1,MLN)": ["isosceles_triangle_property_angle_equal(1,MLN)"]} | |
209 | XiaokaiZhang_2023-03-12 | Geometry3k-211 | 3 | 如图所示，∠WXH=130°，∠YZI=20°，Mul(LengthOfLine(IJ)=LengthOfLine(YJ))。求∠HIJ的大小。 | As shown in the diagram, ∠WXH=130°, ∠YZI=20°, Mul(LengthOfLine(IJ)=LengthOfLine(YJ)). Find the measure of ∠HIJ. | 209.png | [
"Shape(JH,HI,IJ)",
"Shape(HX,XY,YI,IH)",
"Shape(IY,YZ,ZI)",
"Shape(WX,XH)",
"Collinear(WXYZ)",
"Collinear(JHX)",
"Collinear(JIY)",
"Collinear(HIZ)"
] | [
"Equal(MeasureOfAngle(WXH),130)",
"Equal(MeasureOfAngle(YZI),20)",
"Equal(Mul(LengthOfLine(IJ),LengthOfLine(YJ)),Mul(LengthOfLine(HJ),LengthOfLine(XJ)))"
] | [] | Value(MeasureOfAngle(HIJ)) | 50 | [
"adjacent_complementary_angle(1,WXH,HXY)",
"mirror_similar_triangle_judgment_sas(1,JHI,JXY)",
"mirror_similar_triangle_property_angle_equal(1,IJH,XYJ)"
] | {"START": ["adjacent_complementary_angle(1,WXH,HXY)", "mirror_similar_triangle_judgment_sas(1,JHI,JXY)"], "mirror_similar_triangle_judgment_sas(1,JHI,JXY)": ["mirror_similar_triangle_property_angle_equal(1,IJH,XYJ)"]} | |
210 | XiaokaiZhang_2023-03-12 | Geometry3k-212 | 1 | 如图所示，∠BVC=52°，∠VCB=6*x+14*y°，∠ZXY=15*x-8*y°，∠ZXY=∠BVC，CB垂直于VB，XY⊥ZY。求y的值。 | As shown in the diagram, ∠BVC=52°, ∠VCB=6*x+14*y°, ∠ZXY=15*x-8*y°, ∠ZXY=∠BVC, CB⊥VB, XY⊥ZY. Find the value of y. | 210.png | [
"Shape(XY,YZ,ZX)",
"Shape(VC,CB,BV)"
] | [
"Equal(MeasureOfAngle(BVC),52)",
"Equal(MeasureOfAngle(VCB),6*x+14*y)",
"Equal(MeasureOfAngle(ZXY),15*x-8*y)",
"Equal(MeasureOfAngle(ZXY),MeasureOfAngle(BVC))",
"PerpendicularBetweenLine(CB,VB)",
"PerpendicularBetweenLine(XY,ZY)"
] | [
"Equal(MeasureOfAngle(BVC),52)",
"Equal(MeasureOfAngle(VCB),6*x+14*y)",
"Equal(MeasureOfAngle(ZXY),15*x-8*y)",
"Equal(MeasureOfAngle(ZXY),MeasureOfAngle(BVC))",
"PerpendicularBetweenLine(CB,VB)",
"PerpendicularBetweenLine(XY,ZY)"
] | Value(y) | 1 | [
"triangle_property_angle_sum(1,VCB)"
] | {"START": ["triangle_property_angle_sum(1,VCB)"]} | |
211 | XiaokaiZhang_2023-03-12 | Geometry3k-213 | 2 | 如图所示，AB=16，AC=5，BC=x，BC垂直于AC。求x的值。 | As shown in the diagram, AB=16, AC=5, BC=x, BC is perpendicular to AC. Find the value of x. | 211.png | [
"Shape(CA,AB,BC)"
] | [
"Equal(LengthOfLine(AB),16)",
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BC),x)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),16)",
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BC),x)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(x) | sqrt(231) | [
"right_triangle_judgment_angle(1,BCA)",
"right_triangle_property_pythagorean(1,BCA)"
] | {"START": ["right_triangle_judgment_angle(1,BCA)"], "right_triangle_judgment_angle(1,BCA)": ["right_triangle_property_pythagorean(1,BCA)"]} | |
212 | XiaokaiZhang_2023-04-02 | Geometry3k-214 | 4 | 如图所示，弧BRS的角度为103，B是⊙B的圆心，圆O的切线为TR，⊙O的切线为TS。求∠RTS的大小。 | As shown in the diagram, the measure of ⌒BRS is 103, the center of ⊙B is B, TR is the tangent to circle B, the tangent to circle B is TS. Find the measure of ∠RTS. | 212.png | [
"Shape(BS,BSR,RB)",
"Shape(BR,BRS,SB)",
"Shape(BRS,RT,TS)",
"Cocircular(B,RS)"
] | [
"Equal(MeasureOfArc(BRS),103)",
"IsCentreOfCircle(B,B)",
"IsTangentOfCircle(TR,B)",
"IsTangentOfCircle(TS,B)"
] | [
"Equal(MeasureOfArc(BRS),103)",
"IsCentreOfCircle(B,B)",
"IsTangentOfCircle(TR,B)",
"IsTangentOfCircle(TS,B)"
] | Value(MeasureOfAngle(RTS)) | 77 | [
"tangent_of_circle_property_perpendicular(1,TR,B,B)",
"tangent_of_circle_property_perpendicular(2,TS,B,B)",
"arc_property_center_angle(1,BRS,B)",
"quadrilateral_property_angle_sum(1,BRTS)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,TR,B,B)", "tangent_of_circle_property_perpendicular(2,TS,B,B)", "arc_property_center_angle(1,BRS,B)", "quadrilateral_property_angle_sum(1,BRTS)"]} | |
213 | XiaokaiZhang_2023-03-12 | Geometry3k-215 | 1 | 如图所示，∠ABC=76°，∠CAB=Mul(MeasureOfAngle(ABC), 1/2)°。求∠BCA的大小。 | As shown in the diagram, ∠ABC=76°, ∠CAB=Mul(MeasureOfAngle(ABC), 1/2)°. Find the measure of ∠BCA. | 213.png | [
"Shape(CA,AB,BC)"
] | [
"Equal(MeasureOfAngle(ABC),76)",
"Equal(MeasureOfAngle(CAB),Mul(MeasureOfAngle(ABC), 1/2))"
] | [] | Value(MeasureOfAngle(BCA)) | 66 | [
"triangle_property_angle_sum(1,CAB)"
] | {"START": ["triangle_property_angle_sum(1,CAB)"]} | |
214 | XiaokaiZhang_2023-03-12 | Geometry3k-216 | 3 | 如图所示，FJ=FH，GF=GH，∠HFJ=34°。求∠FJH的大小。 | As shown in the diagram, FJ=FH, GF=GH, ∠HFJ=34°. Find the measure of ∠FJH. | 214.png | [
"Shape(FJ,JH,HF)",
"Shape(FH,HG,GF)"
] | [
"Equal(LengthOfLine(FJ),LengthOfLine(FH))",
"Equal(LengthOfLine(GF),LengthOfLine(GH))",
"Equal(MeasureOfAngle(HFJ),34)"
] | [] | Value(MeasureOfAngle(FJH)) | 73 | [
"isosceles_triangle_judgment_line_equal(1,FJH)",
"isosceles_triangle_property_angle_equal(1,FJH)",
"triangle_property_angle_sum(1,FJH)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,FJH)", "triangle_property_angle_sum(1,FJH)"], "isosceles_triangle_judgment_line_equal(1,FJH)": ["isosceles_triangle_property_angle_equal(1,FJH)"]} | |
215 | XiaokaiZhang_2023-04-02 | Geometry3k-217 | 3 | 如图所示，FN=1/4*x+6，IP=PD，NA=2*x-29，PD=16-5*y，PI=12-3*y，IF平行于PN，PN平行于DA。求x的值。 | As shown in the diagram, FN=1/4*x+6, IP=PD, NA=2*x-29, PD=16-5*y, PI=12-3*y, IF is parallel to PN, PN is parallel to DA. Find the value of x. | 215.png | [
"Shape(IP,PN,NF,FI)",
"Shape(PD,DA,AN,NP)",
"Collinear(IPD)",
"Collinear(FNA)"
] | [
"Equal(LengthOfLine(FN),1/4*x+6)",
"Equal(LengthOfLine(IP),LengthOfLine(PD))",
"Equal(LengthOfLine(NA),2*x-29)",
"Equal(LengthOfLine(PD),16-5*y)",
"Equal(LengthOfLine(PI),12-3*y)",
"ParallelBetweenLine(IF,PN)",
"ParallelBetweenLine(PN,DA)"
] | [
"Equal(LengthOfLine(FN),1/4*x+6)",
"Equal(LengthOfLine(IP),LengthOfLine(PD))",
"Equal(LengthOfLine(NA),2*x-29)",
"Equal(LengthOfLine(PD),16-5*y)",
"Equal(LengthOfLine(PI),12-3*y)",
"ParallelBetweenLine(IF,PN)",
"ParallelBetweenLine(PN,DA)"
] | Value(x) | 20 | [
"parallel_judgment_par_par(1,IF,PN,DA)",
"trapezoid_judgment_parallel(1,IDAF)",
"midsegment_of_quadrilateral_judgment_parallel(1,PN,IDAF)"
] | {"START": ["parallel_judgment_par_par(1,IF,PN,DA)"], "parallel_judgment_par_par(1,IF,PN,DA)": ["trapezoid_judgment_parallel(1,IDAF)"], "trapezoid_judgment_parallel(1,IDAF)": ["midsegment_of_quadrilateral_judgment_parallel(1,PN,IDAF)"]} | |
216 | XiaokaiZhang_2023-03-12 | Geometry3k-218 | 3 | 如图所示，TV=TU，∠VTU=74°。求∠TUV的大小。 | As shown in the diagram, TV=TU, ∠VTU=74°. Find the measure of ∠TUV. | 216.png | [
"Shape(TU,UV,VT)"
] | [
"Equal(LengthOfLine(TV),LengthOfLine(TU))",
"Equal(MeasureOfAngle(VTU),74)"
] | [
"Equal(LengthOfLine(TV),LengthOfLine(TU))",
"Equal(MeasureOfAngle(VTU),74)"
] | Value(MeasureOfAngle(TUV)) | 53 | [
"isosceles_triangle_judgment_line_equal(1,TUV)",
"isosceles_triangle_property_angle_equal(1,TUV)",
"triangle_property_angle_sum(1,TUV)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,TUV)", "triangle_property_angle_sum(1,TUV)"], "isosceles_triangle_judgment_line_equal(1,TUV)": ["isosceles_triangle_property_angle_equal(1,TUV)"]} | |
217 | XiaokaiZhang_2023-04-02 | Geometry3k-219 | 12 | 如图所示，AB=22，AC=14，BD=14，CD=22，⊙E的圆心为E，圆O的圆心为O，AC是⊙O的直径，圆E的直径为BD，OC垂直于DC。求四边形ACDB的面积减去扇形OCA和EBD的面积和。 | As shown in the diagram, AB=22, AC=14, BD=14, CD=22, the center of circle E is E, the center of ⊙O is O, AC is the diameter of circle O, BD is the diameter of circle E, OC is perpendicular to DC. Find the area of quadrilateral ACDB minus the sum of the areas of sectors OCA and EBD. | 217.png | [
"Shape(AO,OC,OCA)",
"Shape(OCA,CD,EBD,BA)",
"Shape(EBD,DE,EB)",
"Collinear(AOC)",
"Collinear(BED)",
"Cocircular(O,AC)",
"Cocircular(E,BD)"
] | [
"Equal(LengthOfLine(AB),22)",
"Equal(LengthOfLine(AC),14)",
"Equal(LengthOfLine(BD),14)",
"Equal(LengthOfLine(CD),22)",
"IsCentreOfCircle(E,E)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AC,O)",
"IsDiameterOfCircle(BD,E)",
"PerpendicularBetweenLine(OC,DC)"
] | [
"Equal(LengthOfLine(AB),22)",
"Equal(LengthOfLine(AC),14)",
"Equal(LengthOfLine(BD),14)",
"Equal(LengthOfLine(CD),22)",
"IsCentreOfCircle(E,E)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AC,O)",
"IsDiameterOfCircle(BD,E)",
"PerpendicularBetweenLine(OC,DC)"
] | Value(Sub(AreaOfQuadrilateral(ACDB),Add(AreaOfSector(OCA),AreaOfSector(EBD)))) | 308-49*pi | [
"parallelogram_judgment_equal_and_equal(1,ACDB)",
"parallelogram_area_formula_sine(1,ACDB)",
"diameter_of_circle_property_length_equal(1,AC,O)",
"diameter_of_circle_property_length_equal(1,BD,E)",
"circle_property_length_of_radius_and_diameter(1,O)",
"circle_property_length_of_radius_and_diameter(1,E)",
... | {"START": ["parallelogram_judgment_equal_and_equal(1,ACDB)", "diameter_of_circle_property_length_equal(1,AC,O)", "diameter_of_circle_property_length_equal(1,BD,E)", "circle_property_length_of_radius_and_diameter(1,O)", "circle_property_length_of_radius_and_diameter(1,E)", "flat_angle(1,AOC)", "flat_angle(1,DEB)", "arc_... | |
218 | NaZhu_2023-04-02 | Geometry3k-220 | 4 | 如图所示，∠BDE=109°，∠CBE=24°，∠EAC=95°，∠ECB=33°。求∠EBD的大小。 | As shown in the diagram, ∠BDE=109°, ∠CBE=24°, ∠EAC=95°, ∠ECB=33°. Find the measure of ∠EBD. | 218.png | [
"Shape(AC,CE,EA)",
"Shape(DE,EB,BD)",
"Shape(EC,CB,BE)",
"Collinear(AEB)",
"Collinear(CED)"
] | [
"Equal(MeasureOfAngle(BDE),109)",
"Equal(MeasureOfAngle(CBE),24)",
"Equal(MeasureOfAngle(EAC),95)",
"Equal(MeasureOfAngle(ECB),33)"
] | [
"Equal(MeasureOfAngle(BDE),109)",
"Equal(MeasureOfAngle(CBE),24)",
"Equal(MeasureOfAngle(EAC),95)",
"Equal(MeasureOfAngle(ECB),33)"
] | Value(MeasureOfAngle(EBD)) | 14 | [
"triangle_property_angle_sum(1,ECB)",
"flat_angle(1,DEC)",
"angle_addition(1,DEB,BEC)",
"triangle_property_angle_sum(1,EBD)"
] | {"START": ["triangle_property_angle_sum(1,ECB)", "flat_angle(1,DEC)", "angle_addition(1,DEB,BEC)", "triangle_property_angle_sum(1,EBD)"]} | |
219 | XiaokaiZhang_2023-03-12 | Geometry3k-221 | 4 | 如图所示，AM=MP，AP=sqrt(13)，PD=3*sqrt(13)，PN=ND，∠MAB=∠NDC，△BPA的周长为12，CP垂直于NP。求三角形CPD的周长。 | As shown in the diagram, AM=MP, AP=sqrt(13), PD=3*sqrt(13), PN=ND, ∠MAB=∠NDC, the perimeter of triangle BPA is 12, CP⊥NP. Find the perimeter of △CPD. | 219.png | [
"Shape(AB,BM,MA)",
"Shape(MB,BP,PM)",
"Shape(CP,PN,NC)",
"Shape(CN,ND,DC)",
"Collinear(AMPND)",
"Collinear(CPB)"
] | [
"Equal(LengthOfLine(AM),LengthOfLine(MP))",
"Equal(LengthOfLine(AP),sqrt(13))",
"Equal(LengthOfLine(PD),3*sqrt(13))",
"Equal(LengthOfLine(PN),LengthOfLine(ND))",
"Equal(MeasureOfAngle(MAB),MeasureOfAngle(NDC))",
"Equal(PerimeterOfTriangle(BPA),12)",
"PerpendicularBetweenLine(CP,NP)"
] | [
"Equal(LengthOfLine(AM),LengthOfLine(MP))",
"Equal(LengthOfLine(PN),LengthOfLine(ND))",
"Equal(MeasureOfAngle(MAB),MeasureOfAngle(NDC))",
"PerpendicularBetweenLine(CP,NP)"
] | Value(PerimeterOfTriangle(CPD)) | 36 | [
"vertical_angle(1,BPM,CPN)",
"similar_triangle_judgment_aa(1,BPA,CPD)",
"similar_triangle_property_line_ratio(1,BPA,CPD)",
"similar_triangle_property_perimeter_ratio(1,PAB,PDC)"
] | {"START": ["vertical_angle(1,BPM,CPN)"], "similar_triangle_judgment_aa(1,BPA,CPD)": ["similar_triangle_property_line_ratio(1,BPA,CPD)", "similar_triangle_property_perimeter_ratio(1,PAB,PDC)"], "vertical_angle(1,BPM,CPN)": ["similar_triangle_judgment_aa(1,BPA,CPD)"]} | |
220 | XiaokaiZhang_2023-03-12 | Geometry3k-222 | 2 | 如图所示，AD=15，PF=6，P是△ACE的重心。求直线AP的长度。 | As shown in the diagram, AD=15, PF=6, the centroid of △ACE is P. Find the length of line AP. | 220.png | [
"Shape(AB,BP,PA)",
"Shape(BC,CP,PB)",
"Shape(PC,CD,DP)",
"Shape(PD,DE,EP)",
"Shape(PE,EF,FP)",
"Shape(PF,FA,AP)",
"Collinear(ABC)",
"Collinear(CDE)",
"Collinear(EFA)",
"Collinear(BPE)",
"Collinear(CPF)",
"Collinear(APD)"
] | [
"Equal(LengthOfLine(AD),15)",
"Equal(LengthOfLine(PF),6)",
"IsCentroidOfTriangle(P,ACE)"
] | [] | Value(LengthOfLine(AP)) | 10 | [
"centroid_of_triangle_property_line_ratio(1,P,ACE,D)",
"line_addition(1,AP,PD)"
] | {"START": ["centroid_of_triangle_property_line_ratio(1,P,ACE,D)", "line_addition(1,AP,PD)"]} | |
221 | NaZhu_2023-04-02 | Geometry3k-223 | 1 | 如图所示，∠PSR=x+10°，∠QPS=x°，∠RQP=2*x-16°，∠SRQ=2*x°。求∠PSR的大小。 | As shown in the diagram, ∠PSR=x+10°, ∠QPS=x°, ∠RQP=2*x-16°, ∠SRQ=2*x°. Find the measure of ∠PSR. | 221.png | [
"Shape(QP,PS,SR,RQ)"
] | [
"Equal(MeasureOfAngle(PSR),x+10)",
"Equal(MeasureOfAngle(QPS),x)",
"Equal(MeasureOfAngle(RQP),2*x-16)",
"Equal(MeasureOfAngle(SRQ),2*x)"
] | [
"Equal(MeasureOfAngle(PSR),x+10)",
"Equal(MeasureOfAngle(QPS),x)",
"Equal(MeasureOfAngle(RQP),2*x-16)",
"Equal(MeasureOfAngle(SRQ),2*x)"
] | Value(MeasureOfAngle(PSR)) | 71 | [
"quadrilateral_property_angle_sum(1,QPSR)"
] | {"START": ["quadrilateral_property_angle_sum(1,QPSR)"]} | |
222 | NaZhu_2023-04-02 | Geometry3k-224 | 4 | 如图所示，KL=10，∠LKJ=85°，K是⊙K的圆心。求扇形KLJ的面积。 | As shown in the diagram, KL=10, ∠LKJ=85°, K is the center of circle K. Find the area of the sector KLJ. | 222.png | [
"Shape(KJL,LK,KJ)",
"Shape(KLJ,JK,KL)",
"Cocircular(K,JL)"
] | [
"Equal(LengthOfLine(KL),10)",
"Equal(MeasureOfAngle(LKJ),85)",
"IsCentreOfCircle(K,K)"
] | [
"Equal(LengthOfLine(KL),10)",
"Equal(MeasureOfAngle(LKJ),85)",
"IsCentreOfCircle(K,K)"
] | Value(AreaOfSector(KLJ)) | 1375*pi/18 | [
"radius_of_circle_property_length_equal(1,KL,K)",
"arc_property_center_angle(1,KLJ,K)",
"round_angle(1,LKJ,JKL)",
"sector_area_formula(1,KLJ)"
] | {"START": ["radius_of_circle_property_length_equal(1,KL,K)", "arc_property_center_angle(1,KLJ,K)", "round_angle(1,LKJ,JKL)", "sector_area_formula(1,KLJ)"]} | |
223 | NaZhu_2023-04-02 | Geometry3k-225 | 3 | 如图所示，XW=ZW，XY=ZY，∠XWZ=70°，∠ZYX=56°。求∠YXW的大小。 | As shown in the diagram, XW=ZW, XY=ZY, ∠XWZ=70°, ∠ZYX=56°. Find the measure of ∠YXW. | 223.png | [
"Shape(XW,WZ,ZY,YX)"
] | [
"Equal(LengthOfLine(XW),LengthOfLine(ZW))",
"Equal(LengthOfLine(XY),LengthOfLine(ZY))",
"Equal(MeasureOfAngle(XWZ),70)",
"Equal(MeasureOfAngle(ZYX),56)"
] | [
"Equal(LengthOfLine(XW),LengthOfLine(ZW))",
"Equal(LengthOfLine(XY),LengthOfLine(ZY))",
"Equal(MeasureOfAngle(XWZ),70)",
"Equal(MeasureOfAngle(ZYX),56)"
] | Value(MeasureOfAngle(YXW)) | 117 | [
"kite_judgment_equal_and_equal(1,WZYX)",
"kite_property_opposite_angle_equal(1,WZYX)",
"quadrilateral_property_angle_sum(1,WZYX)"
] | {"START": ["kite_judgment_equal_and_equal(1,WZYX)", "quadrilateral_property_angle_sum(1,WZYX)"], "kite_judgment_equal_and_equal(1,WZYX)": ["kite_property_opposite_angle_equal(1,WZYX)"]} | |
224 | NaZhu_2023-04-02 | Geometry3k-226 | 2 | 如图所示，∠HPM=4*y°，∠MPR=68°，∠PRC=x°，∠SCR=5*z+2°，CM∥RP，CR∥MP。求y的值。 | As shown in the diagram, ∠HPM=4*y°, ∠MPR=68°, ∠PRC=x°, ∠SCR=5*z+2°, CM∥RP, CR is parallel to MP. Find the value of y. | 224.png | [
"Shape(CM,MP,PR,RC)",
"Collinear(NCRI)",
"Collinear(DMPL)",
"Collinear(SCME)",
"Collinear(GRPH)",
"Shape(HP,PM)"
] | [
"Equal(MeasureOfAngle(HPM),4*y)",
"Equal(MeasureOfAngle(MPR),68)",
"Equal(MeasureOfAngle(PRC),x)",
"Equal(MeasureOfAngle(SCR),5*z+2)",
"ParallelBetweenLine(CM,RP)",
"ParallelBetweenLine(CR,MP)"
] | [
"Equal(MeasureOfAngle(HPM),4*y)",
"Equal(MeasureOfAngle(MPR),68)",
"Equal(MeasureOfAngle(PRC),x)",
"Equal(MeasureOfAngle(SCR),5*z+2)",
"ParallelBetweenLine(CM,RP)",
"ParallelBetweenLine(CR,MP)"
] | Value(y) | 28 | [
"flat_angle(1,HPR)",
"angle_addition(1,HPM,MPR)"
] | {"START": ["flat_angle(1,HPR)", "angle_addition(1,HPM,MPR)"]} | |
225 | NaZhu_2023-04-02 | Geometry3k-227 | 2 | 如图所示，CN=8，圆N的圆心为N。求直线DN的长度。 | As shown in the diagram, CN=8, N is the center of circle N. Find the length of line DN. | 225.png | [
"Shape(NDC,CN,ND)",
"Shape(NCF,FN,NC)",
"Shape(NF,FE,EN)",
"Shape(NED,DN,NE)",
"Shape(NFE,EN,NF)",
"Cocircular(N,DCFE)"
] | [
"Equal(LengthOfLine(CN),8)",
"IsCentreOfCircle(N,N)"
] | [
"Equal(LengthOfLine(CN),8)",
"IsCentreOfCircle(N,N)"
] | Value(LengthOfLine(DN)) | 8 | [
"radius_of_circle_property_length_equal(1,ND,N)",
"radius_of_circle_property_length_equal(1,NC,N)"
] | {"START": ["radius_of_circle_property_length_equal(1,ND,N)", "radius_of_circle_property_length_equal(1,NC,N)"]} | |
226 | XiaokaiZhang_2023-03-12 | Geometry3k-228 | 1 | 如图所示，AB=20，AC=10，BC=x，∠ABC=y°，BC⊥AC。求y的值。 | As shown in the diagram, AB=20, AC=10, BC=x, ∠ABC=y°, BC is perpendicular to AC. Find the value of y. | 226.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AB),20)",
"Equal(LengthOfLine(AC),10)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(ABC),y)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),20)",
"Equal(LengthOfLine(AC),10)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(ABC),y)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(y) | 30 | [
"sine_theorem(1,ABC)"
] | {"START": ["sine_theorem(1,ABC)"]} | |
227 | NaZhu_2023-04-02 | Geometry3k-229 | 3 | 如图所示，CB=40，DB=38，SD=28，四边形ACBS是平行四边形，BD垂直于SD。求四边形ACBS的面积。 | As shown in the diagram, CB=40, DB=38, SD=28, quadrilateral ACBS is a parallelogram, BD is perpendicular to SD. Find the area of ACBS. | 227.png | [
"Shape(AC,CB,BS,SA)",
"Shape(BD,DS,SB)",
"Collinear(ASD)"
] | [
"Equal(LengthOfLine(CB),40)",
"Equal(LengthOfLine(DB),38)",
"Equal(LengthOfLine(SD),28)",
"Parallelogram(ACBS)",
"PerpendicularBetweenLine(BD,SD)"
] | [
"Equal(LengthOfLine(CB),40)",
"Equal(LengthOfLine(DB),38)",
"Equal(LengthOfLine(SD),28)",
"PerpendicularBetweenLine(BD,SD)"
] | Value(AreaOfQuadrilateral(ACBS)) | 1520 | [
"altitude_of_quadrilateral_judgment_left_vertex(3,BD,BSAC)",
"parallelogram_property_opposite_line_equal(1,CBSA)",
"parallelogram_area_formula_common(1,BSAC)"
] | {"START": ["altitude_of_quadrilateral_judgment_left_vertex(3,BD,BSAC)", "parallelogram_property_opposite_line_equal(1,CBSA)", "parallelogram_area_formula_common(1,BSAC)"]} | |
228 | XiaokaiZhang_2023-03-12 | Geometry3k-230 | 3 | 如图所示，AB=CB，DA=5*x-6，DC=3*x+4，BD垂直于AD。求直线AC的长度。 | As shown in the diagram, AB=CB, DA=5*x-6, DC=3*x+4, BD⊥AD. Find the length of line AC. | 228.png | [
"Shape(AB,BD,DA)",
"Shape(BC,CD,DB)",
"Collinear(ADC)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(CB))",
"Equal(LengthOfLine(DA),5*x-6)",
"Equal(LengthOfLine(DC),3*x+4)",
"PerpendicularBetweenLine(BD,AD)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(CB))",
"Equal(LengthOfLine(DA),5*x-6)",
"Equal(LengthOfLine(DC),3*x+4)",
"PerpendicularBetweenLine(BD,AD)"
] | Value(LengthOfLine(AC)) | 38 | [
"adjacent_complementary_angle(1,CDB,BDA)",
"perpendicular_bisector_judgment_distance_equal(1,BD,CA)",
"line_addition(1,AD,DC)"
] | {"START": ["adjacent_complementary_angle(1,CDB,BDA)", "line_addition(1,AD,DC)"], "adjacent_complementary_angle(1,CDB,BDA)": ["perpendicular_bisector_judgment_distance_equal(1,BD,CA)"]} | |
229 | XiaokaiZhang_2023-03-12 | Geometry3k-231 | 2 | 如图所示，∠ACB=29°，∠CBA=x°，∠DAB=4*x°。求x的值。 | As shown in the diagram, ∠ACB=29°, ∠CBA=x°, ∠DAB=4*x°. Find the value of x. | 229.png | [
"Shape(AC,CB,BA)",
"Shape(DA,AB)",
"Collinear(DAC)"
] | [
"Equal(MeasureOfAngle(ACB),29)",
"Equal(MeasureOfAngle(CBA),x)",
"Equal(MeasureOfAngle(DAB),4*x)"
] | [
"Equal(MeasureOfAngle(ACB),29)",
"Equal(MeasureOfAngle(CBA),x)",
"Equal(MeasureOfAngle(DAB),4*x)"
] | Value(x) | 29/3 | [
"triangle_property_angle_sum(1,ACB)",
"adjacent_complementary_angle(1,DAB,BAC)"
] | {"START": ["triangle_property_angle_sum(1,ACB)", "adjacent_complementary_angle(1,DAB,BAC)"]} | |
230 | XiaokaiZhang_2023-03-12 | Geometry3k-232 | 5 | 如图所示，NZ=9，XM=4，XN=6，NM平行于ZY。求直线XY的长度。 | As shown in the diagram, NZ=9, XM=4, XN=6, NM∥ZY. Find the length of line XY. | 230.png | [
"Shape(MX,XN,NM)",
"Shape(MN,NZ,ZY,YM)",
"Collinear(XNZ)",
"Collinear(XMY)"
] | [
"Equal(LengthOfLine(NZ),9)",
"Equal(LengthOfLine(XM),4)",
"Equal(LengthOfLine(XN),6)",
"ParallelBetweenLine(NM,ZY)"
] | [
"Equal(LengthOfLine(NZ),9)",
"Equal(LengthOfLine(XM),4)",
"Equal(LengthOfLine(XN),6)",
"ParallelBetweenLine(NM,ZY)"
] | Value(LengthOfLine(XY)) | 10 | [
"parallel_property_corresponding_angle(1,NM,ZY,X)",
"similar_triangle_judgment_aa(1,MXN,YXZ)",
"line_addition(1,XN,NZ)",
"similar_triangle_property_line_ratio(1,MXN,YXZ)",
"similar_triangle_property_line_ratio(1,NMX,ZYX)"
] | {"START": ["parallel_property_corresponding_angle(1,NM,ZY,X)", "line_addition(1,XN,NZ)"], "parallel_property_corresponding_angle(1,NM,ZY,X)": ["similar_triangle_judgment_aa(1,MXN,YXZ)"], "similar_triangle_judgment_aa(1,MXN,YXZ)": ["similar_triangle_property_line_ratio(1,MXN,YXZ)", "similar_triangle_property_line_ratio(... | |
231 | NaZhu_2023-04-02 | Geometry3k-233 | 2 | 如图所示，⌒AHG的角度为78，GB垂直于HB，JH垂直于GH。求∠HGB的大小。 | As shown in the diagram, the measure of ⌒AHG is 78, GB is perpendicular to HB, JH is perpendicular to GH. Find the measure of ∠HGB. | 231.png | [
"Shape(AGF,FB,BG)",
"Shape(AGJ,JB,BF)",
"Shape(GB,BH,HG)",
"Shape(BA,AH,HB)",
"Shape(HA,AJ,JH)",
"Shape(AHG,GH)",
"Shape(AJH,HJ)",
"Collinear(FBH)",
"Collinear(GBAJ)",
"Cocircular(A,GFJH)"
] | [
"Equal(MeasureOfArc(AHG),78)",
"PerpendicularBetweenLine(GB,HB)",
"PerpendicularBetweenLine(JH,GH)"
] | [
"PerpendicularBetweenLine(GB,HB)",
"PerpendicularBetweenLine(JH,GH)"
] | Value(MeasureOfAngle(HGB)) | 51 | [
"arc_property_circumference_angle_external(1,AHG,J)",
"triangle_property_angle_sum(1,GJH)"
] | {"START": ["arc_property_circumference_angle_external(1,AHG,J)", "triangle_property_angle_sum(1,GJH)"]} | |
232 | XiaokaiZhang_2023-03-12 | Geometry3k-234 | 8 | 如图所示，AD=12，BD=4，AD垂直于CD，DE⊥CE，EC⊥AC。求直线DE的长度。 | As shown in the diagram, AD=12, BD=4, AD⊥CD, DE⊥CE, EC⊥AC. Find the length of line DE. | 232.png | [
"Shape(CA,AD,DC)",
"Shape(CD,DE,EC)",
"Shape(ED,DB,BE)",
"Collinear(ADB)",
"Collinear(CEB)"
] | [
"Equal(LengthOfLine(AD),12)",
"Equal(LengthOfLine(BD),4)",
"PerpendicularBetweenLine(AD,CD)",
"PerpendicularBetweenLine(DE,CE)",
"PerpendicularBetweenLine(EC,AC)"
] | [
"PerpendicularBetweenLine(AD,CD)",
"PerpendicularBetweenLine(DE,CE)",
"PerpendicularBetweenLine(EC,AC)"
] | Value(LengthOfLine(DE)) | 2*sqrt(3) | [
"line_addition(1,AD,DB)",
"adjacent_complementary_angle(1,BED,DEC)",
"mirror_similar_triangle_judgment_aa(1,CAD,BCA)",
"mirror_similar_triangle_property_line_ratio(1,CAD,BCA)",
"mirror_similar_triangle_property_line_ratio(1,DCA,CAB)",
"similar_triangle_judgment_aa(1,DBE,ABC)",
"similar_triangle_property... | {"START": ["line_addition(1,AD,DB)", "adjacent_complementary_angle(1,BED,DEC)", "mirror_similar_triangle_judgment_aa(1,CAD,BCA)"], "adjacent_complementary_angle(1,BED,DEC)": ["similar_triangle_judgment_aa(1,DBE,ABC)"], "mirror_similar_triangle_judgment_aa(1,CAD,BCA)": ["mirror_similar_triangle_property_line_ratio(1,CAD... | |
233 | XiaokaiZhang_2023-03-12 | Geometry3k-236 | 2 | 如图所示，WZ=4，XW=4，∠WYX=23°，WZ垂直于YZ，YX⊥WX。求直线XY的长度。 | As shown in the diagram, WZ=4, XW=4, ∠WYX=23°, WZ is perpendicular to YZ, YX is perpendicular to WX. Find the length of line XY. | 233.png | [
"Shape(XW,WY,YX)",
"Shape(WZ,ZY,YW)"
] | [
"Equal(LengthOfLine(WZ),4)",
"Equal(LengthOfLine(XW),4)",
"Equal(MeasureOfAngle(WYX),23)",
"PerpendicularBetweenLine(WZ,YZ)",
"PerpendicularBetweenLine(YX,WX)"
] | [
"Equal(LengthOfLine(WZ),4)",
"Equal(LengthOfLine(XW),4)",
"Equal(MeasureOfAngle(WYX),23)",
"PerpendicularBetweenLine(WZ,YZ)",
"PerpendicularBetweenLine(YX,WX)"
] | Value(LengthOfLine(XY)) | 4/tan(23*pi/180) | [
"triangle_property_angle_sum(1,XWY)",
"sine_theorem(1,XWY)"
] | {"START": ["triangle_property_angle_sum(1,XWY)", "sine_theorem(1,XWY)"]} | |
234 | NaZhu_2023-04-02 | Geometry3k-237 | 3 | 如图所示，∠BEC=57°，DF∥HE，FE平行于DH。求∠IFA的大小。 | As shown in the diagram, ∠BEC=57°, DF is parallel to HE, FE∥DH. Find the measure of ∠IFA. | 234.png | [
"Shape(FD,DH,HE,EF)",
"Collinear(AFDL)",
"Collinear(BEHK)",
"Collinear(IFEC)",
"Collinear(GDHJ)",
"Shape(BE,EC)",
"Shape(IF,FA)"
] | [
"Equal(MeasureOfAngle(BEC),57)",
"ParallelBetweenLine(DF,HE)",
"ParallelBetweenLine(FE,DH)"
] | [
"Equal(MeasureOfAngle(BEC),57)",
"ParallelBetweenLine(DF,HE)",
"ParallelBetweenLine(FE,DH)"
] | Value(MeasureOfAngle(IFA)) | 123 | [
"vertical_angle(1,BEC,HEI)",
"vertical_angle(1,IFA,EFD)",
"parallel_property_ipsilateral_internal_angle(1,EH,FD)"
] | {"START": ["vertical_angle(1,BEC,HEI)", "vertical_angle(1,IFA,EFD)", "parallel_property_ipsilateral_internal_angle(1,EH,FD)"]} | |
235 | NaZhu_2023-04-02 | Geometry3k-238 | 10 | 如图所示，∠NAP=120°，∠PAQ=100°，A是⊙A的圆心。求∠QPN的大小。 | As shown in the diagram, ∠NAP=120°, ∠PAQ=100°, the center of circle A is A. Find the measure of ∠QPN. | 235.png | [
"Shape(APN,NP)",
"Shape(AP,PN,NA)",
"Shape(PA,AQ,QP)",
"Shape(AQP,PQ)",
"Shape(ANQ,QA,AN)",
"Cocircular(A,PNQ)"
] | [
"Equal(MeasureOfAngle(NAP),120)",
"Equal(MeasureOfAngle(PAQ),100)",
"IsCentreOfCircle(A,A)"
] | [
"Equal(MeasureOfAngle(NAP),120)",
"Equal(MeasureOfAngle(PAQ),100)",
"IsCentreOfCircle(A,A)"
] | Value(MeasureOfAngle(QPN)) | 70 | [
"radius_of_circle_property_length_equal(1,AN,A)",
"radius_of_circle_property_length_equal(1,AP,A)",
"radius_of_circle_property_length_equal(1,AQ,A)",
"isosceles_triangle_judgment_line_equal(1,APN)",
"isosceles_triangle_judgment_line_equal(1,AQP)",
"isosceles_triangle_property_angle_equal(1,APN)",
"isosc... | {"START": ["radius_of_circle_property_length_equal(1,AN,A)", "radius_of_circle_property_length_equal(1,AP,A)", "radius_of_circle_property_length_equal(1,AQ,A)", "triangle_property_angle_sum(1,APN)", "triangle_property_angle_sum(1,AQP)", "angle_addition(1,QPA,APN)"], "isosceles_triangle_judgment_line_equal(1,APN)": ["is... | |
236 | XiaokaiZhang_2023-03-12 | Geometry3k-239 | 2 | 如图所示，∠UTV=47°，RU垂直于SU，TV⊥UV，VS垂直于RS。求∠USR的大小。 | As shown in the diagram, ∠UTV=47°, RU⊥SU, TV is perpendicular to UV, VS is perpendicular to RS. Find the measure of ∠USR. | 236.png | [
"Shape(RU,US,SR)",
"Shape(SU,UV,VS)",
"Shape(VU,UT,TV)",
"Collinear(RUT)",
"Collinear(SVT)"
] | [
"Equal(MeasureOfAngle(UTV),47)",
"PerpendicularBetweenLine(RU,SU)",
"PerpendicularBetweenLine(TV,UV)",
"PerpendicularBetweenLine(VS,RS)"
] | [
"PerpendicularBetweenLine(RU,SU)",
"PerpendicularBetweenLine(TV,UV)",
"PerpendicularBetweenLine(VS,RS)"
] | Value(MeasureOfAngle(USR)) | 47 | [
"triangle_property_angle_sum(1,RUS)",
"triangle_property_angle_sum(1,TSR)"
] | {"START": ["triangle_property_angle_sum(1,RUS)", "triangle_property_angle_sum(1,TSR)"]} | |
237 | NaZhu_2023-04-02 | Geometry3k-240 | 1 | 如图所示，FB=9，FC=6，FD=6，FE=x。求x的值。 | As shown in the diagram, FB=9, FC=6, FD=6, FE=x. Find the value of x. | 237.png | [
"Shape(AEC,CF,FE)",
"Shape(ADE,EF,FD)",
"Shape(ACB,BF,FC)",
"Shape(ABD,DF,FB)",
"Collinear(CFD)",
"Collinear(EFAB)",
"Cocircular(A,ECBD)"
] | [
"Equal(LengthOfLine(FB),9)",
"Equal(LengthOfLine(FC),6)",
"Equal(LengthOfLine(FD),6)",
"Equal(LengthOfLine(FE),x)"
] | [
"Equal(LengthOfLine(FB),9)",
"Equal(LengthOfLine(FC),6)",
"Equal(LengthOfLine(FD),6)",
"Equal(LengthOfLine(FE),x)"
] | Value(x) | 4 | [
"circle_property_circular_power_chord_and_chord(1,EFB,CFD,A)"
] | {"START": ["circle_property_circular_power_chord_and_chord(1,EFB,CFD,A)"]} | |
238 | XiaokaiZhang_2023-03-12 | Geometry3k-241 | 4 | 如图所示，KR=RJ，KS=SL，LT=TJ，PT=2。求直线KP的长度。 | As shown in the diagram, KR=RJ, KS=SL, LT=TJ, PT=2. Find the length of line KP. | 238.png | [
"Shape(KR,RP,PK)",
"Shape(PR,RJ,JP)",
"Shape(PJ,JT,TP)",
"Shape(PT,TL,LP)",
"Shape(PL,LS,SP)",
"Shape(PS,SK,KP)",
"Collinear(KRJ)",
"Collinear(JTL)",
"Collinear(LSK)",
"Collinear(KPT)",
"Collinear(RPL)",
"Collinear(JPS)"
] | [
"Equal(LengthOfLine(KR),LengthOfLine(RJ))",
"Equal(LengthOfLine(KS),LengthOfLine(SL))",
"Equal(LengthOfLine(LT),LengthOfLine(TJ))",
"Equal(LengthOfLine(PT),2)"
] | [
"Equal(LengthOfLine(KR),LengthOfLine(RJ))",
"Equal(LengthOfLine(KS),LengthOfLine(SL))",
"Equal(LengthOfLine(LT),LengthOfLine(TJ))"
] | Value(LengthOfLine(KP)) | 4 | [
"median_of_triangle_judgment(1,LR,LKJ)",
"median_of_triangle_judgment(1,KT,KJL)",
"centroid_of_triangle_judgment_intersection(1,P,JLK,T,R)",
"centroid_of_triangle_property_line_ratio(1,P,KJL,T)"
] | {"START": ["median_of_triangle_judgment(1,LR,LKJ)", "median_of_triangle_judgment(1,KT,KJL)"], "centroid_of_triangle_judgment_intersection(1,P,JLK,T,R)": ["centroid_of_triangle_property_line_ratio(1,P,KJL,T)"], "median_of_triangle_judgment(1,KT,KJL)": ["centroid_of_triangle_judgment_intersection(1,P,JLK,T,R)"], "median_... | |
239 | XiaokaiZhang_2023-03-12 | Geometry3k-242 | 3 | 如图所示，TR=ST，∠STR=50°。求∠TRS的大小。 | As shown in the diagram, TR=ST, ∠STR=50°. Find the measure of ∠TRS. | 239.png | [
"Shape(RS,ST,TR)"
] | [
"Equal(LengthOfLine(TR),LengthOfLine(ST))",
"Equal(MeasureOfAngle(STR),50)"
] | [
"Equal(LengthOfLine(TR),LengthOfLine(ST))",
"Equal(MeasureOfAngle(STR),50)"
] | Value(MeasureOfAngle(TRS)) | 65 | [
"triangle_property_angle_sum(1,RST)",
"isosceles_triangle_judgment_line_equal(1,TRS)",
"isosceles_triangle_property_angle_equal(1,TRS)"
] | {"START": ["triangle_property_angle_sum(1,RST)", "isosceles_triangle_judgment_line_equal(1,TRS)"], "isosceles_triangle_judgment_line_equal(1,TRS)": ["isosceles_triangle_property_angle_equal(1,TRS)"]} | |
240 | NaZhu_2023-04-02 | Geometry3k-243 | 2 | 如图所示，AC=9*x-1，AF=2*x+7，四边形ADCB是矩形。求直线AF的长度。 | As shown in the diagram, AC=9*x-1, AF=2*x+7, quadrilateral ADCB is a rectangle. Find the length of line AF. | 240.png | [
"Shape(AD,DF,FA)",
"Shape(AF,FB,BA)",
"Shape(FD,DC,CF)",
"Shape(FC,CB,BF)",
"Collinear(AFC)",
"Collinear(DFB)"
] | [
"Equal(LengthOfLine(AC),9*x-1)",
"Equal(LengthOfLine(AF),2*x+7)",
"Rectangle(ADCB)"
] | [
"Equal(LengthOfLine(AC),9*x-1)",
"Equal(LengthOfLine(AF),2*x+7)"
] | Value(LengthOfLine(AF)) | 13 | [
"line_addition(1,AF,FC)",
"parallelogram_property_diagonal_bisection(1,ADCB,F)"
] | {"START": ["line_addition(1,AF,FC)", "parallelogram_property_diagonal_bisection(1,ADCB,F)"]} | |
241 | XiaokaiZhang_2023-03-12 | Geometry3k-244 | 6 | 如图所示，LR=10，PM=Mul(LengthOfLine(KP),2)，PR∥KL，KN垂直于MN，RM垂直于PM。求直线RM的长度。 | As shown in the diagram, LR=10, PM=Mul(LengthOfLine(KP),2), PR is parallel to KL, KN is perpendicular to MN, RM⊥PM. Find the length of line RM. | 241.png | [
"Shape(LR,RQ,QN,NL)",
"Shape(RM,MQ,QR)",
"Shape(QM,MP,PQ)",
"Shape(NQ,QP,PK,KN)",
"Collinear(LRM)",
"Collinear(MPK)",
"Collinear(LNK)",
"Collinear(RQP)",
"Collinear(NQM)"
] | [
"Equal(LengthOfLine(LR),10)",
"Equal(LengthOfLine(PM),Mul(LengthOfLine(KP),2))",
"ParallelBetweenLine(PR,KL)",
"PerpendicularBetweenLine(KN,MN)",
"PerpendicularBetweenLine(RM,PM)"
] | [] | Value(LengthOfLine(RM)) | 20 | [
"parallel_property_corresponding_angle(1,PR,KL,M)",
"similar_triangle_judgment_aa(1,RMP,LMK)",
"line_addition(1,MP,PK)",
"line_addition(1,MR,RL)",
"similar_triangle_property_line_ratio(1,RMP,LMK)",
"similar_triangle_property_line_ratio(1,PRM,KLM)"
] | {"START": ["parallel_property_corresponding_angle(1,PR,KL,M)", "line_addition(1,MP,PK)", "line_addition(1,MR,RL)"], "parallel_property_corresponding_angle(1,PR,KL,M)": ["similar_triangle_judgment_aa(1,RMP,LMK)"], "similar_triangle_judgment_aa(1,RMP,LMK)": ["similar_triangle_property_line_ratio(1,RMP,LMK)", "similar_tri... | |
242 | XiaokaiZhang_2023-03-12 | Geometry3k-245 | 5 | 如图所示，LR=3，RW=6，TR=8，TS平行于LW。求直线WS的长度。 | As shown in the diagram, LR=3, RW=6, TR=8, TS is parallel to LW. Find the length of line WS. | 242.png | [
"Shape(RW,WL,LR)",
"Shape(LW,WS,ST,TL)",
"Collinear(RLT)",
"Collinear(RWS)"
] | [
"Equal(LengthOfLine(LR),3)",
"Equal(LengthOfLine(RW),6)",
"Equal(LengthOfLine(TR),8)",
"ParallelBetweenLine(TS,LW)"
] | [
"ParallelBetweenLine(TS,LW)"
] | Value(LengthOfLine(WS)) | 10 | [
"parallel_property_corresponding_angle(1,WL,ST,R)",
"similar_triangle_judgment_aa(1,LRW,TRS)",
"line_addition(1,RW,WS)",
"similar_triangle_property_line_ratio(1,LRW,TRS)",
"similar_triangle_property_line_ratio(1,WLR,STR)"
] | {"START": ["parallel_property_corresponding_angle(1,WL,ST,R)", "line_addition(1,RW,WS)"], "parallel_property_corresponding_angle(1,WL,ST,R)": ["similar_triangle_judgment_aa(1,LRW,TRS)"], "similar_triangle_judgment_aa(1,LRW,TRS)": ["similar_triangle_property_line_ratio(1,LRW,TRS)", "similar_triangle_property_line_ratio(... | |
243 | XiaokaiZhang_2023-03-12 | Geometry3k-246 | 5 | 如图所示，AC=AY，AD=x+2，CB=5/3*x+11，DB=3*y-9，DB=YD，YD=2*y+6。求x的值。 | As shown in the diagram, AC=AY, AD=x+2, CB=5/3*x+11, DB=3*y-9, DB=YD, YD=2*y+6. Find the value of x. | 243.png | [
"Shape(CA,AD,DB,BC)",
"Shape(AY,YD,DA)",
"Collinear(CAY)",
"Collinear(BDY)"
] | [
"Equal(LengthOfLine(AC),LengthOfLine(AY))",
"Equal(LengthOfLine(AD),x+2)",
"Equal(LengthOfLine(CB),5/3*x+11)",
"Equal(LengthOfLine(DB),3*y-9)",
"Equal(LengthOfLine(DB),LengthOfLine(YD))",
"Equal(LengthOfLine(YD),2*y+6)"
] | [
"Equal(LengthOfLine(AC),LengthOfLine(AY))",
"Equal(LengthOfLine(AD),x+2)",
"Equal(LengthOfLine(CB),5/3*x+11)",
"Equal(LengthOfLine(DB),3*y-9)",
"Equal(LengthOfLine(DB),LengthOfLine(YD))",
"Equal(LengthOfLine(YD),2*y+6)"
] | Value(x) | 21 | [
"line_addition(1,CA,AY)",
"line_addition(1,BD,DY)",
"similar_triangle_judgment_sas(1,YDA,YBC)",
"similar_triangle_property_line_ratio(1,YDA,YBC)",
"similar_triangle_property_line_ratio(1,AYD,CYB)"
] | {"START": ["line_addition(1,CA,AY)", "line_addition(1,BD,DY)"], "line_addition(1,BD,DY)": ["similar_triangle_judgment_sas(1,YDA,YBC)"], "line_addition(1,CA,AY)": ["similar_triangle_judgment_sas(1,YDA,YBC)"], "similar_triangle_judgment_sas(1,YDA,YBC)": ["similar_triangle_property_line_ratio(1,YDA,YBC)", "similar_triangl... | |
244 | NaZhu_2023-04-02 | Geometry3k-247 | 1 | 如图所示，AB=2*x+1，BC=3*x-7，BA是圆O的切线，BC是圆O的切线。求x的值。 | As shown in the diagram, AB=2*x+1, BC=3*x-7, the tangent to ⊙O is BA, the tangent to ⊙O is BC. Find the value of x. | 244.png | [
"Shape(OCA,CB,BA)",
"Shape(OCA,OAC)",
"Cocircular(O,AC)"
] | [
"Equal(LengthOfLine(AB),2*x+1)",
"Equal(LengthOfLine(BC),3*x-7)",
"IsTangentOfCircle(BA,O)",
"IsTangentOfCircle(BC,O)"
] | [
"Equal(LengthOfLine(AB),2*x+1)",
"Equal(LengthOfLine(BC),3*x-7)"
] | Value(x) | 8 | [
"tangent_of_circle_property_length_equal(1,BA,BC,O)"
] | {"START": ["tangent_of_circle_property_length_equal(1,BA,BC,O)"]} | |
245 | NaZhu_2023-04-02 | Geometry3k-248 | 4 | 如图所示，∠AVS=x°，∠QVA=167°，∠SVT=77°，∠TVU=x°，∠UVQ=26°。求x的值。 | As shown in the diagram, ∠AVS=x°, ∠QVA=167°, ∠SVT=77°, ∠TVU=x°, ∠UVQ=26°. Find the value of x. | 245.png | [
"Shape(VQU,UV,VQ)",
"Shape(VUT,TV,VU)",
"Shape(VTS,SV,VT)",
"Shape(VSA,AV,VS)",
"Shape(VAQ,QV,VA)",
"Cocircular(V,QUTSA)"
] | [
"Equal(MeasureOfAngle(AVS),x)",
"Equal(MeasureOfAngle(QVA),167)",
"Equal(MeasureOfAngle(SVT),77)",
"Equal(MeasureOfAngle(TVU),x)",
"Equal(MeasureOfAngle(UVQ),26)"
] | [
"Equal(MeasureOfAngle(AVS),x)",
"Equal(MeasureOfAngle(QVA),167)",
"Equal(MeasureOfAngle(SVT),77)",
"Equal(MeasureOfAngle(TVU),x)",
"Equal(MeasureOfAngle(UVQ),26)"
] | Value(x) | 45 | [
"angle_addition(1,UVQ,QVA)",
"round_angle(1,UVA,AVU)",
"angle_addition(1,SVT,TVU)",
"angle_addition(1,AVS,SVU)"
] | {"START": ["angle_addition(1,UVQ,QVA)", "round_angle(1,UVA,AVU)", "angle_addition(1,SVT,TVU)", "angle_addition(1,AVS,SVU)"]} | |
246 | XiaokaiZhang_2023-03-12 | Geometry3k-249 | 8 | 如图所示，BC=4，BF=x，∠ABD=30°，∠CBE=30°，∠DBF=30°，∠EBA=30°，BA⊥EA，BD⊥AD，BE⊥CE，BF⊥DF。求x的值。 | As shown in the diagram, BC=4, BF=x, ∠ABD=30°, ∠CBE=30°, ∠DBF=30°, ∠EBA=30°, BA is perpendicular to EA, BD is perpendicular to AD, BE is perpendicular to CE, BF⊥DF. Find the value of x. | 246.png | [
"Shape(CB,BE,EC)",
"Shape(EB,BA,AE)",
"Shape(AB,BD,DA)",
"Shape(DB,BF,FD)"
] | [
"Equal(LengthOfLine(BC),4)",
"Equal(LengthOfLine(BF),x)",
"Equal(MeasureOfAngle(ABD),30)",
"Equal(MeasureOfAngle(CBE),30)",
"Equal(MeasureOfAngle(DBF),30)",
"Equal(MeasureOfAngle(EBA),30)",
"PerpendicularBetweenLine(BA,EA)",
"PerpendicularBetweenLine(BD,AD)",
"PerpendicularBetweenLine(BE,CE)",
"Pe... | [
"Equal(LengthOfLine(BC),4)",
"Equal(LengthOfLine(BF),x)",
"Equal(MeasureOfAngle(ABD),30)",
"Equal(MeasureOfAngle(CBE),30)",
"Equal(MeasureOfAngle(DBF),30)",
"Equal(MeasureOfAngle(EBA),30)",
"PerpendicularBetweenLine(BA,EA)",
"PerpendicularBetweenLine(BD,AD)",
"PerpendicularBetweenLine(BE,CE)",
"Pe... | Value(x) | 9/4 | [
"triangle_property_angle_sum(1,BEC)",
"sine_theorem(1,BEC)",
"triangle_property_angle_sum(1,BAE)",
"sine_theorem(1,BAE)",
"triangle_property_angle_sum(1,BDA)",
"sine_theorem(1,BDA)",
"triangle_property_angle_sum(1,BFD)",
"sine_theorem(1,BFD)"
] | {"START": ["triangle_property_angle_sum(1,BEC)", "sine_theorem(1,BEC)", "triangle_property_angle_sum(1,BAE)", "sine_theorem(1,BAE)", "triangle_property_angle_sum(1,BDA)", "sine_theorem(1,BDA)", "triangle_property_angle_sum(1,BFD)", "sine_theorem(1,BFD)"]} | |
247 | XiaokaiZhang_2023-03-12 | Geometry3k-250 | 2 | 如图所示，AB=x，AC=6，BC=15，BC垂直于AC。求x的值。 | As shown in the diagram, AB=x, AC=6, BC=15, BC⊥AC. Find the value of x. | 247.png | [
"Shape(CA,AB,BC)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),6)",
"Equal(LengthOfLine(BC),15)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),6)",
"Equal(LengthOfLine(BC),15)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(x) | 3*sqrt(29) | [
"right_triangle_judgment_angle(1,BCA)",
"right_triangle_property_pythagorean(1,BCA)"
] | {"START": ["right_triangle_judgment_angle(1,BCA)"], "right_triangle_judgment_angle(1,BCA)": ["right_triangle_property_pythagorean(1,BCA)"]} | |
248 | NaZhu_2023-04-02 | Geometry3k-251 | 1 | 如图所示，∠BGC=40°，∠DGF=53°，CB垂直于GB，FG⊥CG，GF垂直于DF。求∠FGB的大小。 | As shown in the diagram, ∠BGC=40°, ∠DGF=53°, CB is perpendicular to GB, FG is perpendicular to CG, GF⊥DF. Find the measure of ∠FGB. | 248.png | [
"Shape(DG,GF,FD)",
"Shape(FG,GB,BA,AF)",
"Shape(BG,GC,CB)",
"Shape(AB,BC,CA)",
"Collinear(GBA)",
"Collinear(DFA)"
] | [
"Equal(MeasureOfAngle(BGC),40)",
"Equal(MeasureOfAngle(DGF),53)",
"PerpendicularBetweenLine(CB,GB)",
"PerpendicularBetweenLine(FG,CG)",
"PerpendicularBetweenLine(GF,DF)"
] | [
"PerpendicularBetweenLine(CB,GB)",
"PerpendicularBetweenLine(FG,CG)",
"PerpendicularBetweenLine(GF,DF)"
] | Value(MeasureOfAngle(FGB)) | 50 | [
"angle_addition(1,FGA,AGC)"
] | {"START": ["angle_addition(1,FGA,AGC)"]} | |
249 | NaZhu_2023-04-02 | Geometry3k-252 | 1 | 如图所示，FQ=10，FW=x，WQ=4。求x的值。 | As shown in the diagram, FQ=10, FW=x, WQ=4. Find the value of x. | 249.png | [
"Shape(DF,FW,WB,BD)",
"Shape(ND,DB,BW,WQ,QC,CN)",
"Collinear(NDF)",
"Collinear(FWQ)"
] | [
"Equal(LengthOfLine(FQ),10)",
"Equal(LengthOfLine(FW),x)",
"Equal(LengthOfLine(WQ),4)"
] | [
"Equal(LengthOfLine(FQ),10)",
"Equal(LengthOfLine(FW),x)",
"Equal(LengthOfLine(WQ),4)"
] | Value(x) | 6 | [
"line_addition(1,FW,WQ)"
] | {"START": ["line_addition(1,FW,WQ)"]} | |
250 | NaZhu_2023-04-02 | Geometry3k-253 | 9 | 如图所示，AB=2，△DBC为等边△，A是⊙A的圆心。求圆A的面积减去△DBC的面积。 | As shown in the diagram, AB=2, triangle DBC is an equilateral triangle, A is the center of circle A. Find the area of the ⊙A minus the area of △DBC. | 250.png | [
"Shape(ADB,BD)",
"Shape(DB,BA,AD)",
"Shape(AB,BC,CD,DA)",
"Shape(ACD,DC)",
"Shape(ABC,CB)",
"Cocircular(A,DBC)"
] | [
"Equal(LengthOfLine(AB),2)",
"EquilateralTriangle(DBC)",
"IsCentreOfCircle(A,A)"
] | [
"Equal(LengthOfLine(AB),2)",
"IsCentreOfCircle(A,A)"
] | Value(Sub(AreaOfCircle(A),AreaOfTriangle(DBC))) | -3*sqrt(3)+4*pi | [
"equilateral_triangle_property_angle(1,CDB)",
"equilateral_triangle_property_angle(1,DBC)",
"arc_property_center_angle(1,ADB,A)",
"arc_property_circumference_angle_external(1,ADB,C)",
"radius_of_circle_property_length_equal(1,AB,A)",
"radius_of_circle_property_length_equal(1,AD,A)",
"cosine_theorem(1,AD... | {"START": ["equilateral_triangle_property_angle(1,CDB)", "equilateral_triangle_property_angle(1,DBC)", "arc_property_center_angle(1,ADB,A)", "arc_property_circumference_angle_external(1,ADB,C)", "radius_of_circle_property_length_equal(1,AB,A)", "radius_of_circle_property_length_equal(1,AD,A)", "cosine_theorem(1,ADB)", ... | |
251 | NaZhu_2023-04-02 | Geometry3k-254 | 10 | 如图所示，∠TOR=71°，∠UOR=179°，圆O的圆心为O，SR是⊙O的切线。求∠RST的大小。 | As shown in the diagram, ∠TOR=71°, ∠UOR=179°, O is the center of ⊙O, SR is the tangent to ⊙O. Find the measure of ∠RST. | 251.png | [
"Shape(OUR,RO,OU)",
"Shape(OTU,UT)",
"Shape(OT,TU,UO)",
"Shape(ORT,TO,OR)",
"Shape(ORT,RS,ST)",
"Collinear(UTS)",
"Cocircular(O,URT)"
] | [
"Equal(MeasureOfAngle(TOR),71)",
"Equal(MeasureOfAngle(UOR),179)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(SR,O)"
] | [
"Equal(MeasureOfAngle(TOR),71)",
"Equal(MeasureOfAngle(UOR),179)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(SR,O)"
] | Value(MeasureOfAngle(RST)) | 55 | [
"angle_addition(1,UOT,TOR)",
"radius_of_circle_property_length_equal(1,OU,O)",
"radius_of_circle_property_length_equal(1,OT,O)",
"isosceles_triangle_judgment_line_equal(1,OTU)",
"isosceles_triangle_property_angle_equal(1,OTU)",
"triangle_property_angle_sum(1,UOT)",
"flat_angle(1,STU)",
"angle_addition... | {"START": ["angle_addition(1,UOT,TOR)", "radius_of_circle_property_length_equal(1,OU,O)", "radius_of_circle_property_length_equal(1,OT,O)", "triangle_property_angle_sum(1,UOT)", "flat_angle(1,STU)", "angle_addition(1,STO,OTU)", "tangent_of_circle_property_perpendicular(1,SR,O,O)", "quadrilateral_property_angle_sum(1,OR... | |
252 | XiaokaiZhang_2023-03-12 | Geometry3k-255 | 1 | 如图所示，AB=sqrt(481)，AC=16，BC=15，AC⊥BC。求tan(BA)的值。 | As shown in the diagram, AB=sqrt(481), AC=16, BC=15, AC is perpendicular to BC. Find the value of tan(BA). | 252.png | [
"Shape(BA,AC,CB)"
] | [
"Equal(LengthOfLine(AB),sqrt(481))",
"Equal(LengthOfLine(AC),16)",
"Equal(LengthOfLine(BC),15)",
"PerpendicularBetweenLine(AC,BC)"
] | [
"Equal(LengthOfLine(AB),sqrt(481))",
"Equal(LengthOfLine(AC),16)",
"Equal(LengthOfLine(BC),15)",
"PerpendicularBetweenLine(AC,BC)"
] | Value(Tan(MeasureOfAngle(BAC))) | 15/16 | [
"cosine_theorem(1,ACB)"
] | {"START": ["cosine_theorem(1,ACB)"]} | |
253 | NaZhu_2023-04-02 | Geometry3k-256 | 1 | 如图所示，GF=5*x-9，GH=x+7，圆O的切线为GF，圆O的切线为GH。求x的值。 | As shown in the diagram, GF=5*x-9, GH=x+7, GF is the tangent to circle J, the tangent to circle J is GH. Find the value of x. | 253.png | [
"Shape(JHF,HG,GF)",
"Shape(JHF,JFH)",
"Cocircular(J,FH)"
] | [
"Equal(LengthOfLine(GF),5*x-9)",
"Equal(LengthOfLine(GH),x+7)",
"IsTangentOfCircle(GF,J)",
"IsTangentOfCircle(GH,J)"
] | [
"Equal(LengthOfLine(GF),5*x-9)",
"Equal(LengthOfLine(GH),x+7)"
] | Value(x) | 4 | [
"tangent_of_circle_property_length_equal(1,GF,GH,J)"
] | {"START": ["tangent_of_circle_property_length_equal(1,GF,GH,J)"]} | |
254 | XiaokaiZhang_2023-03-12 | Geometry3k-257 | 2 | 如图所示，ZX=3*x+5，ZY=5*x-7，∠ZWX=∠YWZ，WX垂直于ZX，ZY垂直于WY。求直线XZ的长度。 | As shown in the diagram, ZX=3*x+5, ZY=5*x-7, ∠ZWX=∠YWZ, WX⊥ZX, ZY is perpendicular to WY. Find the length of line XZ. | 254.png | [
"Shape(WX,XZ,ZW)",
"Shape(WZ,ZY,YW)"
] | [
"Equal(LengthOfLine(ZX),3*x+5)",
"Equal(LengthOfLine(ZY),5*x-7)",
"Equal(MeasureOfAngle(ZWX),MeasureOfAngle(YWZ))",
"PerpendicularBetweenLine(WX,ZX)",
"PerpendicularBetweenLine(ZY,WY)"
] | [
"Equal(LengthOfLine(ZX),3*x+5)",
"Equal(LengthOfLine(ZY),5*x-7)",
"Equal(MeasureOfAngle(ZWX),MeasureOfAngle(YWZ))",
"PerpendicularBetweenLine(WX,ZX)",
"PerpendicularBetweenLine(ZY,WY)"
] | Value(LengthOfLine(XZ)) | 23 | [
"mirror_congruent_triangle_judgment_aas(3,ZYW,ZWX)",
"mirror_congruent_triangle_property_line_equal(1,WZY,WXZ)"
] | {"START": ["mirror_congruent_triangle_judgment_aas(3,ZYW,ZWX)"], "mirror_congruent_triangle_judgment_aas(3,ZYW,ZWX)": ["mirror_congruent_triangle_property_line_equal(1,WZY,WXZ)"]} | |
255 | XiaokaiZhang_2023-03-12 | Geometry3k-258 | 1 | 如图所示，SR=5，TR=3，TS=4，RT⊥ST。求cos(SR)的值。 | As shown in the diagram, SR=5, TR=3, TS=4, RT⊥ST. Find the value of cos(SR). | 255.png | [
"Shape(TS,SR,RT)"
] | [
"Equal(LengthOfLine(SR),5)",
"Equal(LengthOfLine(TR),3)",
"Equal(LengthOfLine(TS),4)",
"PerpendicularBetweenLine(RT,ST)"
] | [
"Equal(LengthOfLine(SR),5)",
"Equal(LengthOfLine(TR),3)",
"Equal(LengthOfLine(TS),4)",
"PerpendicularBetweenLine(RT,ST)"
] | Value(Cos(MeasureOfAngle(SRT))) | 3/5 | [
"cosine_theorem(1,RTS)"
] | {"START": ["cosine_theorem(1,RTS)"]} | |
256 | XiaokaiZhang_2023-03-12 | Geometry3k-259 | 4 | 如图所示，AC=x，AD=8，BD=y，∠BCA=43°，CA垂直于BA，DB⊥CB。求y的值。 | As shown in the diagram, AC=x, AD=8, BD=y, ∠BCA=43°, CA is perpendicular to BA, DB⊥CB. Find the value of y. | 256.png | [
"Shape(DB,BA,AD)",
"Shape(AB,BC,CA)",
"Collinear(DAC)"
] | [
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(AD),8)",
"Equal(LengthOfLine(BD),y)",
"Equal(MeasureOfAngle(BCA),43)",
"PerpendicularBetweenLine(CA,BA)",
"PerpendicularBetweenLine(DB,CB)"
] | [
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(AD),8)",
"Equal(LengthOfLine(BD),y)",
"Equal(MeasureOfAngle(BCA),43)",
"PerpendicularBetweenLine(CA,BA)",
"PerpendicularBetweenLine(DB,CB)"
] | Value(y) | 8/sin(43*pi/180) | [
"adjacent_complementary_angle(1,CAB,BAD)",
"triangle_property_angle_sum(1,DBA)",
"triangle_property_angle_sum(1,DBC)",
"sine_theorem(1,DBA)"
] | {"START": ["adjacent_complementary_angle(1,CAB,BAD)", "triangle_property_angle_sum(1,DBA)", "triangle_property_angle_sum(1,DBC)", "sine_theorem(1,DBA)"]} | |
257 | XiaokaiZhang_2023-03-12 | Geometry3k-260 | 3 | 如图所示，∠BDF=47°，∠CIF=112°，∠DFB=65°。求∠IAF的大小。 | As shown in the diagram, ∠BDF=47°, ∠CIF=112°, ∠DFB=65°. Find the measure of ∠IAF. | 257.png | [
"Shape(BD,DF,FB)",
"Shape(FI,IA,AF)",
"Shape(DB,BG)",
"Shape(BF,FA)",
"Shape(CI,IF)",
"Collinear(GBFI)",
"Collinear(AFD)",
"Collinear(AIC)"
] | [
"Equal(MeasureOfAngle(BDF),47)",
"Equal(MeasureOfAngle(CIF),112)",
"Equal(MeasureOfAngle(DFB),65)"
] | [
"Equal(MeasureOfAngle(BDF),47)",
"Equal(MeasureOfAngle(CIF),112)",
"Equal(MeasureOfAngle(DFB),65)"
] | Value(MeasureOfAngle(IAF)) | 47 | [
"vertical_angle(1,DFB,AFI)",
"adjacent_complementary_angle(1,CIF,FIA)",
"triangle_property_angle_sum(1,FIA)"
] | {"START": ["vertical_angle(1,DFB,AFI)", "adjacent_complementary_angle(1,CIF,FIA)", "triangle_property_angle_sum(1,FIA)"]} | |
258 | NaZhu_2023-04-02 | Geometry3k-261 | 2 | 如图所示，∠ACE=y°，∠BDA=68°，∠EAC=2*x°，∠EBD=3*x-15°，EA∥BD。求y的值。 | As shown in the diagram, ∠ACE=y°, ∠BDA=68°, ∠EAC=2*x°, ∠EBD=3*x-15°, EA∥BD. Find the value of y. | 258.png | [
"Shape(BD,DA,AE,EB)",
"Shape(EA,AC,CE)",
"Collinear(BEC)",
"Collinear(DAC)"
] | [
"Equal(MeasureOfAngle(ACE),y)",
"Equal(MeasureOfAngle(BDA),68)",
"Equal(MeasureOfAngle(EAC),2*x)",
"Equal(MeasureOfAngle(EBD),3*x-15)",
"ParallelBetweenLine(EA,BD)"
] | [
"Equal(MeasureOfAngle(ACE),y)",
"Equal(MeasureOfAngle(BDA),68)",
"Equal(MeasureOfAngle(EAC),2*x)",
"Equal(MeasureOfAngle(EBD),3*x-15)",
"ParallelBetweenLine(EA,BD)"
] | Value(y) | 25 | [
"parallel_property_corresponding_angle(2,DB,AE,C)",
"triangle_property_angle_sum(1,BDC)"
] | {"START": ["parallel_property_corresponding_angle(2,DB,AE,C)", "triangle_property_angle_sum(1,BDC)"]} | |
259 | XiaokaiZhang_2023-03-12 | Geometry3k-262 | 4 | 如图所示，AB=10，AE=25/4，BC=x+2，DE=x-1，三角形ABE相似于三角形ACD。求直线BC的长度。 | As shown in the diagram, AB=10, AE=25/4, BC=x+2, DE=x-1, △ABE is similar to △ACD.. Find the length of line BC. | 259.png | [
"Shape(AB,BE,EA)",
"Shape(BC,CD,DE,EB)",
"Collinear(ABC)",
"Collinear(AED)"
] | [
"Equal(LengthOfLine(AB),10)",
"Equal(LengthOfLine(AE),25/4)",
"Equal(LengthOfLine(BC),x+2)",
"Equal(LengthOfLine(DE),x-1)",
"SimilarBetweenTriangle(ABE,ACD)"
] | [
"Equal(LengthOfLine(AB),10)",
"Equal(LengthOfLine(AE),25/4)",
"Equal(LengthOfLine(BC),x+2)",
"Equal(LengthOfLine(DE),x-1)"
] | Value(LengthOfLine(BC)) | 8 | [
"similar_triangle_property_line_ratio(1,EAB,DAC)",
"similar_triangle_property_line_ratio(1,BEA,CDA)",
"line_addition(1,AB,BC)",
"line_addition(1,AE,ED)"
] | {"START": ["similar_triangle_property_line_ratio(1,EAB,DAC)", "similar_triangle_property_line_ratio(1,BEA,CDA)", "line_addition(1,AB,BC)", "line_addition(1,AE,ED)"]} | |
260 | NaZhu_2023-04-02 | Geometry3k-263 | 3 | 如图所示，AD=5+x，BD=x，BF=5，CF=5+x。求x的值。 | As shown in the diagram, AD=5+x, BD=x, BF=5, CF=5+x. Find the value of x. | 260.png | [
"Shape(EDF,DB,BF)",
"Shape(EAD,DA)",
"Shape(EDF,FC,ECA,AD)",
"Shape(EFC,CF)",
"Collinear(BDA)",
"Collinear(BFC)",
"Cocircular(E,DFCA)"
] | [
"Equal(LengthOfLine(AD),5+x)",
"Equal(LengthOfLine(BD),x)",
"Equal(LengthOfLine(BF),5)",
"Equal(LengthOfLine(CF),5+x)"
] | [
"Equal(LengthOfLine(AD),5+x)",
"Equal(LengthOfLine(BD),x)",
"Equal(LengthOfLine(BF),5)",
"Equal(LengthOfLine(CF),5+x)"
] | Value(x) | 5 | [
"line_addition(1,BD,DA)",
"line_addition(1,BF,FC)",
"circle_property_circular_power_segment_and_segment_line(1,BDA,BFC,E)"
] | {"START": ["line_addition(1,BD,DA)", "line_addition(1,BF,FC)", "circle_property_circular_power_segment_and_segment_line(1,BDA,BFC,E)"]} | |
261 | NaZhu_2023-04-02 | Geometry3k-264 | 2 | 如图所示，AG=12，圆A的圆心为A。求直线LA的长度。 | As shown in the diagram, AG=12, the center of ⊙A is A. Find the length of line LA. | 261.png | [
"Shape(APD,DA,AP)",
"Shape(ADL,LA,AD)",
"Shape(ALG,GA,AL)",
"Shape(AGF,FA,AG)",
"Shape(AFP,PA,AF)",
"Cocircular(A,PDLGF)"
] | [
"Equal(LengthOfLine(AG),12)",
"IsCentreOfCircle(A,A)"
] | [
"Equal(LengthOfLine(AG),12)",
"IsCentreOfCircle(A,A)"
] | Value(LengthOfLine(LA)) | 12 | [
"radius_of_circle_property_length_equal(1,AG,A)",
"radius_of_circle_property_length_equal(1,AL,A)"
] | {"START": ["radius_of_circle_property_length_equal(1,AG,A)", "radius_of_circle_property_length_equal(1,AL,A)"]} | |
262 | NaZhu_2023-04-02 | Geometry3k-265 | 6 | 如图所示，∠FJH=82°，GFJH是菱形。求∠JHK的大小。 | As shown in the diagram, ∠FJH=82°, GFJH is a rhombus. Find the measure of ∠JHK. | 262.png | [
"Shape(GF,FK,KG)",
"Shape(GK,KH,HG)",
"Shape(FJ,JK,KF)",
"Shape(KJ,JH,HK)",
"Collinear(FKH)",
"Collinear(GKJ)"
] | [
"Equal(MeasureOfAngle(FJH),82)",
"Rhombus(GFJH)"
] | [] | Value(MeasureOfAngle(JHK)) | 49 | [
"kite_property_diagonal_perpendicular_bisection(1,JHGF,K)",
"altitude_of_triangle_judgment(1,JK,JHF)",
"isosceles_triangle_judgment_line_equal(1,JHF)",
"isosceles_triangle_property_line_coincidence(1,JHF,K)",
"angle_addition(1,FJK,KJH)",
"triangle_property_angle_sum(1,JHK)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,JHGF,K)", "isosceles_triangle_judgment_line_equal(1,JHF)", "angle_addition(1,FJK,KJH)", "triangle_property_angle_sum(1,JHK)"], "altitude_of_triangle_judgment(1,JK,JHF)": ["isosceles_triangle_property_line_coincidence(1,JHF,K)"], "isosceles_triangle_judgment_l... | |
263 | XiaokaiZhang_2023-03-12 | Geometry3k-266 | 3 | 如图所示，AB=13，AC=7，CB=10，FE=14，三角形ACB镜像相似于三角形DFE。求三角形DFE的周长。 | As shown in the diagram, AB=13, AC=7, CB=10, FE=14, △ACB is mirror similar to △DFE.. Find the perimeter of △DFE. | 263.png | [
"Shape(AC,CB,BA)",
"Shape(DF,FE,ED)"
] | [
"Equal(LengthOfLine(AB),13)",
"Equal(LengthOfLine(AC),7)",
"Equal(LengthOfLine(CB),10)",
"Equal(LengthOfLine(FE),14)",
"MirrorSimilarBetweenTriangle(ACB,DFE)"
] | [
"Equal(LengthOfLine(AB),13)",
"Equal(LengthOfLine(AC),7)",
"Equal(LengthOfLine(CB),10)",
"Equal(LengthOfLine(FE),14)"
] | Value(PerimeterOfTriangle(DFE)) | 42 | [
"triangle_perimeter_formula(1,ACB)",
"mirror_similar_triangle_property_line_ratio(1,ACB,DFE)",
"mirror_similar_triangle_property_perimeter_ratio(1,ACB,DFE)"
] | {"START": ["triangle_perimeter_formula(1,ACB)", "mirror_similar_triangle_property_line_ratio(1,ACB,DFE)", "mirror_similar_triangle_property_perimeter_ratio(1,ACB,DFE)"]} | |
264 | NaZhu_2023-04-02 | Geometry3k-267 | 1 | 如图所示，∠CPD=3*x-15°，DP垂直于AP，ADCB是菱形。求x的值。 | As shown in the diagram, ∠CPD=3*x-15°, DP is perpendicular to AP, quadrilateral ADCB is a rhombus. Find the value of x. | 264.png | [
"Shape(AD,DP,PA)",
"Shape(AP,PB,BA)",
"Shape(PD,DC,CP)",
"Shape(PC,CB,BP)",
"Collinear(APC)",
"Collinear(DPB)"
] | [
"Equal(MeasureOfAngle(CPD),3*x-15)",
"PerpendicularBetweenLine(DP,AP)",
"Rhombus(ADCB)"
] | [
"PerpendicularBetweenLine(DP,AP)"
] | Value(x) | 35 | [
"kite_property_diagonal_perpendicular_bisection(1,CBAD,P)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,CBAD,P)"]} | |
265 | NaZhu_2023-04-02 | Geometry3k-268 | 13 | 如图所示，∠KFD=x°，⌒ECG的角度为45，⌒EDC的角度为130，圆E的圆心为E。求x的值。 | As shown in the diagram, ∠KFD=x°, the measure of ⌒ECG is 45, the measure of arc EDC is 130, E is the center of circle E. Find the value of x. | 265.png | [
"Shape(EDC,CD)",
"Shape(DC,CE,ED)",
"Shape(ECG,GE,EC)",
"Shape(EGK,KE,EG)",
"Shape(EKD,DE,EK)",
"Shape(EKD,KF,FD)",
"Collinear(GEKF)",
"Collinear(CDF)",
"Cocircular(E,CGKD)"
] | [
"Equal(MeasureOfAngle(KFD),x)",
"Equal(MeasureOfArc(ECG),45)",
"Equal(MeasureOfArc(EDC),130)",
"IsCentreOfCircle(E,E)"
] | [
"Equal(MeasureOfAngle(KFD),x)",
"Equal(MeasureOfArc(ECG),45)",
"Equal(MeasureOfArc(EDC),130)",
"IsCentreOfCircle(E,E)"
] | Value(x) | 20 | [
"arc_property_center_angle(1,EDC,E)",
"arc_property_center_angle(1,ECG,E)",
"radius_of_circle_property_length_equal(1,EC,E)",
"radius_of_circle_property_length_equal(1,ED,E)",
"isosceles_triangle_judgment_line_equal(1,EDC)",
"isosceles_triangle_property_angle_equal(1,EDC)",
"triangle_property_angle_sum(... | {"START": ["arc_property_center_angle(1,EDC,E)", "arc_property_center_angle(1,ECG,E)", "radius_of_circle_property_length_equal(1,EC,E)", "radius_of_circle_property_length_equal(1,ED,E)", "triangle_property_angle_sum(1,DCE)", "flat_angle(1,FDC)", "angle_addition(1,FDE,EDC)", "flat_angle(1,GEK)", "angle_addition(1,GEC,CE... | |
266 | XiaokaiZhang_2023-03-12 | Geometry3k-269 | 1 | 如图所示，AB=c，AC=b，CB=a，a=14，b=48，c=50，BC⊥AC。求cos(AB)的值。 | As shown in the diagram, AB=c, AC=b, CB=a, a=14, b=48, c=50, BC⊥AC. Find the value of cos(AB). | 266.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(AC),b)",
"Equal(LengthOfLine(CB),a)",
"Equal(a,14)",
"Equal(b,48)",
"Equal(c,50)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(AC),b)",
"Equal(LengthOfLine(CB),a)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(Cos(MeasureOfAngle(ABC))) | 7/25 | [
"cosine_theorem(1,BCA)"
] | {"START": ["cosine_theorem(1,BCA)"]} | |
267 | NaZhu_2023-04-02 | Geometry3k-270 | 3 | 如图所示，AF=x+4，BD=x，BF=8，DC=2*x。求x的值。 | As shown in the diagram, AF=x+4, BD=x, BF=8, DC=2*x. Find the value of x. | 267.png | [
"Shape(EAF,FA)",
"Shape(ECA,AF,EFD,DC)",
"Shape(EDC,CD)",
"Shape(EFD,FB,BD)",
"Collinear(CDB)",
"Collinear(AFB)",
"Cocircular(E,AFDC)"
] | [
"Equal(LengthOfLine(AF),x+4)",
"Equal(LengthOfLine(BD),x)",
"Equal(LengthOfLine(BF),8)",
"Equal(LengthOfLine(DC),2*x)"
] | [
"Equal(LengthOfLine(AF),x+4)",
"Equal(LengthOfLine(BD),x)",
"Equal(LengthOfLine(BF),8)",
"Equal(LengthOfLine(DC),2*x)"
] | Value(x) | 4/3+4*sqrt(19)/3 | [
"line_addition(1,CD,DB)",
"line_addition(1,AF,FB)",
"circle_property_circular_power_segment_and_segment_line(1,BFA,BDC,E)"
] | {"START": ["line_addition(1,CD,DB)", "line_addition(1,AF,FB)", "circle_property_circular_power_segment_and_segment_line(1,BFA,BDC,E)"]} | |
268 | NaZhu_2023-04-02 | Geometry3k-271 | 2 | 如图所示，∠BFD=165°，∠CFB=x°，∠DFC=145°。求x的值。 | As shown in the diagram, ∠BFD=165°, ∠CFB=x°, ∠DFC=145°. Find the value of x. | 268.png | [
"Shape(FDB,BF,FD)",
"Shape(FBC,CF,FB)",
"Shape(FCD,DF,FC)",
"Cocircular(F,DBC)"
] | [
"Equal(MeasureOfAngle(BFD),165)",
"Equal(MeasureOfAngle(CFB),x)",
"Equal(MeasureOfAngle(DFC),145)"
] | [
"Equal(MeasureOfAngle(BFD),165)",
"Equal(MeasureOfAngle(CFB),x)",
"Equal(MeasureOfAngle(DFC),145)"
] | Value(x) | 50 | [
"angle_addition(1,BFD,DFC)",
"round_angle(1,CFB,BFC)"
] | {"START": ["angle_addition(1,BFD,DFC)", "round_angle(1,CFB,BFC)"]} | |
269 | NaZhu_2023-04-02 | Geometry3k-272 | 8 | 如图所示，KL=AJ，弧CKL的角度为5*x，弧DJA的角度为3*x+54，⊙C的半径与⊙D的半径相等，C是圆C的圆心，⊙D的圆心为D。求x的值。 | As shown in the diagram, KL=AJ, the measure of arc CKL is 5*x, the measure of arc DJA is 3*x+54, the radius of ⊙C is equal to the radius of circle D, C is the center of circle C, D is the center of circle D. Find the value of x. | 269.png | [
"Shape(CKL,LK)",
"Shape(KL,LC,CK)",
"Shape(CLK,KC,CL)",
"Shape(DJA,AJ)",
"Shape(DJ,JA,AD)",
"Shape(DAJ,JD,DA)",
"Cocircular(C,KL)",
"Cocircular(D,JA)"
] | [
"Equal(LengthOfLine(KL),LengthOfLine(AJ))",
"Equal(MeasureOfArc(CKL),5*x)",
"Equal(MeasureOfArc(DJA),3*x+54)",
"Equal(RadiusOfCircle(C),RadiusOfCircle(D))",
"IsCentreOfCircle(C,C)",
"IsCentreOfCircle(D,D)"
] | [
"Equal(LengthOfLine(KL),LengthOfLine(AJ))",
"Equal(MeasureOfArc(CKL),5*x)",
"Equal(MeasureOfArc(DJA),3*x+54)",
"IsCentreOfCircle(C,C)",
"IsCentreOfCircle(D,D)"
] | Value(x) | 27 | [
"arc_property_center_angle(1,CKL,C)",
"arc_property_center_angle(1,DJA,D)",
"radius_of_circle_property_length_equal(1,CK,C)",
"radius_of_circle_property_length_equal(1,CL,C)",
"radius_of_circle_property_length_equal(1,DJ,D)",
"radius_of_circle_property_length_equal(1,DA,D)",
"congruent_triangle_judgment... | {"START": ["arc_property_center_angle(1,CKL,C)", "arc_property_center_angle(1,DJA,D)", "radius_of_circle_property_length_equal(1,CK,C)", "radius_of_circle_property_length_equal(1,CL,C)", "radius_of_circle_property_length_equal(1,DJ,D)", "radius_of_circle_property_length_equal(1,DA,D)"], "congruent_triangle_judgment_sss... | |
270 | XiaokaiZhang_2023-03-12 | Geometry3k-273 | 1 | 如图所示，AB=c，AC=b，CB=a，a=14，b=48，c=50，BC⊥AC。求tan(CA)的值。 | As shown in the diagram, AB=c, AC=b, CB=a, a=14, b=48, c=50, BC is perpendicular to AC. Find the value of tan(CA). | 270.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(AC),b)",
"Equal(LengthOfLine(CB),a)",
"Equal(a,14)",
"Equal(b,48)",
"Equal(c,50)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(AC),b)",
"Equal(LengthOfLine(CB),a)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(Tan(MeasureOfAngle(CAB))) | 7/24 | [
"cosine_theorem(1,ABC)"
] | {"START": ["cosine_theorem(1,ABC)"]} | |
271 | NaZhu_2023-04-02 | Geometry3k-274 | 3 | 如图所示，∠FEC=a°，∠GDA=b°，DH⊥EH。求a+b-90的值。 | As shown in the diagram, ∠FEC=a°, ∠GDA=b°, DH⊥EH. Find the value of a+b-90. | 271.png | [
"Shape(DH,HE,ED)",
"Collinear(ADH)",
"Collinear(HEF)",
"Collinear(GDEC)",
"Shape(GD,DA)",
"Shape(FE,EC)"
] | [
"Equal(MeasureOfAngle(FEC),a)",
"Equal(MeasureOfAngle(GDA),b)",
"PerpendicularBetweenLine(DH,EH)"
] | [
"Equal(MeasureOfAngle(FEC),a)",
"Equal(MeasureOfAngle(GDA),b)",
"PerpendicularBetweenLine(DH,EH)"
] | Value(a+b-90) | 0 | [
"vertical_angle(1,HEG,FEC)",
"vertical_angle(1,GDA,EDH)",
"triangle_property_angle_sum(1,DHE)"
] | {"START": ["vertical_angle(1,HEG,FEC)", "vertical_angle(1,GDA,EDH)", "triangle_property_angle_sum(1,DHE)"]} | |
272 | NaZhu_2023-04-02 | Geometry3k-275 | 3 | 如图所示，∠UZY=2*x+24°，∠VZU=4*x°，∠XZW=∠YZX，Z是⊙Z的圆心。求⌒ZYU的角度。 | As shown in the diagram, ∠UZY=2*x+24°, ∠VZU=4*x°, ∠XZW=∠YZX, Z is the center of circle Z. Find the measure of ⌒ZYU. | 272.png | [
"Shape(ZVW,WZ,ZV)",
"Shape(ZWX,XZ,ZW)",
"Shape(ZXY,YZ,ZX)",
"Shape(ZYU,UZ,ZY)",
"Shape(ZUV,VZ,ZU)",
"Collinear(VZY)",
"Collinear(WZU)",
"Cocircular(Z,VWXYU)"
] | [
"Equal(MeasureOfAngle(UZY),2*x+24)",
"Equal(MeasureOfAngle(VZU),4*x)",
"Equal(MeasureOfAngle(XZW),MeasureOfAngle(YZX))",
"IsCentreOfCircle(Z,Z)"
] | [
"IsCentreOfCircle(Z,Z)"
] | Value(MeasureOfArc(ZYU)) | 76 | [
"flat_angle(1,VZY)",
"angle_addition(1,VZU,UZY)",
"arc_property_center_angle(1,ZYU,Z)"
] | {"START": ["flat_angle(1,VZY)", "angle_addition(1,VZU,UZY)", "arc_property_center_angle(1,ZYU,Z)"]} | |
273 | NaZhu_2023-04-02 | Geometry3k-276 | 2 | 如图所示，∠ABG=47°，∠ACG=136°，∠BED=63°，∠DFB=∠BDF，∠EBA=69°。求∠BCA的大小。 | As shown in the diagram, ∠ABG=47°, ∠ACG=136°, ∠BED=63°, ∠DFB=∠BDF, ∠EBA=69°. Find the measure of ∠BCA. | 273.png | [
"Shape(ED,DB,BE)",
"Shape(BD,DF,FB)",
"Shape(AB,BC,CA)",
"Collinear(DBCG)",
"Collinear(EBF)",
"Shape(EB,BA)",
"Shape(AC,CG)"
] | [
"Equal(MeasureOfAngle(ABG),47)",
"Equal(MeasureOfAngle(ACG),136)",
"Equal(MeasureOfAngle(BED),63)",
"Equal(MeasureOfAngle(DFB),MeasureOfAngle(BDF))",
"Equal(MeasureOfAngle(EBA),69)"
] | [
"Equal(MeasureOfAngle(ABG),47)",
"Equal(MeasureOfAngle(ACG),136)",
"Equal(MeasureOfAngle(BED),63)",
"Equal(MeasureOfAngle(DFB),MeasureOfAngle(BDF))",
"Equal(MeasureOfAngle(EBA),69)"
] | Value(MeasureOfAngle(BCA)) | 44 | [
"flat_angle(1,BCG)",
"angle_addition(1,BCA,ACG)"
] | {"START": ["flat_angle(1,BCG)", "angle_addition(1,BCA,ACG)"]} | |
274 | NaZhu_2023-04-02 | Geometry3k-277 | 2 | 如图所示，∠ADC=60°，D是⊙D的圆心。求∠ABC的大小。 | As shown in the diagram, ∠ADC=60°, D is the center of circle D. Find the measure of ∠ABC. | 274.png | [
"Shape(DAB,BA)",
"Shape(DBC,CB)",
"Shape(DCA,AD,DC)",
"Shape(DA,AB,BC,CD)",
"Cocircular(D,ABC)"
] | [
"Equal(MeasureOfAngle(ADC),60)",
"IsCentreOfCircle(D,D)"
] | [
"Equal(MeasureOfAngle(ADC),60)",
"IsCentreOfCircle(D,D)"
] | Value(MeasureOfAngle(ABC)) | 30 | [
"arc_property_center_angle(1,DCA,D)",
"arc_property_circumference_angle_external(1,DCA,B)"
] | {"START": ["arc_property_center_angle(1,DCA,D)", "arc_property_circumference_angle_external(1,DCA,B)"]} | |
275 | XiaokaiZhang_2023-03-12 | Geometry3k-278 | 4 | 如图所示，DC=6-x，DG=2，JL=4，JM=x，DG是△DDC的高，JM是三角形JJL的高，DG⊥CG，JM垂直于LM，△KLJ相似于△ECD。求直线DC的长度。 | As shown in the diagram, DC=6-x, DG=2, JL=4, JM=x, DG is the altitude of triangle DDC, JM is the altitude of triangle JJL, DG is perpendicular to CG, JM is perpendicular to LM, triangle KLJ is similar to triangle ECD.. Find the length of line DC. | 275.png | [
"Shape(JK,KM,MJ)",
"Shape(JM,ML,LJ)",
"Shape(DE,EG,GD)",
"Shape(DG,GC,CD)",
"Collinear(KML)",
"Collinear(EGC)"
] | [
"Equal(LengthOfLine(DC),6-x)",
"Equal(LengthOfLine(DG),2)",
"Equal(LengthOfLine(JL),4)",
"Equal(LengthOfLine(JM),x)",
"IsAltitudeOfTriangle(DG,EDC)",
"IsAltitudeOfTriangle(JM,KJL)",
"PerpendicularBetweenLine(DG,CG)",
"PerpendicularBetweenLine(JM,LM)",
"SimilarBetweenTriangle(KLJ,ECD)"
] | [
"Equal(LengthOfLine(DC),6-x)",
"Equal(LengthOfLine(DG),2)",
"Equal(LengthOfLine(JL),4)",
"Equal(LengthOfLine(JM),x)",
"PerpendicularBetweenLine(DG,CG)",
"PerpendicularBetweenLine(JM,LM)"
] | Value(LengthOfLine(DC)) | 2 | [
"similar_triangle_property_angle_equal(1,LJK,CDE)",
"similar_triangle_judgment_aa(1,JML,DGC)",
"similar_triangle_property_line_ratio(1,LJM,CDG)",
"similar_triangle_property_line_ratio(1,MLJ,GCD)"
] | {"START": ["similar_triangle_property_angle_equal(1,LJK,CDE)"], "similar_triangle_judgment_aa(1,JML,DGC)": ["similar_triangle_property_line_ratio(1,MLJ,GCD)", "similar_triangle_property_line_ratio(1,LJM,CDG)"], "similar_triangle_property_angle_equal(1,LJK,CDE)": ["similar_triangle_judgment_aa(1,JML,DGC)"]} | |
276 | XiaokaiZhang_2023-03-12 | Geometry3k-280 | 3 | 如图所示，BA=3，BD=x-1，CE=x+2，EF=8，∠GAB=∠EFG，AB垂直于CB，DE⊥FE。求直线EC的长度。 | As shown in the diagram, BA=3, BD=x-1, CE=x+2, EF=8, ∠GAB=∠EFG, AB⊥CB, DE is perpendicular to FE. Find the length of line EC. | 276.png | [
"Shape(AB,BC,CG,GA)",
"Shape(GC,CD,DG)",
"Shape(GD,DE,EF,FG)",
"Collinear(BCDE)",
"Collinear(AGD)",
"Collinear(CGF)"
] | [
"Equal(LengthOfLine(BA),3)",
"Equal(LengthOfLine(BD),x-1)",
"Equal(LengthOfLine(CE),x+2)",
"Equal(LengthOfLine(EF),8)",
"Equal(MeasureOfAngle(GAB),MeasureOfAngle(EFG))",
"PerpendicularBetweenLine(AB,CB)",
"PerpendicularBetweenLine(DE,FE)"
] | [
"Equal(LengthOfLine(BA),3)",
"Equal(LengthOfLine(BD),x-1)",
"Equal(LengthOfLine(CE),x+2)",
"Equal(LengthOfLine(EF),8)",
"Equal(MeasureOfAngle(GAB),MeasureOfAngle(EFG))",
"PerpendicularBetweenLine(AB,CB)",
"PerpendicularBetweenLine(DE,FE)"
] | Value(LengthOfLine(EC)) | 24/5 | [
"mirror_similar_triangle_judgment_aa(1,DAB,CEF)",
"mirror_similar_triangle_property_line_ratio(1,DAB,CEF)",
"mirror_similar_triangle_property_line_ratio(1,ABD,FCE)"
] | {"START": ["mirror_similar_triangle_judgment_aa(1,DAB,CEF)"], "mirror_similar_triangle_judgment_aa(1,DAB,CEF)": ["mirror_similar_triangle_property_line_ratio(1,DAB,CEF)", "mirror_similar_triangle_property_line_ratio(1,ABD,FCE)"]} | |
277 | NaZhu_2023-03-12 | Geometry3k-281 | 2 | 如图所示，AC=5，BC=3，AC⊥BC。求直线AB的长度。 | As shown in the diagram, AC=5, BC=3, AC is perpendicular to BC. Find the length of line AB. | 277.png | [
"Shape(AC,CB,BA)"
] | [
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BC),3)",
"PerpendicularBetweenLine(AC,BC)"
] | [
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BC),3)",
"PerpendicularBetweenLine(AC,BC)"
] | Value(LengthOfLine(AB)) | sqrt(34) | [
"right_triangle_judgment_angle(1,ACB)",
"right_triangle_property_pythagorean(1,ACB)"
] | {"START": ["right_triangle_judgment_angle(1,ACB)"], "right_triangle_judgment_angle(1,ACB)": ["right_triangle_property_pythagorean(1,ACB)"]} | |
278 | NaZhu_2023-03-12 | Geometry3k-282 | 2 | 如图所示，△CEA的面积为52，CD=b+5，EA=b，ED垂直于CD。求b的值。 | As shown in the diagram, the area of △CEA is 52, CD=b+5, EA=b, ED⊥CD. Find the value of b. | 278.png | [
"Shape(CE,ED,DC)",
"Shape(CD,DA,AC)",
"Collinear(EDA)"
] | [
"Equal(AreaOfTriangle(CEA),52)",
"Equal(LengthOfLine(CD),b+5)",
"Equal(LengthOfLine(EA),b)",
"PerpendicularBetweenLine(ED,CD)"
] | [
"Equal(LengthOfLine(CD),b+5)",
"Equal(LengthOfLine(EA),b)",
"PerpendicularBetweenLine(ED,CD)"
] | Value(b) | 8 | [
"altitude_of_triangle_judgment(1,CD,CEA)",
"triangle_area_formula_common(1,CEA)"
] | {"START": ["altitude_of_triangle_judgment(1,CD,CEA)", "triangle_area_formula_common(1,CEA)"]} | |
279 | NaZhu_2023-04-02 | Geometry3k-283 | 6 | 如图所示，AI=18，FH=37，GB=9，GB垂直于HB，IA⊥FA。求三角形IHF的面积与△FHG的面积之和。 | As shown in the diagram, AI=18, FH=37, GB=9, GB is perpendicular to HB, IA is perpendicular to FA. Find the sum of the area of △IHF and the area of △FHG. | 279.png | [
"Shape(GF,FA,AB,BG)",
"Shape(FI,IA,AF)",
"Shape(AI,IH,HB,BA)",
"Shape(GB,BH,HG)",
"Collinear(FABH)"
] | [
"Equal(LengthOfLine(AI),18)",
"Equal(LengthOfLine(FH),37)",
"Equal(LengthOfLine(GB),9)",
"PerpendicularBetweenLine(GB,HB)",
"PerpendicularBetweenLine(IA,FA)"
] | [
"Equal(LengthOfLine(AI),18)",
"Equal(LengthOfLine(FH),37)",
"Equal(LengthOfLine(GB),9)",
"PerpendicularBetweenLine(GB,HB)",
"PerpendicularBetweenLine(IA,FA)"
] | Value(Add(AreaOfTriangle(IHF),AreaOfTriangle(FHG))) | 999/2 | [
"adjacent_complementary_angle(1,BAI,IAF)",
"adjacent_complementary_angle(1,FBG,GBH)",
"altitude_of_triangle_judgment(1,GB,GFH)",
"altitude_of_triangle_judgment(1,IA,IHF)",
"triangle_area_formula_common(1,GFH)",
"triangle_area_formula_common(1,IHF)"
] | {"START": ["adjacent_complementary_angle(1,BAI,IAF)", "adjacent_complementary_angle(1,FBG,GBH)", "triangle_area_formula_common(1,GFH)", "triangle_area_formula_common(1,IHF)"], "adjacent_complementary_angle(1,BAI,IAF)": ["altitude_of_triangle_judgment(1,IA,IHF)"], "adjacent_complementary_angle(1,FBG,GBH)": ["altitude_of... | |
280 | NaZhu_2023-03-12 | Geometry3k-284 | 2 | 如图所示，AC=9，AY=18，PO=x，QP=y，RO=14，YB=21，四边形RQPO相似于四边形BCAY。求y的值。 | As shown in the diagram, AC=9, AY=18, PO=x, QP=y, RO=14, YB=21, quadrilateral RQPO is similar to quadrilateral BCAY. Find the value of y. | 280.png | [
"Shape(RQ,QP,PO,OR)",
"Shape(AY,YB,BC,CA)"
] | [
"Equal(LengthOfLine(AC),9)",
"Equal(LengthOfLine(AY),18)",
"Equal(LengthOfLine(PO),x)",
"Equal(LengthOfLine(QP),y)",
"Equal(LengthOfLine(RO),14)",
"Equal(LengthOfLine(YB),21)",
"SimilarBetweenQuadrilateral(RQPO,BCAY)"
] | [
"Equal(LengthOfLine(AC),9)",
"Equal(LengthOfLine(AY),18)",
"Equal(LengthOfLine(PO),x)",
"Equal(LengthOfLine(QP),y)",
"Equal(LengthOfLine(RO),14)",
"Equal(LengthOfLine(YB),21)"
] | Value(y) | 6 | [
"similar_quadrilateral_property_line_ratio(1,QPOR,CAYB)",
"similar_quadrilateral_property_line_ratio(1,ORQP,YBCA)"
] | {"START": ["similar_quadrilateral_property_line_ratio(1,QPOR,CAYB)", "similar_quadrilateral_property_line_ratio(1,ORQP,YBCA)"]} | |
281 | NaZhu_2023-03-12 | Geometry3k-285 | 4 | 如图所示，RT=2*x+6，VT=10，WR=x+6，WS=8，∠SWR=∠VTR。求直线RT的长度。 | As shown in the diagram, RT=2*x+6, VT=10, WR=x+6, WS=8, ∠SWR=∠VTR. Find the length of line RT. | 281.png | [
"Shape(SW,WR,RS)",
"Shape(VT,TR,RV)",
"Collinear(SRV)",
"Collinear(WRT)"
] | [
"Equal(LengthOfLine(RT),2*x+6)",
"Equal(LengthOfLine(VT),10)",
"Equal(LengthOfLine(WR),x+6)",
"Equal(LengthOfLine(WS),8)",
"Equal(MeasureOfAngle(SWR),MeasureOfAngle(VTR))"
] | [
"Equal(LengthOfLine(RT),2*x+6)",
"Equal(LengthOfLine(VT),10)",
"Equal(LengthOfLine(WR),x+6)",
"Equal(LengthOfLine(WS),8)",
"Equal(MeasureOfAngle(SWR),MeasureOfAngle(VTR))"
] | Value(LengthOfLine(RT)) | 10 | [
"vertical_angle(1,WRS,TRV)",
"similar_triangle_judgment_aa(1,SWR,VTR)",
"similar_triangle_property_line_ratio(1,SWR,VTR)",
"similar_triangle_property_line_ratio(1,RSW,RVT)"
] | {"START": ["vertical_angle(1,WRS,TRV)"], "similar_triangle_judgment_aa(1,SWR,VTR)": ["similar_triangle_property_line_ratio(1,SWR,VTR)", "similar_triangle_property_line_ratio(1,RSW,RVT)"], "vertical_angle(1,WRS,TRV)": ["similar_triangle_judgment_aa(1,SWR,VTR)"]} | |
282 | NaZhu_2023-04-02 | Geometry3k-286 | 3 | 如图所示，∠EFA=63°，F是圆F的圆心，DF⊥EF。求⌒FDA的角度。 | As shown in the diagram, ∠EFA=63°, the center of circle F is F, DF is perpendicular to EF. Find the measure of arc FDA. | 282.png | [
"Shape(FCB,BF,FC)",
"Shape(FBA,AF,FB)",
"Shape(FAE,EF,FA)",
"Shape(FED,DF,FE)",
"Shape(FDC,CF,FD)",
"Cocircular(F,AEDCB)"
] | [
"Equal(MeasureOfAngle(EFA),63)",
"IsCentreOfCircle(F,F)",
"PerpendicularBetweenLine(DF,EF)"
] | [
"Equal(MeasureOfAngle(EFA),63)",
"IsCentreOfCircle(F,F)",
"PerpendicularBetweenLine(DF,EF)"
] | Value(MeasureOfArc(FDA)) | 207 | [
"angle_addition(1,DFE,EFA)",
"round_angle(1,DFA,AFD)",
"arc_property_center_angle(1,FDA,F)"
] | {"START": ["angle_addition(1,DFE,EFA)", "round_angle(1,DFA,AFD)", "arc_property_center_angle(1,FDA,F)"]} | |
283 | NaZhu_2023-03-12 | Geometry3k-287 | 2 | 如图所示，AB=y，BC=x，CA=7*sqrt(2)，∠CAB=45°，BC⊥AC。求y的值。 | As shown in the diagram, AB=y, BC=x, CA=7*sqrt(2), ∠CAB=45°, BC is perpendicular to AC. Find the value of y. | 283.png | [
"Shape(AB,BC,CA)"
] | [
"Equal(LengthOfLine(AB),y)",
"Equal(LengthOfLine(BC),x)",
"Equal(LengthOfLine(CA),7*sqrt(2))",
"Equal(MeasureOfAngle(CAB),45)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),y)",
"Equal(LengthOfLine(BC),x)",
"Equal(LengthOfLine(CA),7*sqrt(2))",
"Equal(MeasureOfAngle(CAB),45)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(y) | 14 | [
"triangle_property_angle_sum(1,BCA)",
"sine_theorem(1,ABC)"
] | {"START": ["triangle_property_angle_sum(1,BCA)", "sine_theorem(1,ABC)"]} | |
284 | NaZhu_2023-04-02 | Geometry3k-288 | 1 | 如图所示，∠FBE=4*x°，弧ACD的角度为9*x+26，⌒AFE的角度为35。求x的值。 | As shown in the diagram, ∠FBE=4*x°, the measure of ⌒ACD is 9*x+26, the measure of arc AFE is 35. Find the value of x. | 284.png | [
"Shape(AEC,CE)",
"Shape(ACD,DF,AFE,EC)",
"Shape(ADF,FD)",
"Shape(BE,AFE,FB)",
"Collinear(CEB)",
"Collinear(DFB)",
"Cocircular(A,ECDF)"
] | [
"Equal(MeasureOfAngle(FBE),4*x)",
"Equal(MeasureOfArc(ACD),9*x+26)",
"Equal(MeasureOfArc(AFE),35)"
] | [
"Equal(MeasureOfAngle(FBE),4*x)",
"Equal(MeasureOfArc(ACD),9*x+26)",
"Equal(MeasureOfArc(AFE),35)"
] | Value(x) | 9 | [
"circle_property_circular_power_segment_and_segment_angle(1,BFD,BEC,A)"
] | {"START": ["circle_property_circular_power_segment_and_segment_angle(1,BFD,BEC,A)"]} | |
285 | NaZhu_2023-04-02 | Geometry3k-289 | 4 | 如图所示，BD=22，DN=18，∠NAD=40°，ACBD是▱，DN垂直于AN。求ACBD的面积。 | As shown in the diagram, BD=22, DN=18, ∠NAD=40°, AD and CB are opposite sides of the parallelogram ACBD, DN⊥AN. Find the area of ACBD. | 285.png | [
"Shape(CB,BD,DA,AC)",
"Shape(AD,DN,NA)",
"Collinear(BDN)"
] | [
"Equal(LengthOfLine(BD),22)",
"Equal(LengthOfLine(DN),18)",
"Equal(MeasureOfAngle(NAD),40)",
"Parallelogram(ACBD)",
"PerpendicularBetweenLine(DN,AN)"
] | [
"Equal(LengthOfLine(BD),22)",
"Equal(LengthOfLine(DN),18)",
"Equal(MeasureOfAngle(NAD),40)",
"PerpendicularBetweenLine(DN,AN)"
] | Value(AreaOfQuadrilateral(ACBD)) | 396/tan(2*pi/9) | [
"triangle_property_angle_sum(1,ADN)",
"sine_theorem(1,NAD)",
"altitude_of_quadrilateral_judgment_right_vertex(5,AN,CBDA)",
"parallelogram_area_formula_common(1,CBDA)"
] | {"START": ["triangle_property_angle_sum(1,ADN)", "sine_theorem(1,NAD)", "altitude_of_quadrilateral_judgment_right_vertex(5,AN,CBDA)", "parallelogram_area_formula_common(1,CBDA)"]} | |
286 | NaZhu_2023-04-02 | Geometry3k-290 | 4 | 如图所示，AC=8，AD=4，CB=10，AC⊥BC，DA垂直于CA。求四边形ACBD的面积。 | As shown in the diagram, AC=8, AD=4, CB=10, AC⊥BC, DA is perpendicular to CA. Find the area of ACBD. | 286.png | [
"Shape(BD,DA,AC,CB)"
] | [
"Equal(LengthOfLine(AC),8)",
"Equal(LengthOfLine(AD),4)",
"Equal(LengthOfLine(CB),10)",
"PerpendicularBetweenLine(AC,BC)",
"PerpendicularBetweenLine(DA,CA)"
] | [
"Equal(LengthOfLine(AC),8)",
"Equal(LengthOfLine(AD),4)",
"Equal(LengthOfLine(CB),10)",
"PerpendicularBetweenLine(AC,BC)",
"PerpendicularBetweenLine(DA,CA)"
] | Value(AreaOfQuadrilateral(ACBD)) | 56 | [
"parallel_judgment_ipsilateral_internal_angle(1,AD,CB)",
"trapezoid_judgment_parallel(1,ACBD)",
"right_trapezoid_judgment_right_angle(1,ACBD)",
"right_trapezoid_area_formular(1,ACBD)"
] | {"START": ["parallel_judgment_ipsilateral_internal_angle(1,AD,CB)"], "parallel_judgment_ipsilateral_internal_angle(1,AD,CB)": ["trapezoid_judgment_parallel(1,ACBD)"], "right_trapezoid_judgment_right_angle(1,ACBD)": ["right_trapezoid_area_formular(1,ACBD)"], "trapezoid_judgment_parallel(1,ACBD)": ["right_trapezoid_judgm... | |
287 | NaZhu_2023-03-12 | Geometry3k-291 | 1 | 如图所示，AB=13，AC=12，BC=15。求∠BAC的大小。 | As shown in the diagram, AB=13, AC=12, BC=15. Find the measure of ∠BAC. | 287.png | [
"Shape(AC,CB,BA)"
] | [
"Equal(LengthOfLine(AB),13)",
"Equal(LengthOfLine(AC),12)",
"Equal(LengthOfLine(BC),15)"
] | [
"Equal(LengthOfLine(AB),13)",
"Equal(LengthOfLine(AC),12)",
"Equal(LengthOfLine(BC),15)"
] | Value(MeasureOfAngle(BAC)) | 180*acos(11/39)/pi | [
"cosine_theorem(1,ACB)"
] | {"START": ["cosine_theorem(1,ACB)"]} | |
288 | NaZhu_2023-04-02 | Geometry3k-292 | 3 | 如图所示，BE=15，CB=12，CE=x，E是⊙E的圆心，BC是圆O的切线。求x的值。 | As shown in the diagram, BE=15, CB=12, CE=x, the center of ⊙E is E, BC is the tangent to ⊙E. Find the value of x. | 288.png | [
"Shape(ECF,FE,EC)",
"Shape(EFC,CE,EF)",
"Shape(ECF,CB,BF)",
"Collinear(BFE)",
"Cocircular(E,CF)"
] | [
"Equal(LengthOfLine(BE),15)",
"Equal(LengthOfLine(CB),12)",
"Equal(LengthOfLine(CE),x)",
"IsCentreOfCircle(E,E)",
"IsTangentOfCircle(BC,E)"
] | [
"Equal(LengthOfLine(BE),15)",
"Equal(LengthOfLine(CB),12)",
"Equal(LengthOfLine(CE),x)",
"IsCentreOfCircle(E,E)"
] | Value(x) | 9 | [
"tangent_of_circle_property_perpendicular(1,BC,E,E)",
"right_triangle_judgment_angle(1,ECB)",
"right_triangle_property_pythagorean(1,ECB)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,BC,E,E)"], "right_triangle_judgment_angle(1,ECB)": ["right_triangle_property_pythagorean(1,ECB)"], "tangent_of_circle_property_perpendicular(1,BC,E,E)": ["right_triangle_judgment_angle(1,ECB)"]} | |
289 | NaZhu_2023-04-02 | Geometry3k-293 | 4 | 如图所示，∠BGC=100°，∠DCF=75°。求∠GAC的大小。 | As shown in the diagram, ∠BGC=100°, ∠DCF=75°. Find the measure of ∠GAC. | 289.png | [
"Shape(GA,AC,CG)",
"Collinear(EGCF)",
"Collinear(AGB)",
"Collinear(ACD)",
"Shape(BG,GC)",
"Shape(DC,CF)"
] | [
"Equal(MeasureOfAngle(BGC),100)",
"Equal(MeasureOfAngle(DCF),75)"
] | [
"Equal(MeasureOfAngle(BGC),100)",
"Equal(MeasureOfAngle(DCF),75)"
] | Value(MeasureOfAngle(GAC)) | 25 | [
"flat_angle(1,BGA)",
"angle_addition(1,BGC,CGA)",
"vertical_angle(1,ACE,DCF)",
"triangle_property_angle_sum(1,GAC)"
] | {"START": ["flat_angle(1,BGA)", "angle_addition(1,BGC,CGA)", "vertical_angle(1,ACE,DCF)", "triangle_property_angle_sum(1,GAC)"]} | |
290 | NaZhu_2023-04-02 | Geometry3k-294 | 2 | 如图所示，∠EFA=63°，F是⊙F的圆心，DF垂直于EF。求⌒FAD的角度。 | As shown in the diagram, ∠EFA=63°, F is the center of circle F, DF⊥EF. Find the measure of arc FAD. | 290.png | [
"Shape(FCB,BF,FC)",
"Shape(FBA,AF,FB)",
"Shape(FAE,EF,FA)",
"Shape(FED,DF,FE)",
"Shape(FDC,CF,FD)",
"Cocircular(F,CBAED)"
] | [
"Equal(MeasureOfAngle(EFA),63)",
"IsCentreOfCircle(F,F)",
"PerpendicularBetweenLine(DF,EF)"
] | [
"Equal(MeasureOfAngle(EFA),63)",
"IsCentreOfCircle(F,F)",
"PerpendicularBetweenLine(DF,EF)"
] | Value(MeasureOfArc(FAD)) | 153 | [
"angle_addition(1,DFE,EFA)",
"arc_property_center_angle(1,FAD,F)"
] | {"START": ["angle_addition(1,DFE,EFA)", "arc_property_center_angle(1,FAD,F)"]} | |
291 | NaZhu_2023-03-12 | Geometry3k-295 | 7 | 如图所示，AE=2*x+1，CD=DB，EB=3*x-5，CA垂直于EA，DE垂直于BE。求x的值。 | As shown in the diagram, AE=2*x+1, CD=DB, EB=3*x-5, CA⊥EA, DE is perpendicular to BE. Find the value of x. | 291.png | [
"Shape(BD,DE,EB)",
"Shape(DC,CA,AE,ED)",
"Collinear(CDB)",
"Collinear(AEB)"
] | [
"Equal(LengthOfLine(AE),2*x+1)",
"Equal(LengthOfLine(CD),LengthOfLine(DB))",
"Equal(LengthOfLine(EB),3*x-5)",
"PerpendicularBetweenLine(CA,EA)",
"PerpendicularBetweenLine(DE,BE)"
] | [
"Equal(LengthOfLine(AE),2*x+1)",
"Equal(LengthOfLine(CD),LengthOfLine(DB))",
"Equal(LengthOfLine(EB),3*x-5)",
"PerpendicularBetweenLine(CA,EA)",
"PerpendicularBetweenLine(DE,BE)"
] | Value(x) | 6 | [
"parallel_judgment_corresponding_angle(2,AC,ED,B)",
"parallel_property_corresponding_angle(1,DE,CA,B)",
"similar_triangle_judgment_aa(1,EBD,ABC)",
"line_addition(1,BD,DC)",
"line_addition(1,BE,EA)",
"similar_triangle_property_line_ratio(1,EBD,ABC)",
"similar_triangle_property_line_ratio(1,DEB,CAB)"
] | {"START": ["parallel_judgment_corresponding_angle(2,AC,ED,B)", "line_addition(1,BD,DC)", "line_addition(1,BE,EA)"], "parallel_judgment_corresponding_angle(2,AC,ED,B)": ["parallel_property_corresponding_angle(1,DE,CA,B)"], "parallel_property_corresponding_angle(1,DE,CA,B)": ["similar_triangle_judgment_aa(1,EBD,ABC)"], "... | |
292 | NaZhu_2023-03-12 | Geometry3k-296 | 4 | 如图所示，AB=BY，AB=x，AY=8，∠YAB=y°，AB⊥YB。求x的值。 | As shown in the diagram, AB=BY, AB=x, AY=8, ∠YAB=y°, AB⊥YB. Find the value of x. | 292.png | [
"Shape(AB,BY,YA)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(BY))",
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AY),8)",
"Equal(MeasureOfAngle(YAB),y)",
"PerpendicularBetweenLine(AB,YB)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(BY))",
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AY),8)",
"Equal(MeasureOfAngle(YAB),y)",
"PerpendicularBetweenLine(AB,YB)"
] | Value(x) | 4*sqrt(2) | [
"isosceles_triangle_judgment_line_equal(1,BYA)",
"isosceles_triangle_property_angle_equal(1,BYA)",
"triangle_property_angle_sum(1,ABY)",
"sine_theorem(1,ABY)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,BYA)", "triangle_property_angle_sum(1,ABY)", "sine_theorem(1,ABY)"], "isosceles_triangle_judgment_line_equal(1,BYA)": ["isosceles_triangle_property_angle_equal(1,BYA)"]} | |
293 | NaZhu_2023-04-02 | Geometry3k-297 | 0 | 如图所示，JH=5，JH=y+2，LV=2*x+6，ML=20-5*x，ML=LV，MN=4，QO=3。求x的值。 | As shown in the diagram, JH=5, JH=y+2, LV=2*x+6, ML=20-5*x, ML=LV, MN=4, QO=3. Find the value of x. | 293.png | [
"Shape(ML,LQ,QB,BM)",
"Shape(QL,LV,VU,UQ)",
"Collinear(NMLVS)",
"Collinear(ABQUY)",
"Collinear(GMBP)",
"Collinear(CLQO)",
"Collinear(JVUH)"
] | [
"Equal(LengthOfLine(JH),5)",
"Equal(LengthOfLine(JH),y+2)",
"Equal(LengthOfLine(LV),2*x+6)",
"Equal(LengthOfLine(ML),20-5*x)",
"Equal(LengthOfLine(ML),LengthOfLine(LV))",
"Equal(LengthOfLine(MN),4)",
"Equal(LengthOfLine(QO),3)"
] | [
"Equal(LengthOfLine(JH),5)",
"Equal(LengthOfLine(JH),y+2)",
"Equal(LengthOfLine(LV),2*x+6)",
"Equal(LengthOfLine(ML),20-5*x)",
"Equal(LengthOfLine(ML),LengthOfLine(LV))",
"Equal(LengthOfLine(MN),4)",
"Equal(LengthOfLine(QO),3)"
] | Value(x) | 2 | [] | {"START": []} | |
294 | NaZhu_2023-04-02 | Geometry3k-298 | 0 | 如图所示，∠GJI=∠LON，∠GJI=y+30°，∠HGJ=87°，∠IHG=98°，∠IHG=∠NML，∠JIH=∠ONM，∠LON=60°，∠MLO=x-4°，HGJI与MLON相似。求y的值。 | As shown in the diagram, ∠GJI=∠LON, ∠GJI=y+30°, ∠HGJ=87°, ∠IHG=98°, ∠IHG=∠NML, ∠JIH=∠ONM, ∠LON=60°, ∠MLO=x-4°, HGJI is similar to MLON. Find the value of y. | 294.png | [
"Shape(IH,HG,GJ,JI)",
"Shape(NM,ML,LO,ON)"
] | [
"Equal(MeasureOfAngle(GJI),MeasureOfAngle(LON))",
"Equal(MeasureOfAngle(GJI),y+30)",
"Equal(MeasureOfAngle(HGJ),87)",
"Equal(MeasureOfAngle(IHG),98)",
"Equal(MeasureOfAngle(IHG),MeasureOfAngle(NML))",
"Equal(MeasureOfAngle(JIH),MeasureOfAngle(ONM))",
"Equal(MeasureOfAngle(LON),60)",
"Equal(MeasureOfAn... | [
"Equal(MeasureOfAngle(GJI),MeasureOfAngle(LON))",
"Equal(MeasureOfAngle(GJI),y+30)",
"Equal(MeasureOfAngle(HGJ),87)",
"Equal(MeasureOfAngle(IHG),98)",
"Equal(MeasureOfAngle(IHG),MeasureOfAngle(NML))",
"Equal(MeasureOfAngle(JIH),MeasureOfAngle(ONM))",
"Equal(MeasureOfAngle(LON),60)",
"Equal(MeasureOfAn... | Value(y) | 30 | [] | {"START": []} | |
295 | NaZhu_2023-04-02 | Geometry3k-299 | 4 | 如图所示，∠ACE=∠EAC，∠DBG=136°，∠DEB=47°，∠EFA=63°，∠FED=69°。求∠FAE的大小。 | As shown in the diagram, ∠ACE=∠EAC, ∠DBG=136°, ∠DEB=47°, ∠EFA=63°, ∠FED=69°. Find the measure of ∠FAE. | 295.png | [
"Shape(FA,AE,EF)",
"Shape(AC,CE,EA)",
"Shape(DE,EB,BD)",
"Collinear(AEBG)",
"Collinear(FEC)",
"Shape(FE,ED)",
"Shape(FE,EB)",
"Shape(DB,BG)"
] | [
"Equal(MeasureOfAngle(ACE),MeasureOfAngle(EAC))",
"Equal(MeasureOfAngle(DBG),136)",
"Equal(MeasureOfAngle(DEB),47)",
"Equal(MeasureOfAngle(EFA),63)",
"Equal(MeasureOfAngle(FED),69)"
] | [
"Equal(MeasureOfAngle(DBG),136)",
"Equal(MeasureOfAngle(DEB),47)",
"Equal(MeasureOfAngle(EFA),63)",
"Equal(MeasureOfAngle(FED),69)"
] | Value(MeasureOfAngle(FAE)) | 53 | [
"flat_angle(1,AEB)",
"angle_addition(1,FED,DEB)",
"angle_addition(1,AEF,FEB)",
"triangle_property_angle_sum(1,FAE)"
] | {"START": ["flat_angle(1,AEB)", "angle_addition(1,FED,DEB)", "angle_addition(1,AEF,FEB)", "triangle_property_angle_sum(1,FAE)"]} | |
296 | NaZhu_2023-04-02 | Geometry3k-300 | 6 | 如图所示，RC=x，ST=20，TC=12，R是⊙R的圆心，TS是⊙O的切线。求x的值。 | As shown in the diagram, RC=x, ST=20, TC=12, the center of ⊙R is R, TS is the tangent to ⊙R. Find the value of x. | 296.png | [
"Shape(RSC,CR,RS)",
"Shape(RCS,SR,RC)",
"Shape(RCS,CT,TS)",
"Collinear(RCT)",
"Cocircular(R,SC)"
] | [
"Equal(LengthOfLine(RC),x)",
"Equal(LengthOfLine(ST),20)",
"Equal(LengthOfLine(TC),12)",
"IsCentreOfCircle(R,R)",
"IsTangentOfCircle(TS,R)"
] | [
"Equal(LengthOfLine(RC),x)",
"Equal(LengthOfLine(ST),20)",
"Equal(LengthOfLine(TC),12)",
"IsCentreOfCircle(R,R)"
] | Value(x) | 32/3 | [
"tangent_of_circle_property_perpendicular(2,TS,R,R)",
"right_triangle_judgment_angle(1,TSR)",
"line_addition(1,RC,CT)",
"radius_of_circle_property_length_equal(1,RS,R)",
"radius_of_circle_property_length_equal(1,RC,R)",
"right_triangle_property_pythagorean(1,TSR)"
] | {"START": ["tangent_of_circle_property_perpendicular(2,TS,R,R)", "line_addition(1,RC,CT)", "radius_of_circle_property_length_equal(1,RS,R)", "radius_of_circle_property_length_equal(1,RC,R)"], "right_triangle_judgment_angle(1,TSR)": ["right_triangle_property_pythagorean(1,TSR)"], "tangent_of_circle_property_perpendicula... | |
297 | NaZhu_2023-03-12 | Geometry3k-301 | 1 | 如图所示，AB=x，AC=11，BC=9，∠BCA=28°。求x的值。 | As shown in the diagram, AB=x, AC=11, BC=9, ∠BCA=28°. Find the value of x. | 297.png | [
"Shape(AB,BC,CA)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),11)",
"Equal(LengthOfLine(BC),9)",
"Equal(MeasureOfAngle(BCA),28)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),11)",
"Equal(LengthOfLine(BC),9)",
"Equal(MeasureOfAngle(BCA),28)"
] | Value(x) | sqrt(202-198*cos(7*pi/45)) | [
"cosine_theorem(1,CAB)"
] | {"START": ["cosine_theorem(1,CAB)"]} | |
298 | NaZhu_2023-04-02 | Geometry3k-302 | 1 | 如图所示，AC=5*y，AY=2*x-5，CB=3*x-18，YB=2*y+12，AC和YB是平行四边形AYBC的一组对边。求y的值。 | As shown in the diagram, AC=5*y, AY=2*x-5, CB=3*x-18, YB=2*y+12, AC and YB are opposite sides of the parallelogram AYBC. Find the value of y. | 298.png | [
"Shape(AY,YB,BC,CA)"
] | [
"Equal(LengthOfLine(AC),5*y)",
"Equal(LengthOfLine(AY),2*x-5)",
"Equal(LengthOfLine(CB),3*x-18)",
"Equal(LengthOfLine(YB),2*y+12)",
"Parallelogram(AYBC)"
] | [
"Equal(LengthOfLine(AC),5*y)",
"Equal(LengthOfLine(AY),2*x-5)",
"Equal(LengthOfLine(CB),3*x-18)",
"Equal(LengthOfLine(YB),2*y+12)"
] | Value(y) | 4 | [
"parallelogram_property_opposite_line_equal(1,YBCA)"
] | {"START": ["parallelogram_property_opposite_line_equal(1,YBCA)"]} | |
299 | NaZhu_2023-04-02 | Geometry3k-303 | 3 | 如图所示，∠BAD=65°，∠DAE=110°，∠EAC=x°，CA⊥BA。求x的值。 | As shown in the diagram, ∠BAD=65°, ∠DAE=110°, ∠EAC=x°, CA is perpendicular to BA. Find the value of x. | 299.png | [
"Shape(ABC,CA,AB)",
"Shape(ADB,BA,AD)",
"Shape(ACE,EA,AC)",
"Shape(AED,DA,AE)",
"Cocircular(A,BCED)"
] | [
"Equal(MeasureOfAngle(BAD),65)",
"Equal(MeasureOfAngle(DAE),110)",
"Equal(MeasureOfAngle(EAC),x)",
"PerpendicularBetweenLine(CA,BA)"
] | [
"Equal(MeasureOfAngle(BAD),65)",
"Equal(MeasureOfAngle(DAE),110)",
"Equal(MeasureOfAngle(EAC),x)",
"PerpendicularBetweenLine(CA,BA)"
] | Value(x) | 95 | [
"angle_addition(1,CAB,BAD)",
"angle_addition(1,CAD,DAE)",
"round_angle(1,EAC,CAE)"
] | {"START": ["angle_addition(1,CAB,BAD)", "angle_addition(1,CAD,DAE)", "round_angle(1,EAC,CAE)"]} | |
300 | NaZhu_2023-03-12 | Geometry3k-304 | 6 | 如图所示，ML=3，PJ=x，HP是三角形MLK的中位线，JH是三角形LKM的中位线，PJ是三角形KML的中位线。求x的值。 | As shown in the diagram, ML=3, PJ=x, HP is the midsegment of △ MLK, JH is the midsegment of triangle LKM, PJ is the midsegment of △ KML. Find the value of x. | 300.png | [
"Shape(KP,PJ,JK)",
"Shape(JP,PH,HJ)",
"Shape(PM,MH,HP)",
"Shape(JH,HL,LJ)",
"Collinear(KPM)",
"Collinear(MHL)",
"Collinear(KJL)"
] | [
"Equal(LengthOfLine(ML),3)",
"Equal(LengthOfLine(PJ),x)",
"IsMidsegmentOfTriangle(HP,MLK)",
"IsMidsegmentOfTriangle(JH,LKM)",
"IsMidsegmentOfTriangle(PJ,KML)"
] | [
"Equal(LengthOfLine(ML),3)",
"Equal(LengthOfLine(PJ),x)"
] | Value(x) | 3/2 | [
"midsegment_of_triangle_property_parallel(1,PJ,KML)",
"parallel_property_corresponding_angle(1,PJ,ML,K)",
"similar_triangle_judgment_aa(1,JKP,LKM)",
"line_addition(1,KP,PM)",
"similar_triangle_property_line_ratio(1,KPJ,KML)",
"similar_triangle_property_line_ratio(1,JKP,LKM)"
] | {"START": ["midsegment_of_triangle_property_parallel(1,PJ,KML)", "line_addition(1,KP,PM)"], "midsegment_of_triangle_property_parallel(1,PJ,KML)": ["parallel_property_corresponding_angle(1,PJ,ML,K)"], "parallel_property_corresponding_angle(1,PJ,ML,K)": ["similar_triangle_judgment_aa(1,JKP,LKM)"], "similar_triangle_judgm... |
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