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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
301 | NaZhu_2023-03-12 | Geometry3k-305 | 9 | 如图所示，QP=24，RT=16，ST=9，PQ平行于TR。求直线PS的长度。 | As shown in the diagram, QP=24, RT=16, ST=9, PQ∥TR. Find the length of line PS. | 301.png | [
"Shape(RQ,QP,PT,TR)",
"Shape(SR,RT,TS)",
"Collinear(PTS)",
"Collinear(QRS)"
] | [
"Equal(LengthOfLine(QP),24)",
"Equal(LengthOfLine(RT),16)",
"Equal(LengthOfLine(ST),9)",
"ParallelBetweenLine(PQ,TR)"
] | [
"ParallelBetweenLine(PQ,TR)"
] | Value(LengthOfLine(PS)) | 27/2 | [
"parallel_property_ipsilateral_internal_angle(1,PQ,TR)",
"parallel_property_ipsilateral_internal_angle(1,RT,QP)",
"flat_angle(1,PTS)",
"flat_angle(1,SRQ)",
"angle_addition(1,PTR,RTS)",
"angle_addition(1,SRT,TRQ)",
"similar_triangle_judgment_aa(1,SRT,SQP)",
"similar_triangle_property_line_ratio(1,RTS,Q... | {"START": ["parallel_property_ipsilateral_internal_angle(1,PQ,TR)", "parallel_property_ipsilateral_internal_angle(1,RT,QP)", "flat_angle(1,PTS)", "flat_angle(1,SRQ)", "angle_addition(1,PTR,RTS)", "angle_addition(1,SRT,TRQ)"], "angle_addition(1,PTR,RTS)": ["similar_triangle_judgment_aa(1,SRT,SQP)"], "angle_addition(1,SR... | |
302 | NaZhu_2023-04-02 | Geometry3k-306 | 2 | 如图所示，AB=10，弧OBA的角度为60，⊙O的半径为10，O是⊙O的圆心，BX垂直于OX。求直线AX的长度。 | As shown in the diagram, AB=10, the measure of arc OBA is 60, the radius of ⊙O is 10, O is the center of ⊙O, BX⊥OX. Find the length of line AX. | 302.png | [
"Shape(OBY,YX,XB)",
"Shape(OYA,AX,XY)",
"Shape(OAB,BX,XA)",
"Shape(BX,XO)",
"Shape(OX,XA)",
"Collinear(AXB)",
"Collinear(YXO)",
"Cocircular(O,BYA)"
] | [
"Equal(LengthOfLine(AB),10)",
"Equal(MeasureOfArc(OBA),60)",
"Equal(RadiusOfCircle(O),10)",
"IsCentreOfCircle(O,O)",
"PerpendicularBetweenLine(BX,OX)"
] | [
"Equal(LengthOfLine(AB),10)",
"Equal(MeasureOfArc(OBA),60)",
"Equal(RadiusOfCircle(O),10)",
"IsCentreOfCircle(O,O)",
"PerpendicularBetweenLine(BX,OX)"
] | Value(LengthOfLine(AX)) | 5 | [
"circle_property_chord_perpendicular_bisect_chord(1,O,OX,BA)",
"line_addition(1,AX,XB)"
] | {"START": ["circle_property_chord_perpendicular_bisect_chord(1,O,OX,BA)", "line_addition(1,AX,XB)"]} | |
303 | NaZhu_2023-04-02 | Geometry3k-307 | 2 | 如图所示，PM=20，QR=6*x，TS=2*x，M平分线段RS，P平分线段QT，QT和SR是梯形QTSR的腰。求x的值。 | As shown in the diagram, PM=20, QR=6*x, TS=2*x, M bisects segment RS, P bisects segment QT, quadrilateral QTSR is a trapezoid. Find the value of x. | 303.png | [
"Shape(QP,PM,MR,RQ)",
"Shape(PT,TS,SM,MP)",
"Collinear(QPT)",
"Collinear(RMS)"
] | [
"Equal(LengthOfLine(PM),20)",
"Equal(LengthOfLine(QR),6*x)",
"Equal(LengthOfLine(TS),2*x)",
"IsMidpointOfLine(M,RS)",
"IsMidpointOfLine(P,QT)",
"Trapezoid(QTSR)"
] | [] | Value(x) | 5 | [
"midsegment_of_quadrilateral_judgment_midpoint(1,PM,QTSR)",
"midsegment_of_quadrilateral_property_length(1,PM,QTSR)"
] | {"START": ["midsegment_of_quadrilateral_judgment_midpoint(1,PM,QTSR)"], "midsegment_of_quadrilateral_judgment_midpoint(1,PM,QTSR)": ["midsegment_of_quadrilateral_property_length(1,PM,QTSR)"]} | |
304 | NaZhu_2023-04-02 | Geometry3k-308 | 6 | 如图所示，∠GCH=2*x°，∠HCD=6*x+28°，C是圆C的圆心，FC⊥GC。求⌒CFE的角度。 | As shown in the diagram, ∠GCH=2*x°, ∠HCD=6*x+28°, the center of circle C is C, FC⊥GC. Find the measure of ⌒CFE. | 304.png | [
"Shape(CHG,GC,CH)",
"Shape(CGF,FC,CG)",
"Shape(CFE,EC,CF)",
"Shape(CED,DC,CE)",
"Shape(CDH,HC,CD)",
"Collinear(HCE)",
"Collinear(GCD)",
"Cocircular(C,HGFED)"
] | [
"Equal(MeasureOfAngle(GCH),2*x)",
"Equal(MeasureOfAngle(HCD),6*x+28)",
"IsCentreOfCircle(C,C)",
"PerpendicularBetweenLine(FC,GC)"
] | [
"IsCentreOfCircle(C,C)",
"PerpendicularBetweenLine(FC,GC)"
] | Value(MeasureOfArc(CFE)) | 52 | [
"flat_angle(1,GCD)",
"angle_addition(1,GCH,HCD)",
"vertical_angle(1,GCH,DCE)",
"adjacent_complementary_angle(1,DCF,FCG)",
"angle_addition(1,DCE,ECF)",
"arc_property_center_angle(1,CFE,C)"
] | {"START": ["flat_angle(1,GCD)", "angle_addition(1,GCH,HCD)", "vertical_angle(1,GCH,DCE)", "adjacent_complementary_angle(1,DCF,FCG)", "angle_addition(1,DCE,ECF)", "arc_property_center_angle(1,CFE,C)"]} | |
305 | NaZhu_2023-03-12 | Geometry3k-309 | 3 | 如图所示，RT=x，ST=7，∠RTS=30°，∠TSR=120°。求x的值。 | As shown in the diagram, RT=x, ST=7, ∠RTS=30°, ∠TSR=120°. Find the value of x. | 305.png | [
"Shape(SR,RT,TS)"
] | [
"Equal(LengthOfLine(RT),x)",
"Equal(LengthOfLine(ST),7)",
"Equal(MeasureOfAngle(RTS),30)",
"Equal(MeasureOfAngle(TSR),120)"
] | [
"Equal(LengthOfLine(RT),x)",
"Equal(LengthOfLine(ST),7)",
"Equal(MeasureOfAngle(RTS),30)",
"Equal(MeasureOfAngle(TSR),120)"
] | Value(x) | 7*sqrt(3) | [
"triangle_property_angle_sum(1,SRT)",
"sine_theorem(1,SRT)",
"sine_theorem(1,RTS)"
] | {"START": ["triangle_property_angle_sum(1,SRT)", "sine_theorem(1,SRT)", "sine_theorem(1,RTS)"]} | |
306 | NaZhu_2023-04-02 | Geometry3k-310 | 3 | 如图所示，弧SRT的长度为8，∠TSR=70°，S是圆S的圆心。求圆S的周长。 | As shown in the diagram, the length of arc SRT is 8, ∠TSR=70°, S is the center of ⊙S. Find the circumference of the circle S. | 306.png | [
"Shape(STR,RS,ST)",
"Shape(SRT,TS,SR)",
"Cocircular(S,TR)"
] | [
"Equal(LengthOfArc(SRT),8)",
"Equal(MeasureOfAngle(TSR),70)",
"IsCentreOfCircle(S,S)"
] | [
"Equal(LengthOfArc(SRT),8)",
"Equal(MeasureOfAngle(TSR),70)",
"IsCentreOfCircle(S,S)"
] | Value(PerimeterOfCircle(S)) | 288/7 | [
"arc_property_center_angle(1,SRT,S)",
"arc_length_formula(1,SRT)",
"circle_perimeter_formula(1,S)"
] | {"START": ["arc_property_center_angle(1,SRT,S)", "arc_length_formula(1,SRT)", "circle_perimeter_formula(1,S)"]} | |
307 | NaZhu_2023-03-12 | Geometry3k-311 | 2 | 如图所示，∠ABD=45°，∠BCD=70°，∠DAB=40°。求∠DBC的大小。 | As shown in the diagram, ∠ABD=45°, ∠BCD=70°, ∠DAB=40°. Find the measure of ∠DBC. | 307.png | [
"Shape(BC,CD,DB)",
"Shape(BD,DA,AB)",
"Collinear(CDA)"
] | [
"Equal(MeasureOfAngle(ABD),45)",
"Equal(MeasureOfAngle(BCD),70)",
"Equal(MeasureOfAngle(DAB),40)"
] | [
"Equal(MeasureOfAngle(ABD),45)",
"Equal(MeasureOfAngle(BCD),70)",
"Equal(MeasureOfAngle(DAB),40)"
] | Value(MeasureOfAngle(DBC)) | 25 | [
"triangle_property_angle_sum(1,CAB)",
"angle_addition(1,ABD,DBC)"
] | {"START": ["triangle_property_angle_sum(1,CAB)", "angle_addition(1,ABD,DBC)"]} | |
308 | NaZhu_2023-04-02 | Geometry3k-312 | 5 | 如图所示，NQ=2*x+3，QK=5*x-9，KJ垂直于NJ，NM⊥KM，四边形JNMK是矩形。求直线JQ的长度。 | As shown in the diagram, NQ=2*x+3, QK=5*x-9, KJ⊥NJ, NM is perpendicular to KM, quadrilateral JNMK is a rectangle. Find the length of line JQ. | 308.png | [
"Shape(JN,NQ,QJ)",
"Shape(JQ,QK,KJ)",
"Shape(KQ,QM,MK)",
"Shape(QN,NM,MQ)",
"Collinear(JQM)",
"Collinear(NQK)"
] | [
"Equal(LengthOfLine(NQ),2*x+3)",
"Equal(LengthOfLine(QK),5*x-9)",
"PerpendicularBetweenLine(KJ,NJ)",
"PerpendicularBetweenLine(NM,KM)",
"Rectangle(JNMK)"
] | [] | Value(LengthOfLine(JQ)) | 11 | [
"parallelogram_property_diagonal_bisection(1,JNMK,Q)",
"parallelogram_property_diagonal_bisection(1,NMKJ,Q)",
"rectangle_property_diagonal_equal(1,JNMK)",
"line_addition(1,JQ,QM)",
"line_addition(1,NQ,QK)"
] | {"START": ["parallelogram_property_diagonal_bisection(1,JNMK,Q)", "parallelogram_property_diagonal_bisection(1,NMKJ,Q)", "rectangle_property_diagonal_equal(1,JNMK)", "line_addition(1,JQ,QM)", "line_addition(1,NQ,QK)"]} | |
309 | NaZhu_2023-03-12 | Geometry3k-313 | 3 | 如图所示，AB=5*x-11，AD=3*x+5，BC=15，CD=15，AC垂直于DC。求直线AB的长度。 | As shown in the diagram, AB=5*x-11, AD=3*x+5, BC=15, CD=15, AC is perpendicular to DC. Find the length of line AB. | 309.png | [
"Shape(AB,BC,CA)",
"Shape(AC,CD,DA)",
"Collinear(BCD)"
] | [
"Equal(LengthOfLine(AB),5*x-11)",
"Equal(LengthOfLine(AD),3*x+5)",
"Equal(LengthOfLine(BC),15)",
"Equal(LengthOfLine(CD),15)",
"PerpendicularBetweenLine(AC,DC)"
] | [
"Equal(LengthOfLine(AB),5*x-11)",
"Equal(LengthOfLine(AD),3*x+5)",
"Equal(LengthOfLine(BC),15)",
"Equal(LengthOfLine(CD),15)",
"PerpendicularBetweenLine(AC,DC)"
] | Value(LengthOfLine(AB)) | 29 | [
"adjacent_complementary_angle(1,BCA,ACD)",
"perpendicular_bisector_judgment_per_and_mid(1,AC,BD)",
"perpendicular_bisector_property_distance_equal(1,AC,BD)"
] | {"START": ["adjacent_complementary_angle(1,BCA,ACD)"], "adjacent_complementary_angle(1,BCA,ACD)": ["perpendicular_bisector_judgment_per_and_mid(1,AC,BD)"], "perpendicular_bisector_judgment_per_and_mid(1,AC,BD)": ["perpendicular_bisector_property_distance_equal(1,AC,BD)"]} | |
310 | NaZhu_2023-04-02 | Geometry3k-314 | 1 | 如图所示，BE=x，DE=x+10，EA=x+1，EC=x+3。求x的值。 | As shown in the diagram, BE=x, DE=x+10, EA=x+1, EC=x+3. Find the value of x. | 310.png | [
"Shape(FBA,AE,EB)",
"Shape(FAD,DF,FE,EA)",
"Shape(FDC,CE,EF,FD)",
"Shape(FCB,BE,EC)",
"Collinear(BEFD)",
"Collinear(AEC)",
"Cocircular(F,CBAD)"
] | [
"Equal(LengthOfLine(BE),x)",
"Equal(LengthOfLine(DE),x+10)",
"Equal(LengthOfLine(EA),x+1)",
"Equal(LengthOfLine(EC),x+3)"
] | [
"Equal(LengthOfLine(BE),x)",
"Equal(LengthOfLine(DE),x+10)",
"Equal(LengthOfLine(EA),x+1)",
"Equal(LengthOfLine(EC),x+3)"
] | Value(x) | 1/2 | [
"circle_property_circular_power_chord_and_chord(1,CEA,BED,F)"
] | {"START": ["circle_property_circular_power_chord_and_chord(1,CEA,BED,F)"]} | |
311 | NaZhu_2023-03-12 | Geometry3k-315 | 0 | 如图所示，KJ=6，KL=12，RQ=4，RS=8，∠JLK=30°，∠JLK=∠QSR，∠KJL=∠RQS，∠LKJ=∠SRQ，∠LKJ=y°，∠QSR=x°，∠RQS=80°。求x的值。 | As shown in the diagram, KJ=6, KL=12, RQ=4, RS=8, ∠JLK=30°, ∠JLK=∠QSR, ∠KJL=∠RQS, ∠LKJ=∠SRQ, ∠LKJ=y°, ∠QSR=x°, ∠RQS=80°. Find the value of x. | 311.png | [
"Shape(KJ,JL,LK)",
"Shape(RQ,QS,SR)"
] | [
"Equal(LengthOfLine(KJ),6)",
"Equal(LengthOfLine(KL),12)",
"Equal(LengthOfLine(RQ),4)",
"Equal(LengthOfLine(RS),8)",
"Equal(MeasureOfAngle(JLK),30)",
"Equal(MeasureOfAngle(JLK),MeasureOfAngle(QSR))",
"Equal(MeasureOfAngle(KJL),MeasureOfAngle(RQS))",
"Equal(MeasureOfAngle(LKJ),MeasureOfAngle(SRQ))",
... | [
"Equal(LengthOfLine(KJ),6)",
"Equal(LengthOfLine(KL),12)",
"Equal(LengthOfLine(RQ),4)",
"Equal(LengthOfLine(RS),8)",
"Equal(MeasureOfAngle(JLK),30)",
"Equal(MeasureOfAngle(JLK),MeasureOfAngle(QSR))",
"Equal(MeasureOfAngle(KJL),MeasureOfAngle(RQS))",
"Equal(MeasureOfAngle(LKJ),MeasureOfAngle(SRQ))",
... | Value(x) | 30 | [] | {"START": []} | |
312 | NaZhu_2023-03-12 | Geometry3k-316 | 3 | 如图所示，AB=x，BC=5，DE=x-4，FE=3，∠ABC=∠FED，∠CAB=∠EDF。求直线DE的长度。 | As shown in the diagram, AB=x, BC=5, DE=x-4, FE=3, ∠ABC=∠FED, ∠CAB=∠EDF. Find the length of line DE. | 312.png | [
"Shape(AB,BC,CA)",
"Shape(DF,FE,ED)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(BC),5)",
"Equal(LengthOfLine(DE),x-4)",
"Equal(LengthOfLine(FE),3)",
"Equal(MeasureOfAngle(ABC),MeasureOfAngle(FED))",
"Equal(MeasureOfAngle(CAB),MeasureOfAngle(EDF))"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(BC),5)",
"Equal(LengthOfLine(DE),x-4)",
"Equal(LengthOfLine(FE),3)",
"Equal(MeasureOfAngle(ABC),MeasureOfAngle(FED))",
"Equal(MeasureOfAngle(CAB),MeasureOfAngle(EDF))"
] | Value(LengthOfLine(DE)) | 6 | [
"mirror_similar_triangle_judgment_aa(1,CAB,FED)",
"mirror_similar_triangle_property_line_ratio(1,CAB,FED)",
"mirror_similar_triangle_property_line_ratio(1,ABC,DFE)"
] | {"START": ["mirror_similar_triangle_judgment_aa(1,CAB,FED)"], "mirror_similar_triangle_judgment_aa(1,CAB,FED)": ["mirror_similar_triangle_property_line_ratio(1,CAB,FED)", "mirror_similar_triangle_property_line_ratio(1,ABC,DFE)"]} | |
313 | NaZhu_2023-03-12 | Geometry3k-317 | 2 | 如图所示，AC=x，BA=7，BC=19，BA垂直于CA。求x的值。 | As shown in the diagram, AC=x, BA=7, BC=19, BA⊥CA. Find the value of x. | 313.png | [
"Shape(BA,AC,CB)"
] | [
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(BA),7)",
"Equal(LengthOfLine(BC),19)",
"PerpendicularBetweenLine(BA,CA)"
] | [
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(BA),7)",
"Equal(LengthOfLine(BC),19)",
"PerpendicularBetweenLine(BA,CA)"
] | Value(x) | 2*sqrt(78) | [
"right_triangle_judgment_angle(1,BAC)",
"right_triangle_property_pythagorean(1,BAC)"
] | {"START": ["right_triangle_judgment_angle(1,BAC)"], "right_triangle_judgment_angle(1,BAC)": ["right_triangle_property_pythagorean(1,BAC)"]} | |
314 | NaZhu_2023-04-02 | Geometry3k-318 | 1 | 如图所示，RP=y+4，RS=27，TP=2*y-5，TQ=5*x，∠PQT=95°，∠RQP=33°，∠TSP=3*z°，QTSR是平行四边形。求x的值。 | As shown in the diagram, RP=y+4, RS=27, TP=2*y-5, TQ=5*x, ∠PQT=95°, ∠RQP=33°, ∠TSP=3*z°, QR and TS are opposite sides of the ▱ QTSR. Find the value of x. | 314.png | [
"Shape(QT,TP,PQ)",
"Shape(PT,TS,SP)",
"Shape(QP,PR,RQ)",
"Shape(RP,PS,SR)",
"Collinear(QPS)",
"Collinear(TPR)"
] | [
"Equal(LengthOfLine(RP),y+4)",
"Equal(LengthOfLine(RS),27)",
"Equal(LengthOfLine(TP),2*y-5)",
"Equal(LengthOfLine(TQ),5*x)",
"Equal(MeasureOfAngle(PQT),95)",
"Equal(MeasureOfAngle(RQP),33)",
"Equal(MeasureOfAngle(TSP),3*z)",
"Parallelogram(QTSR)"
] | [
"Equal(LengthOfLine(RP),y+4)",
"Equal(LengthOfLine(RS),27)",
"Equal(LengthOfLine(TP),2*y-5)",
"Equal(LengthOfLine(TQ),5*x)",
"Equal(MeasureOfAngle(PQT),95)",
"Equal(MeasureOfAngle(RQP),33)",
"Equal(MeasureOfAngle(TSP),3*z)"
] | Value(x) | 27/5 | [
"parallelogram_property_opposite_line_equal(1,QTSR)"
] | {"START": ["parallelogram_property_opposite_line_equal(1,QTSR)"]} | |
315 | NaZhu_2023-04-02 | Geometry3k-319 | 8 | 如图所示，EN=3，三角形DCB为等边三角形，N是圆N的圆心，三角形DCB的重心为O，N是△DCB的内心，BG是圆O的切线，CF是圆O的切线，⊙O的切线为DE。求三角形DCB的面积减去圆N的面积。 | As shown in the diagram, EN=3, △DCB is an equilateral △, N is the center of ⊙N, the centroid of △DCB is N, the incenter of triangle DCB is N, BG is the tangent to ⊙N, CF is the tangent to circle N, the tangent to circle N is DE. Find the area of triangle DCB minus the area of the ⊙N. | 315.png | [
"Shape(NQG,NGE,EN,NQ)",
"Shape(NEF,NFQ,QN,NE)",
"Shape(NGE,GD,DE)",
"Shape(NEF,EC,CF)",
"Shape(NFQ,FB,BQ)",
"Shape(NQG,QB,BG)",
"Collinear(ENQB)",
"Collinear(DGB)",
"Collinear(CFB)",
"Collinear(DEC)",
"Cocircular(N,GEFQ)"
] | [
"Equal(LengthOfLine(EN),3)",
"EquilateralTriangle(DCB)",
"IsCentreOfCircle(N,N)",
"IsCentroidOfTriangle(N,DCB)",
"IsIncenterOfTriangle(N,DCB)",
"IsTangentOfCircle(BG,N)",
"IsTangentOfCircle(CF,N)",
"IsTangentOfCircle(DE,N)"
] | [
"Equal(LengthOfLine(EN),3)",
"IsCentreOfCircle(N,N)"
] | Value(Sub(AreaOfTriangle(DCB),AreaOfCircle(N))) | -9*pi+27*sqrt(3) | [
"tangent_of_circle_property_perpendicular(2,DE,N,N)",
"equilateral_triangle_property_angle(1,DCB)",
"centroid_of_triangle_property_line_ratio(1,N,BDC,E)",
"line_addition(1,EN,NB)",
"sine_theorem(1,BDE)",
"triangle_area_formula_sine(1,DCB)",
"radius_of_circle_property_length_equal(1,NE,N)",
"circle_are... | {"START": ["tangent_of_circle_property_perpendicular(2,DE,N,N)", "equilateral_triangle_property_angle(1,DCB)", "centroid_of_triangle_property_line_ratio(1,N,BDC,E)", "line_addition(1,EN,NB)", "sine_theorem(1,BDE)", "triangle_area_formula_sine(1,DCB)", "radius_of_circle_property_length_equal(1,NE,N)", "circle_area_formu... | |
316 | NaZhu_2023-04-02 | Geometry3k-320 | 2 | 如图所示，AH=y，DA=8，DH=x，HF=6，∠DAH=30°，∠FBC=45°，DC∥HF，AH⊥DH，CF垂直于BF。求y的值。 | As shown in the diagram, AH=y, DA=8, DH=x, HF=6, ∠DAH=30°, ∠FBC=45°, DC∥HF, AH⊥DH, CF is perpendicular to BF. Find the value of y. | 316.png | [
"Shape(DA,AH,HD)",
"Shape(DH,HF,FC,CD)",
"Shape(CF,FB,BC)",
"Collinear(AHFB)"
] | [
"Equal(LengthOfLine(AH),y)",
"Equal(LengthOfLine(DA),8)",
"Equal(LengthOfLine(DH),x)",
"Equal(LengthOfLine(HF),6)",
"Equal(MeasureOfAngle(DAH),30)",
"Equal(MeasureOfAngle(FBC),45)",
"ParallelBetweenLine(DC,HF)",
"PerpendicularBetweenLine(AH,DH)",
"PerpendicularBetweenLine(CF,BF)"
] | [
"Equal(LengthOfLine(AH),y)",
"Equal(LengthOfLine(DA),8)",
"Equal(LengthOfLine(DH),x)",
"Equal(LengthOfLine(HF),6)",
"Equal(MeasureOfAngle(DAH),30)",
"Equal(MeasureOfAngle(FBC),45)",
"ParallelBetweenLine(DC,HF)",
"PerpendicularBetweenLine(AH,DH)",
"PerpendicularBetweenLine(CF,BF)"
] | Value(y) | 4*sqrt(3) | [
"triangle_property_angle_sum(1,AHD)",
"sine_theorem(1,AHD)"
] | {"START": ["triangle_property_angle_sum(1,AHD)", "sine_theorem(1,AHD)"]} | |
317 | NaZhu_2023-04-02 | Geometry3k-321 | 4 | 如图所示，BG=10，EG=10，圆G的半径为26，G是⊙G的圆心，CB⊥GB，FE⊥GE。求直线DE的长度。 | As shown in the diagram, BG=10, EG=10, the radius of ⊙G is 26, G is the center of circle G, CB⊥GB, FE is perpendicular to GE. Find the length of line DE. | 317.png | [
"Shape(GAF,FG,GA)",
"Shape(GF,FE,EG)",
"Shape(AG,GB,BA)",
"Shape(GFD,DE,EF)",
"Shape(GCA,AB,BC)",
"Shape(GE,ED,GDC,CB,BG)",
"Collinear(FED)",
"Collinear(ABC)",
"Cocircular(G,AFDC)"
] | [
"Equal(LengthOfLine(BG),10)",
"Equal(LengthOfLine(EG),10)",
"Equal(RadiusOfCircle(G),26)",
"IsCentreOfCircle(G,G)",
"PerpendicularBetweenLine(CB,GB)",
"PerpendicularBetweenLine(FE,GE)"
] | [
"Equal(LengthOfLine(BG),10)",
"Equal(LengthOfLine(EG),10)",
"Equal(RadiusOfCircle(G),26)",
"IsCentreOfCircle(G,G)"
] | Value(LengthOfLine(DE)) | 24 | [
"radius_of_circle_property_length_equal(1,GF,G)",
"right_triangle_judgment_angle(1,FEG)",
"right_triangle_property_pythagorean(1,FEG)",
"circle_property_chord_perpendicular_bisect_chord(1,G,GE,FD)"
] | {"START": ["radius_of_circle_property_length_equal(1,GF,G)", "right_triangle_judgment_angle(1,FEG)", "circle_property_chord_perpendicular_bisect_chord(1,G,GE,FD)"], "right_triangle_judgment_angle(1,FEG)": ["right_triangle_property_pythagorean(1,FEG)"]} | |
318 | NaZhu_2023-03-12 | Geometry3k-322 | 8 | 如图所示，RA=3，TA=8，UB=x+2，WB=3*x-6，TA是三角形TRS的中线，WB是三角形WUV的中线，三角形RST相似于三角形UVW。求直线UB的长度。 | As shown in the diagram, RA=3, TA=8, UB=x+2, WB=3*x-6, TA is the median of △ TRS, WB is the median of △ WUV, triangle RST is similar to triangle UVW.. Find the length of line UB. | 318.png | [
"Shape(TR,RA,AT)",
"Shape(TA,AS,ST)",
"Shape(WU,UB,BW)",
"Shape(WB,BV,VW)",
"Collinear(RAS)",
"Collinear(UBV)"
] | [
"Equal(LengthOfLine(RA),3)",
"Equal(LengthOfLine(TA),8)",
"Equal(LengthOfLine(UB),x+2)",
"Equal(LengthOfLine(WB),3*x-6)",
"IsMedianOfTriangle(TA,TRS)",
"IsMedianOfTriangle(WB,WUV)",
"SimilarBetweenTriangle(RST,UVW)"
] | [] | Value(LengthOfLine(UB)) | 36 | [
"similar_triangle_property_angle_equal(1,RST,UVW)",
"similar_triangle_property_line_ratio(1,TRS,WUV)",
"similar_triangle_property_line_ratio(1,STR,VWU)",
"line_addition(1,RA,AS)",
"line_addition(1,UB,BV)",
"similar_triangle_judgment_sas(1,RAT,UBW)",
"similar_triangle_property_line_ratio(1,RAT,UBW)",
"... | {"START": ["similar_triangle_property_angle_equal(1,RST,UVW)", "similar_triangle_property_line_ratio(1,TRS,WUV)", "similar_triangle_property_line_ratio(1,STR,VWU)", "line_addition(1,RA,AS)", "line_addition(1,UB,BV)"], "line_addition(1,RA,AS)": ["similar_triangle_judgment_sas(1,RAT,UBW)"], "line_addition(1,UB,BV)": ["si... | |
319 | NaZhu_2023-04-02 | Geometry3k-323 | 1 | 如图所示，AE=x+7，BE=4，EC=9，EF=x。求x的值。 | As shown in the diagram, AE=x+7, BE=4, EC=9, EF=x. Find the value of x. | 319.png | [
"Shape(DAC,CE,EA)",
"Shape(DCB,BE,EC)",
"Shape(DBF,FE,EB)",
"Shape(DFA,AE,EF)",
"Collinear(AEB)",
"Collinear(CEF)",
"Cocircular(D,ACBF)"
] | [
"Equal(LengthOfLine(AE),x+7)",
"Equal(LengthOfLine(BE),4)",
"Equal(LengthOfLine(EC),9)",
"Equal(LengthOfLine(EF),x)"
] | [
"Equal(LengthOfLine(AE),x+7)",
"Equal(LengthOfLine(BE),4)",
"Equal(LengthOfLine(EC),9)",
"Equal(LengthOfLine(EF),x)"
] | Value(x) | 28/5 | [
"circle_property_circular_power_chord_and_chord(1,AEB,CEF,D)"
] | {"START": ["circle_property_circular_power_chord_and_chord(1,AEB,CEF,D)"]} | |
320 | NaZhu_2023-04-02 | Geometry3k-325 | 2 | 如图所示，TR=44，VA=21，M平分线段TV，N平分线段RA，四边形TVAR是梯形。求直线MN的长度。 | As shown in the diagram, TR=44, VA=21, M is the midpoint of segment TV, N is the midpoint of segment RA, TV and AR are the non-parallel sides (legs) of trapezoid TVAR. Find the length of line MN. | 320.png | [
"Shape(TM,MN,NR,RT)",
"Shape(MV,VA,AN,NM)",
"Collinear(TMV)",
"Collinear(RNA)"
] | [
"Equal(LengthOfLine(TR),44)",
"Equal(LengthOfLine(VA),21)",
"IsMidpointOfLine(M,TV)",
"IsMidpointOfLine(N,RA)",
"Trapezoid(TVAR)"
] | [] | Value(LengthOfLine(MN)) | 65/2 | [
"midsegment_of_quadrilateral_judgment_midpoint(1,MN,TVAR)",
"midsegment_of_quadrilateral_property_length(1,MN,TVAR)"
] | {"START": ["midsegment_of_quadrilateral_judgment_midpoint(1,MN,TVAR)"], "midsegment_of_quadrilateral_judgment_midpoint(1,MN,TVAR)": ["midsegment_of_quadrilateral_property_length(1,MN,TVAR)"]} | |
321 | NaZhu_2023-03-12 | Geometry3k-326 | 0 | 如图所示，AC=5*sqrt(26)，AD=25，CB=sqrt(26)，CD=5，DB=1，AD垂直于CD，BC⊥AC。求直线CD的长度与直线DB的长度之比。 | As shown in the diagram, AC=5*sqrt(26), AD=25, CB=sqrt(26), CD=5, DB=1, AD is perpendicular to CD, BC is perpendicular to AC. Find the ratio of the length of line CD to the length of line DB. | 321.png | [
"Shape(CA,AD,DC)",
"Shape(CD,DB,BC)",
"Collinear(ADB)"
] | [
"Equal(LengthOfLine(AC),5*sqrt(26))",
"Equal(LengthOfLine(AD),25)",
"Equal(LengthOfLine(CB),sqrt(26))",
"Equal(LengthOfLine(CD),5)",
"Equal(LengthOfLine(DB),1)",
"PerpendicularBetweenLine(AD,CD)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AC),5*sqrt(26))",
"Equal(LengthOfLine(AD),25)",
"Equal(LengthOfLine(CB),sqrt(26))",
"Equal(LengthOfLine(CD),5)",
"Equal(LengthOfLine(DB),1)",
"PerpendicularBetweenLine(AD,CD)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(Div(LengthOfLine(CD),LengthOfLine(DB))) | 5 | [] | {"START": []} | |
322 | NaZhu_2023-04-02 | Geometry3k-327 | 1 | 如图所示，∠AGB=30°，G是圆G的圆心，CG垂直于DG。求弧GBA的角度。 | As shown in the diagram, ∠AGB=30°, the center of circle G is G, CG is perpendicular to DG. Find the measure of ⌒GBA. | 322.png | [
"Shape(GBA,AG,GB)",
"Shape(GCB,BG,GC)",
"Shape(GDC,CG,GD)",
"Shape(GFD,DG,GF)",
"Shape(GAF,FG,GA)",
"Collinear(AGD)",
"Collinear(BGF)",
"Cocircular(G,CBAFD)"
] | [
"Equal(MeasureOfAngle(AGB),30)",
"IsCentreOfCircle(G,G)",
"PerpendicularBetweenLine(CG,DG)"
] | [
"IsCentreOfCircle(G,G)",
"PerpendicularBetweenLine(CG,DG)"
] | Value(MeasureOfArc(GBA)) | 30 | [
"arc_property_center_angle(1,GBA,G)"
] | {"START": ["arc_property_center_angle(1,GBA,G)"]} | |
323 | NaZhu_2023-03-12 | Geometry3k-328 | 6 | 如图所示，UY=32，WY=40，WZ=3*x-6，ZU=x+6，∠UWZ=∠UYW，ZU垂直于WU。求直线UZ的长度。 | As shown in the diagram, UY=32, WY=40, WZ=3*x-6, ZU=x+6, ∠UWZ=∠UYW, ZU is perpendicular to WU. Find the length of line UZ. | 323.png | [
"Shape(WZ,ZU,UW)",
"Shape(WU,UY,YW)",
"Collinear(ZUY)"
] | [
"Equal(LengthOfLine(UY),32)",
"Equal(LengthOfLine(WY),40)",
"Equal(LengthOfLine(WZ),3*x-6)",
"Equal(LengthOfLine(ZU),x+6)",
"Equal(MeasureOfAngle(UWZ),MeasureOfAngle(UYW))",
"PerpendicularBetweenLine(ZU,WU)"
] | [
"Equal(LengthOfLine(UY),32)",
"Equal(LengthOfLine(WY),40)",
"Equal(LengthOfLine(WZ),3*x-6)",
"Equal(LengthOfLine(ZU),x+6)",
"Equal(MeasureOfAngle(UWZ),MeasureOfAngle(UYW))",
"PerpendicularBetweenLine(ZU,WU)"
] | Value(LengthOfLine(UZ)) | 18 | [
"adjacent_complementary_angle(1,ZUW,WUY)",
"similar_triangle_judgment_aa(1,ZUW,WUY)",
"right_triangle_judgment_angle(1,WUY)",
"right_triangle_property_pythagorean(1,WUY)",
"similar_triangle_property_line_ratio(1,ZUW,WUY)",
"similar_triangle_property_line_ratio(1,WZU,YWU)"
] | {"START": ["adjacent_complementary_angle(1,ZUW,WUY)"], "adjacent_complementary_angle(1,ZUW,WUY)": ["similar_triangle_judgment_aa(1,ZUW,WUY)", "right_triangle_judgment_angle(1,WUY)"], "right_triangle_judgment_angle(1,WUY)": ["right_triangle_property_pythagorean(1,WUY)"], "similar_triangle_judgment_aa(1,ZUW,WUY)": ["simi... | |
324 | NaZhu_2023-04-02 | Geometry3k-329 | 7 | 如图所示，NC=10，ND=2，圆M的圆心为M，⊙O的切线为NC。求△MNC的周长。 | As shown in the diagram, NC=10, ND=2, M is the center of ⊙M, NC is the tangent to ⊙M. Find the perimeter of triangle MNC. | 324.png | [
"Shape(MCD,DM,MC)",
"Shape(MDC,CM,MD)",
"Shape(MDC,DN,NC)",
"Collinear(NDM)",
"Cocircular(M,DC)"
] | [
"Equal(LengthOfLine(NC),10)",
"Equal(LengthOfLine(ND),2)",
"IsCentreOfCircle(M,M)",
"IsTangentOfCircle(NC,M)"
] | [
"Equal(LengthOfLine(NC),10)",
"Equal(LengthOfLine(ND),2)",
"IsCentreOfCircle(M,M)"
] | Value(PerimeterOfTriangle(MNC)) | 60 | [
"tangent_of_circle_property_perpendicular(2,NC,M,M)",
"line_addition(1,ND,DM)",
"radius_of_circle_property_length_equal(1,MD,M)",
"radius_of_circle_property_length_equal(1,MC,M)",
"right_triangle_judgment_angle(1,NCM)",
"right_triangle_property_pythagorean(1,NCM)",
"triangle_perimeter_formula(1,NCM)"
] | {"START": ["tangent_of_circle_property_perpendicular(2,NC,M,M)", "line_addition(1,ND,DM)", "radius_of_circle_property_length_equal(1,MD,M)", "radius_of_circle_property_length_equal(1,MC,M)", "triangle_perimeter_formula(1,NCM)"], "right_triangle_judgment_angle(1,NCM)": ["right_triangle_property_pythagorean(1,NCM)"], "ta... | |
325 | NaZhu_2023-04-02 | Geometry3k-330 | 2 | 如图所示，∠BAC=124°。求∠CAE的大小。 | As shown in the diagram, ∠BAC=124°. Find the measure of ∠CAE. | 325.png | [
"Shape(BA,AC)",
"Shape(DA,AB)",
"Shape(CA,AE)",
"Shape(EA,AD)",
"Collinear(BAE)",
"Collinear(CAD)"
] | [
"Equal(MeasureOfAngle(BAC),124)"
] | [
"Equal(MeasureOfAngle(BAC),124)"
] | Value(MeasureOfAngle(CAE)) | 56 | [
"flat_angle(1,BAE)",
"angle_addition(1,BAC,CAE)"
] | {"START": ["flat_angle(1,BAE)", "angle_addition(1,BAC,CAE)"]} | |
326 | NaZhu_2023-03-12 | Geometry3k-331 | 1 | 如图所示，NM=39，PM=36，PN=15，MP⊥NP。求cos(NM)的值。 | As shown in the diagram, NM=39, PM=36, PN=15, MP⊥NP. Find the value of cos(NM). | 326.png | [
"Shape(PN,NM,MP)"
] | [
"Equal(LengthOfLine(NM),39)",
"Equal(LengthOfLine(PM),36)",
"Equal(LengthOfLine(PN),15)",
"PerpendicularBetweenLine(MP,NP)"
] | [
"Equal(LengthOfLine(NM),39)",
"Equal(LengthOfLine(PM),36)",
"Equal(LengthOfLine(PN),15)",
"PerpendicularBetweenLine(MP,NP)"
] | Value(Cos(MeasureOfAngle(NMP))) | 12/13 | [
"cosine_theorem(1,MPN)"
] | {"START": ["cosine_theorem(1,MPN)"]} | |
327 | NaZhu_2023-04-02 | Geometry3k-332 | 1 | 如图所示，AE=2，BE=x，DE=4，EC=5。求x的值。 | As shown in the diagram, AE=2, BE=x, DE=4, EC=5. Find the value of x. | 327.png | [
"Shape(XAD,DE,EA)",
"Shape(XDB,BX,XE,ED)",
"Shape(XBC,CE,EX,XB)",
"Shape(XCA,AE,EC)",
"Collinear(AEB)",
"Collinear(DEC)",
"Cocircular(X,CADB)"
] | [
"Equal(LengthOfLine(AE),2)",
"Equal(LengthOfLine(BE),x)",
"Equal(LengthOfLine(DE),4)",
"Equal(LengthOfLine(EC),5)"
] | [
"Equal(LengthOfLine(AE),2)",
"Equal(LengthOfLine(BE),x)",
"Equal(LengthOfLine(DE),4)",
"Equal(LengthOfLine(EC),5)"
] | Value(x) | 10 | [
"circle_property_circular_power_chord_and_chord(1,CED,AEB,X)"
] | {"START": ["circle_property_circular_power_chord_and_chord(1,CED,AEB,X)"]} | |
328 | NaZhu_2023-04-02 | Geometry3k-333 | 3 | 如图所示，∠GID=4*x°，∠HIM=8*x-12°，∠KHA=6*y+10°，HO平行于IM。求y的值。 | As shown in the diagram, ∠GID=4*x°, ∠HIM=8*x-12°, ∠KHA=6*y+10°, HO is parallel to IM. Find the value of y. | 328.png | [
"Shape(KH,HA)",
"Shape(AH,HO)",
"Shape(DI,IH)",
"Shape(HI,IM)",
"Shape(IH,HK)",
"Shape(OH,HI)",
"Shape(GI,ID)",
"Shape(MI,IG)",
"Collinear(KHO)",
"Collinear(DIM)",
"Collinear(AHIG)"
] | [
"Equal(MeasureOfAngle(GID),4*x)",
"Equal(MeasureOfAngle(HIM),8*x-12)",
"Equal(MeasureOfAngle(KHA),6*y+10)",
"ParallelBetweenLine(HO,IM)"
] | [
"Equal(MeasureOfAngle(GID),4*x)",
"Equal(MeasureOfAngle(HIM),8*x-12)",
"Equal(MeasureOfAngle(KHA),6*y+10)",
"ParallelBetweenLine(HO,IM)"
] | Value(y) | 79/3 | [
"vertical_angle(1,AIM,GID)",
"parallel_property_ipsilateral_internal_angle(1,HO,IM)",
"vertical_angle(1,KHA,OHI)"
] | {"START": ["vertical_angle(1,AIM,GID)", "parallel_property_ipsilateral_internal_angle(1,HO,IM)", "vertical_angle(1,KHA,OHI)"]} | |
329 | NaZhu_2023-04-02 | Geometry3k-334 | 5 | 如图所示，∠FOE=118°，∠JDA=104°，OD∥FI。求∠IFJ的大小。 | As shown in the diagram, ∠FOE=118°, ∠JDA=104°, OD is parallel to FI. Find the measure of ∠IFJ. | 329.png | [
"Shape(JD,DA)",
"Shape(FO,OE)",
"Shape(IF,FJ)",
"Shape(JF,FK)",
"Collinear(JDIC)",
"Collinear(HOFJ)",
"Collinear(ADOE)",
"Collinear(BIFK)"
] | [
"Equal(MeasureOfAngle(FOE),118)",
"Equal(MeasureOfAngle(JDA),104)",
"ParallelBetweenLine(OD,FI)"
] | [
"Equal(MeasureOfAngle(FOE),118)",
"Equal(MeasureOfAngle(JDA),104)",
"ParallelBetweenLine(OD,FI)"
] | Value(MeasureOfAngle(IFJ)) | 62 | [
"parallel_property_collinear_extend(1,OD,FI,E)",
"parallel_property_collinear_extend(2,IF,OE,K)",
"parallel_property_corresponding_angle(1,FK,OE,J)",
"flat_angle(1,IFK)",
"angle_addition(1,IFJ,JFK)"
] | {"START": ["parallel_property_collinear_extend(1,OD,FI,E)", "flat_angle(1,IFK)", "angle_addition(1,IFJ,JFK)"], "parallel_property_collinear_extend(1,OD,FI,E)": ["parallel_property_collinear_extend(2,IF,OE,K)"], "parallel_property_collinear_extend(2,IF,OE,K)": ["parallel_property_corresponding_angle(1,FK,OE,J)"]} | |
330 | NaZhu_2023-03-12 | Geometry3k-335 | 2 | 如图所示，DC=DA，∠ACD=66°，∠DBA=24°。求∠DAC的大小。 | As shown in the diagram, DC=DA, ∠ACD=66°, ∠DBA=24°. Find the measure of ∠DAC. | 330.png | [
"Shape(DB,BA,AD)",
"Shape(CD,DA,AC)",
"Collinear(BDC)"
] | [
"Equal(LengthOfLine(DC),LengthOfLine(DA))",
"Equal(MeasureOfAngle(ACD),66)",
"Equal(MeasureOfAngle(DBA),24)"
] | [
"Equal(LengthOfLine(DC),LengthOfLine(DA))",
"Equal(MeasureOfAngle(ACD),66)",
"Equal(MeasureOfAngle(DBA),24)"
] | Value(MeasureOfAngle(DAC)) | 66 | [
"isosceles_triangle_judgment_line_equal(1,DAC)",
"isosceles_triangle_property_angle_equal(1,DAC)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,DAC)"], "isosceles_triangle_judgment_line_equal(1,DAC)": ["isosceles_triangle_property_angle_equal(1,DAC)"]} | |
331 | NaZhu_2023-04-02 | Geometry3k-336 | 2 | 如图所示，∠BTS=3*x°，∠BVU=y+16°，∠TSB=3*y°，∠VUB=2*x+15°。求∠TSB的大小。 | As shown in the diagram, ∠BTS=3*x°, ∠BVU=y+16°, ∠TSB=3*y°, ∠VUB=2*x+15°. Find the measure of ∠TSB. | 331.png | [
"Shape(ATS,ST)",
"Shape(AVU,UV)",
"Shape(AUT,TB,BU)",
"Shape(TS,SB,BT)",
"Shape(UB,BV,VU)",
"Shape(ASV,VA,AS)",
"Shape(AV,VB,BS,SA)",
"Collinear(TBV)",
"Collinear(UBS)",
"Cocircular(A,UTSV)"
] | [
"Equal(MeasureOfAngle(BTS),3*x)",
"Equal(MeasureOfAngle(BVU),y+16)",
"Equal(MeasureOfAngle(TSB),3*y)",
"Equal(MeasureOfAngle(VUB),2*x+15)"
] | [
"Equal(MeasureOfAngle(BTS),3*x)",
"Equal(MeasureOfAngle(BVU),y+16)",
"Equal(MeasureOfAngle(TSB),3*y)",
"Equal(MeasureOfAngle(VUB),2*x+15)"
] | Value(MeasureOfAngle(TSB)) | 24 | [
"arc_property_circumference_angle_external(1,AUT,S)",
"arc_property_circumference_angle_external(1,AUT,V)"
] | {"START": ["arc_property_circumference_angle_external(1,AUT,S)", "arc_property_circumference_angle_external(1,AUT,V)"]} | |
332 | NaZhu_2023-03-12 | Geometry3k-337 | 4 | 如图所示，∠CDE=57°，∠DBE=42°，∠EBA=36°，∠ECD=28°。求∠BAE的大小。 | As shown in the diagram, ∠CDE=57°, ∠DBE=42°, ∠EBA=36°, ∠ECD=28°. Find the measure of ∠BAE. | 332.png | [
"Shape(AE,EB,BA)",
"Shape(BE,ED,DB)",
"Shape(EC,CD,DE)",
"Collinear(AED)",
"Collinear(CEB)"
] | [
"Equal(MeasureOfAngle(CDE),57)",
"Equal(MeasureOfAngle(DBE),42)",
"Equal(MeasureOfAngle(EBA),36)",
"Equal(MeasureOfAngle(ECD),28)"
] | [
"Equal(MeasureOfAngle(CDE),57)",
"Equal(MeasureOfAngle(DBE),42)",
"Equal(MeasureOfAngle(EBA),36)",
"Equal(MeasureOfAngle(ECD),28)"
] | Value(MeasureOfAngle(BAE)) | 49 | [
"triangle_property_angle_sum(1,ADB)",
"triangle_property_angle_sum(1,BCD)",
"angle_addition(1,DBE,EBA)",
"angle_addition(1,CDE,EDB)"
] | {"START": ["triangle_property_angle_sum(1,ADB)", "triangle_property_angle_sum(1,BCD)", "angle_addition(1,DBE,EBA)", "angle_addition(1,CDE,EDB)"]} | |
333 | NaZhu_2023-03-12 | Geometry3k-338 | 1 | 如图所示，SR=5，ST=4，TR=3，ST垂直于RT。求tan(SR)的值。 | As shown in the diagram, SR=5, ST=4, TR=3, ST⊥RT. Find the value of tan(SR). | 333.png | [
"Shape(TS,SR,RT)"
] | [
"Equal(LengthOfLine(SR),5)",
"Equal(LengthOfLine(ST),4)",
"Equal(LengthOfLine(TR),3)",
"PerpendicularBetweenLine(ST,RT)"
] | [
"Equal(LengthOfLine(SR),5)",
"Equal(LengthOfLine(ST),4)",
"Equal(LengthOfLine(TR),3)",
"PerpendicularBetweenLine(ST,RT)"
] | Value(Tan(MeasureOfAngle(SRT))) | 4/3 | [
"cosine_theorem(1,RTS)"
] | {"START": ["cosine_theorem(1,RTS)"]} | |
334 | NaZhu_2023-04-02 | Geometry3k-339 | 2 | 如图所示，∠ADC=105°，∠BAD=74°，∠DCB=x°，四边形ADCB是风筝形。求∠DCB的大小。 | As shown in the diagram, ∠ADC=105°, ∠BAD=74°, ∠DCB=x°, quadrilateral ADCB is a kite. Find the measure of ∠DCB. | 334.png | [
"Shape(BA,AD,DC,CB)"
] | [
"Equal(MeasureOfAngle(ADC),105)",
"Equal(MeasureOfAngle(BAD),74)",
"Equal(MeasureOfAngle(DCB),x)",
"Kite(ADCB)"
] | [
"Equal(MeasureOfAngle(ADC),105)",
"Equal(MeasureOfAngle(BAD),74)",
"Equal(MeasureOfAngle(DCB),x)"
] | Value(MeasureOfAngle(DCB)) | 76 | [
"kite_property_opposite_angle_equal(1,ADCB)",
"quadrilateral_property_angle_sum(1,ADCB)"
] | {"START": ["kite_property_opposite_angle_equal(1,ADCB)", "quadrilateral_property_angle_sum(1,ADCB)"]} | |
335 | NaZhu_2023-04-02 | Geometry3k-340 | 2 | 如图所示，CE=24，DE=24，EA=12，EB=x。求直线AB的长度。 | As shown in the diagram, CE=24, DE=24, EA=12, EB=x. Find the length of line AB. | 335.png | [
"Shape(FAD,DE,EA)",
"Shape(FCA,AE,EC)",
"Shape(FDB,BF,FE,ED)",
"Shape(FBC,CE,EF,FB)",
"Collinear(DEC)",
"Collinear(AEFB)",
"Cocircular(F,ADBC)"
] | [
"Equal(LengthOfLine(CE),24)",
"Equal(LengthOfLine(DE),24)",
"Equal(LengthOfLine(EA),12)",
"Equal(LengthOfLine(EB),x)"
] | [
"Equal(LengthOfLine(CE),24)",
"Equal(LengthOfLine(DE),24)",
"Equal(LengthOfLine(EA),12)",
"Equal(LengthOfLine(EB),x)"
] | Value(LengthOfLine(AB)) | 60 | [
"circle_property_circular_power_chord_and_chord(1,AEB,DEC,F)",
"line_addition(1,AE,EB)"
] | {"START": ["circle_property_circular_power_chord_and_chord(1,AEB,DEC,F)", "line_addition(1,AE,EB)"]} | |
336 | NaZhu_2023-03-12 | Geometry3k-341 | 3 | 如图所示，RS=x，RT=y+3，TS=49，VU=21，WU=29，WV=20，∠RST=∠UWV，∠STR=∠WVU，∠TRS=∠VUW。求x的值。 | As shown in the diagram, RS=x, RT=y+3, TS=49, VU=21, WU=29, WV=20, ∠RST=∠UWV, ∠STR=∠WVU, ∠TRS=∠VUW. Find the value of x. | 336.png | [
"Shape(RS,ST,TR)",
"Shape(UW,WV,VU)"
] | [
"Equal(LengthOfLine(RS),x)",
"Equal(LengthOfLine(RT),y+3)",
"Equal(LengthOfLine(TS),49)",
"Equal(LengthOfLine(VU),21)",
"Equal(LengthOfLine(WU),29)",
"Equal(LengthOfLine(WV),20)",
"Equal(MeasureOfAngle(RST),MeasureOfAngle(UWV))",
"Equal(MeasureOfAngle(STR),MeasureOfAngle(WVU))",
"Equal(MeasureOfAngl... | [
"Equal(LengthOfLine(RS),x)",
"Equal(LengthOfLine(RT),y+3)",
"Equal(LengthOfLine(TS),49)",
"Equal(LengthOfLine(VU),21)",
"Equal(LengthOfLine(WU),29)",
"Equal(LengthOfLine(WV),20)",
"Equal(MeasureOfAngle(RST),MeasureOfAngle(UWV))",
"Equal(MeasureOfAngle(STR),MeasureOfAngle(WVU))",
"Equal(MeasureOfAngl... | Value(x) | 1421/20 | [
"similar_triangle_judgment_aa(1,RST,UWV)",
"similar_triangle_property_line_ratio(1,RST,UWV)",
"similar_triangle_property_line_ratio(1,TRS,VUW)"
] | {"START": ["similar_triangle_judgment_aa(1,RST,UWV)"], "similar_triangle_judgment_aa(1,RST,UWV)": ["similar_triangle_property_line_ratio(1,RST,UWV)", "similar_triangle_property_line_ratio(1,TRS,VUW)"]} | |
337 | JiaZou_2023-04-09 | Geometry3k-342 | 1 | 如图所示，DA=8，DH=x，HF=6，∠ABC=45°，∠DAB=30°，DC∥AB，AH垂直于DH，CF垂直于BF。求x的值。 | As shown in the diagram, DA=8, DH=x, HF=6, ∠ABC=45°, ∠DAB=30°, DC∥AB, AH is perpendicular to DH, CF is perpendicular to BF. Find the value of x. | 337.png | [
"Shape(DA,AH,HD)",
"Shape(DH,HF,FC,CD)",
"Shape(CF,FB,BC)",
"Collinear(AHFB)"
] | [
"Equal(LengthOfLine(DA),8)",
"Equal(LengthOfLine(DH),x)",
"Equal(LengthOfLine(HF),6)",
"Equal(MeasureOfAngle(ABC),45)",
"Equal(MeasureOfAngle(DAB),30)",
"ParallelBetweenLine(DC,AB)",
"PerpendicularBetweenLine(AH,DH)",
"PerpendicularBetweenLine(CF,BF)"
] | [
"Equal(LengthOfLine(DA),8)",
"Equal(LengthOfLine(DH),x)",
"Equal(LengthOfLine(HF),6)",
"Equal(MeasureOfAngle(ABC),45)",
"Equal(MeasureOfAngle(DAB),30)",
"ParallelBetweenLine(DC,AB)",
"PerpendicularBetweenLine(AH,DH)",
"PerpendicularBetweenLine(CF,BF)"
] | Value(x) | 4 | [
"sine_theorem(1,DAH)"
] | {"START": ["sine_theorem(1,DAH)"]} | |
338 | JiaZou_2023-04-09 | Geometry3k-344 | 2 | 如图所示，BD=7，FE=6，OE=9，四边形FBOD是风筝形。求四边形FBOD的面积。 | As shown in the diagram, BD=7, FE=6, OE=9, OB and OD are one pair of adjacent sides of the kite FBOD. Find the area of quadrilateral FBOD. | 338.png | [
"Shape(FB,BE,EF)",
"Shape(FE,ED,DF)",
"Shape(BO,OE,EB)",
"Shape(EO,OD,DE)",
"Collinear(FEO)",
"Collinear(BED)"
] | [
"Equal(LengthOfLine(BD),7)",
"Equal(LengthOfLine(FE),6)",
"Equal(LengthOfLine(OE),9)",
"Kite(FBOD)"
] | [
"Equal(LengthOfLine(BD),7)",
"Equal(LengthOfLine(FE),6)",
"Equal(LengthOfLine(OE),9)"
] | Value(AreaOfQuadrilateral(FBOD)) | 105/2 | [
"line_addition(1,FE,EO)",
"kite_area_formula_diagonal(1,FBOD)"
] | {"START": ["line_addition(1,FE,EO)", "kite_area_formula_diagonal(1,FBOD)"]} | |
339 | JiaZou_2023-04-09 | Geometry3k-345 | 4 | 如图所示，∠KGL=x°，弧AJH的角度为47，⌒ALK的角度为116。求x的值。 | As shown in the diagram, ∠KGL=x°, the measure of ⌒AJH is 47, the measure of ⌒ALK is 116. Find the value of x. | 339.png | [
"Shape(LG,GH,AHL)",
"Shape(HG,GJ,AJH)",
"Shape(KG,GL,ALK)",
"Shape(JG,GK,KJ)",
"Shape(JK,AKJ)",
"Collinear(LGJ)",
"Collinear(HGK)",
"Cocircular(A,HLKJ)"
] | [
"Equal(MeasureOfAngle(KGL),x)",
"Equal(MeasureOfArc(AJH),47)",
"Equal(MeasureOfArc(ALK),116)"
] | [
"Equal(MeasureOfAngle(KGL),x)",
"Equal(MeasureOfArc(AJH),47)",
"Equal(MeasureOfArc(ALK),116)"
] | Value(x) | 163/2 | [
"arc_property_circumference_angle_external(1,ALK,J)",
"arc_property_circumference_angle_external(1,AJH,K)",
"triangle_property_angle_sum(1,JGK)",
"adjacent_complementary_angle(1,JGK,KGL)"
] | {"START": ["arc_property_circumference_angle_external(1,ALK,J)", "arc_property_circumference_angle_external(1,AJH,K)", "triangle_property_angle_sum(1,JGK)", "adjacent_complementary_angle(1,JGK,KGL)"]} | |
340 | NaZhu_2023-03-12 | Geometry3k-346 | 0 | 如图所示，∠APC=72+x°，CP是△CAB的高，CQ是∠BCA的角平分线，R平分线段AB。求x的值。 | As shown in the diagram, ∠APC=72+x°, CP is the altitude of △ CAB, CQ is the angle bisector of ∠BCA, R bisects segment AB. Find the value of x. | 340.png | [
"Shape(CA,AP,PC)",
"Shape(CP,PQ,QC)",
"Shape(CQ,QR,RC)",
"Shape(CR,RB,BC)",
"Collinear(APQRB)"
] | [
"Equal(MeasureOfAngle(APC),72+x)",
"IsAltitudeOfTriangle(CP,CAB)",
"IsBisectorOfAngle(CQ,BCA)",
"IsMidpointOfLine(R,AB)"
] | [] | Value(x) | 18 | [] | {"START": []} | |
341 | JiaZou_2023-04-09 | Geometry3k-347 | 7 | 如图所示，AZ=y，QZ=z，RQ=12，RS=10，RZ=x，∠AQR=30°，∠SPA=45°，SR平行于AZ，PA垂直于SA，RZ⊥QZ。求y的值。 | As shown in the diagram, AZ=y, QZ=z, RQ=12, RS=10, RZ=x, ∠AQR=30°, ∠SPA=45°, SR is parallel to AZ, PA is perpendicular to SA, RZ⊥QZ. Find the value of y. | 341.png | [
"Shape(SP,PA,AS)",
"Shape(SA,AZ,ZR,RS)",
"Shape(RZ,ZQ,QR)",
"Collinear(PAZQ)"
] | [
"Equal(LengthOfLine(AZ),y)",
"Equal(LengthOfLine(QZ),z)",
"Equal(LengthOfLine(RQ),12)",
"Equal(LengthOfLine(RS),10)",
"Equal(LengthOfLine(RZ),x)",
"Equal(MeasureOfAngle(AQR),30)",
"Equal(MeasureOfAngle(SPA),45)",
"ParallelBetweenLine(SR,AZ)",
"PerpendicularBetweenLine(PA,SA)",
"PerpendicularBetwee... | [
"Equal(LengthOfLine(AZ),y)",
"Equal(LengthOfLine(QZ),z)",
"Equal(LengthOfLine(RQ),12)",
"Equal(LengthOfLine(RS),10)",
"Equal(LengthOfLine(RZ),x)",
"Equal(MeasureOfAngle(AQR),30)",
"Equal(MeasureOfAngle(SPA),45)",
"ParallelBetweenLine(SR,AZ)",
"PerpendicularBetweenLine(PA,SA)",
"PerpendicularBetwee... | Value(y) | 10 | [
"adjacent_complementary_angle(1,PAS,SAZ)",
"adjacent_complementary_angle(1,PZR,RZQ)",
"perpendicular_judgment_angle(1,SA,ZA)",
"perpendicular_judgment_angle(1,AZ,RZ)",
"parallel_judgment_per_per(1,AS,ZR)",
"parallelogram_judgment_parallel_and_parallel(1,SAZR)",
"parallelogram_property_opposite_line_equa... | {"START": ["adjacent_complementary_angle(1,PAS,SAZ)", "adjacent_complementary_angle(1,PZR,RZQ)"], "adjacent_complementary_angle(1,PAS,SAZ)": ["perpendicular_judgment_angle(1,SA,ZA)"], "adjacent_complementary_angle(1,PZR,RZQ)": ["perpendicular_judgment_angle(1,AZ,RZ)"], "parallel_judgment_per_per(1,AS,ZR)": ["parallelog... | |
342 | NaZhu_2023-03-12 | Geometry3k-348 | 0 | 如图所示，AC=x-3，BA=2*x-7，BC=4*x-21，三角形BAC为等腰三角形。求直线AC的长度。 | As shown in the diagram, AC=x-3, BA=2*x-7, BC=4*x-21, BA and BC are the legs of the isosceles △ BAC. Find the length of line AC. | 342.png | [
"Shape(BA,AC,CB)"
] | [
"Equal(LengthOfLine(AC),x-3)",
"Equal(LengthOfLine(BA),2*x-7)",
"Equal(LengthOfLine(BC),4*x-21)",
"IsoscelesTriangle(BAC)"
] | [
"Equal(LengthOfLine(AC),x-3)",
"Equal(LengthOfLine(BA),2*x-7)",
"Equal(LengthOfLine(BC),4*x-21)"
] | Value(LengthOfLine(AC)) | 4 | [] | {"START": []} | |
343 | NaZhu_2023-03-12 | Geometry3k-349 | 8 | 如图所示，BC=z，BX=6*x，CA=y，CX=36，XA=x，AC垂直于BC，BX垂直于CX。求y的值。 | As shown in the diagram, BC=z, BX=6*x, CA=y, CX=36, XA=x, AC is perpendicular to BC, BX⊥CX. Find the value of y. | 343.png | [
"Shape(CB,BX,XC)",
"Shape(CX,XA,AC)",
"Collinear(BXA)"
] | [
"Equal(LengthOfLine(BC),z)",
"Equal(LengthOfLine(BX),6*x)",
"Equal(LengthOfLine(CA),y)",
"Equal(LengthOfLine(CX),36)",
"Equal(LengthOfLine(XA),x)",
"PerpendicularBetweenLine(AC,BC)",
"PerpendicularBetweenLine(BX,CX)"
] | [
"Equal(LengthOfLine(BC),z)",
"Equal(LengthOfLine(BX),6*x)",
"Equal(LengthOfLine(CA),y)",
"Equal(LengthOfLine(CX),36)",
"Equal(LengthOfLine(XA),x)",
"PerpendicularBetweenLine(AC,BC)",
"PerpendicularBetweenLine(BX,CX)"
] | Value(y) | 6*sqrt(42) | [
"adjacent_complementary_angle(1,BXC,CXA)",
"angle_addition(1,ACX,XCB)",
"triangle_property_angle_sum(1,CBX)",
"similar_triangle_judgment_aa(1,CBX,ACX)",
"similar_triangle_property_line_ratio(1,CBX,ACX)",
"similar_triangle_property_line_ratio(1,BXC,CXA)",
"right_triangle_judgment_angle(1,CXA)",
"right_... | {"START": ["adjacent_complementary_angle(1,BXC,CXA)", "angle_addition(1,ACX,XCB)", "triangle_property_angle_sum(1,CBX)"], "adjacent_complementary_angle(1,BXC,CXA)": ["similar_triangle_judgment_aa(1,CBX,ACX)", "right_triangle_judgment_angle(1,CXA)"], "angle_addition(1,ACX,XCB)": ["similar_triangle_judgment_aa(1,CBX,ACX)... | |
344 | JiaZou_2023-04-09 | Geometry3k-350 | 4 | 如图所示，∠GOI=3*y+1°，∠HBI=3*x+11°，∠OIE=4*x-5°，GE∥OI，HG∥BE，OI∥HB。求y的值。 | As shown in the diagram, ∠GOI=3*y+1°, ∠HBI=3*x+11°, ∠OIE=4*x-5°, GE∥OI, HG∥BE, OI∥HB. Find the value of y. | 344.png | [
"Shape(GO,OI,IE,EG)",
"Shape(OH,HB,BI,IO)",
"Collinear(GOH)",
"Collinear(EIB)"
] | [
"Equal(MeasureOfAngle(GOI),3*y+1)",
"Equal(MeasureOfAngle(HBI),3*x+11)",
"Equal(MeasureOfAngle(OIE),4*x-5)",
"ParallelBetweenLine(GE,OI)",
"ParallelBetweenLine(HG,BE)",
"ParallelBetweenLine(OI,HB)"
] | [
"Equal(MeasureOfAngle(GOI),3*y+1)",
"Equal(MeasureOfAngle(HBI),3*x+11)",
"Equal(MeasureOfAngle(OIE),4*x-5)",
"ParallelBetweenLine(GE,OI)",
"ParallelBetweenLine(HG,BE)",
"ParallelBetweenLine(OI,HB)"
] | Value(y) | 40 | [
"parallel_property_corresponding_angle(2,BH,IO,E)",
"parallel_property_collinear_extend(3,HG,BE,O)",
"parallel_property_collinear_extend(3,EB,GO,I)",
"parallel_property_ipsilateral_internal_angle(1,OG,IE)"
] | {"START": ["parallel_property_corresponding_angle(2,BH,IO,E)", "parallel_property_collinear_extend(3,HG,BE,O)"], "parallel_property_collinear_extend(3,EB,GO,I)": ["parallel_property_ipsilateral_internal_angle(1,OG,IE)"], "parallel_property_collinear_extend(3,HG,BE,O)": ["parallel_property_collinear_extend(3,EB,GO,I)"]} | |
345 | JiaZou_2023-04-09 | Geometry3k-352 | 9 | 如图所示，∠EAB=40°，AB⊥DB，四边形ABDC是长方形。求∠DEB的大小。 | As shown in the diagram, ∠EAB=40°, AB⊥DB, ABDC is a rectangle. Find the measure of ∠DEB. | 345.png | [
"Shape(AE,EC,CA)",
"Shape(BE,EA,AB)",
"Shape(EB,BD,DE)",
"Shape(CE,ED,DC)",
"Collinear(AED)",
"Collinear(BEC)"
] | [
"Equal(MeasureOfAngle(EAB),40)",
"PerpendicularBetweenLine(AB,DB)",
"Rectangle(ABDC)"
] | [
"Equal(MeasureOfAngle(EAB),40)",
"PerpendicularBetweenLine(AB,DB)",
"Rectangle(ABDC)"
] | Value(MeasureOfAngle(DEB)) | 80 | [
"rectangle_property_diagonal_equal(1,ABDC)",
"parallelogram_property_diagonal_bisection(1,ABDC,E)",
"parallelogram_property_diagonal_bisection(1,BDCA,E)",
"line_addition(1,AE,ED)",
"line_addition(1,BE,EC)",
"isosceles_triangle_judgment_line_equal(1,EBD)",
"isosceles_triangle_property_angle_equal(1,EBD)"... | {"START": ["rectangle_property_diagonal_equal(1,ABDC)", "parallelogram_property_diagonal_bisection(1,ABDC,E)", "parallelogram_property_diagonal_bisection(1,BDCA,E)", "line_addition(1,AE,ED)", "line_addition(1,BE,EC)", "triangle_property_angle_sum(1,ABD)", "triangle_property_angle_sum(1,EBD)"], "isosceles_triangle_judgm... | |
346 | NaZhu_2023-03-12 | Geometry3k-353 | 4 | 如图所示，AB=5，AC=7，BC=6，DE=3，三角形ABC与三角形DEF是相似三角形。求三角形DEF的周长。 | As shown in the diagram, AB=5, AC=7, BC=6, DE=3, △ABC is similar to △DEF.. Find the perimeter of △DEF. | 346.png | [
"Shape(AB,BC,CA)",
"Shape(DE,EF,FD)"
] | [
"Equal(LengthOfLine(AB),5)",
"Equal(LengthOfLine(AC),7)",
"Equal(LengthOfLine(BC),6)",
"Equal(LengthOfLine(DE),3)",
"SimilarBetweenTriangle(ABC,DEF)"
] | [
"Equal(LengthOfLine(AB),5)",
"Equal(LengthOfLine(AC),7)",
"Equal(LengthOfLine(BC),6)",
"Equal(LengthOfLine(DE),3)"
] | Value(PerimeterOfTriangle(DEF)) | 54/5 | [
"triangle_perimeter_formula(1,DEF)",
"similar_triangle_property_line_ratio(1,ABC,DEF)",
"similar_triangle_property_line_ratio(1,CAB,FDE)",
"similar_triangle_property_line_ratio(1,BCA,EFD)"
] | {"START": ["triangle_perimeter_formula(1,DEF)", "similar_triangle_property_line_ratio(1,ABC,DEF)", "similar_triangle_property_line_ratio(1,CAB,FDE)", "similar_triangle_property_line_ratio(1,BCA,EFD)"]} | |
347 | NaZhu_2023-03-12 | Geometry3k-354 | 2 | 如图所示，AB=x，BC=y，CA=14，∠CAB=30°，AB垂直于CB。求x的值。 | As shown in the diagram, AB=x, BC=y, CA=14, ∠CAB=30°, AB⊥CB. Find the value of x. | 347.png | [
"Shape(CA,AB,BC)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(BC),y)",
"Equal(LengthOfLine(CA),14)",
"Equal(MeasureOfAngle(CAB),30)",
"PerpendicularBetweenLine(AB,CB)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(BC),y)",
"Equal(LengthOfLine(CA),14)",
"Equal(MeasureOfAngle(CAB),30)",
"PerpendicularBetweenLine(AB,CB)"
] | Value(x) | 7*sqrt(3) | [
"triangle_property_angle_sum(1,ABC)",
"sine_theorem(1,ABC)"
] | {"START": ["triangle_property_angle_sum(1,ABC)", "sine_theorem(1,ABC)"]} | |
348 | JiaZou_2023-04-09 | Geometry3k-355 | 3 | 如图所示，∠BCF=120°，∠DBC=5*y°，∠DFE=108°，∠FDB=2*x°，BD∥CE。求y的值。 | As shown in the diagram, ∠BCF=120°, ∠DBC=5*y°, ∠DFE=108°, ∠FDB=2*x°, BD is parallel to CE. Find the value of y. | 348.png | [
"Shape(BC,CF,FD,DB)",
"Collinear(CFE)"
] | [
"Equal(MeasureOfAngle(BCF),120)",
"Equal(MeasureOfAngle(DBC),5*y)",
"Equal(MeasureOfAngle(DFE),108)",
"Equal(MeasureOfAngle(FDB),2*x)",
"ParallelBetweenLine(BD,CE)"
] | [
"Equal(MeasureOfAngle(BCF),120)",
"Equal(MeasureOfAngle(DBC),5*y)",
"Equal(MeasureOfAngle(DFE),108)",
"Equal(MeasureOfAngle(FDB),2*x)",
"ParallelBetweenLine(BD,CE)"
] | Value(y) | 12 | [
"parallel_property_collinear_extend(3,EC,DB,F)",
"parallel_property_ipsilateral_internal_angle(1,FC,DB)",
"quadrilateral_property_angle_sum(1,BCFD)"
] | {"START": ["parallel_property_collinear_extend(3,EC,DB,F)", "quadrilateral_property_angle_sum(1,BCFD)"], "parallel_property_collinear_extend(3,EC,DB,F)": ["parallel_property_ipsilateral_internal_angle(1,FC,DB)"]} | |
349 | JiaZou_2023-04-09 | Geometry3k-356 | 7 | 如图所示，AF=10*z-40，CF=18-6*x，DA=2*z，DB=12*y-4，EB=4*y，EC=3*x，AD是⊙O的切线，⊙O的切线为AF，BD是⊙O的切线，BE是⊙O的切线，CE是圆O的切线，CF是⊙O的切线。求△ABC的周长。 | As shown in the diagram, AF=10*z-40, CF=18-6*x, DA=2*z, DB=12*y-4, EB=4*y, EC=3*x, AD is the tangent to circle G, the tangent to ⊙G is AF, the tangent to ⊙G is BD, BE is the tangent to circle G, the tangent to circle G is CE, the tangent to circle G is CF. Find the perimeter of △ABC. | 349.png | [
"Shape(GFD,FA,AD)",
"Shape(GDE,DB,BE)",
"Shape(GEF,EC,CF)",
"Shape(GFD,GDE,GEF)",
"Collinear(AFC)",
"Collinear(CEB)",
"Collinear(BDA)",
"Cocircular(G,FDE)"
] | [
"Equal(LengthOfLine(AF),10*z-40)",
"Equal(LengthOfLine(CF),18-6*x)",
"Equal(LengthOfLine(DA),2*z)",
"Equal(LengthOfLine(DB),12*y-4)",
"Equal(LengthOfLine(EB),4*y)",
"Equal(LengthOfLine(EC),3*x)",
"IsTangentOfCircle(AD,G)",
"IsTangentOfCircle(AF,G)",
"IsTangentOfCircle(BD,G)",
"IsTangentOfCircle(BE... | [
"Equal(LengthOfLine(AF),10*z-40)",
"Equal(LengthOfLine(CF),18-6*x)",
"Equal(LengthOfLine(DA),2*z)",
"Equal(LengthOfLine(DB),12*y-4)",
"Equal(LengthOfLine(EB),4*y)",
"Equal(LengthOfLine(EC),3*x)",
"IsTangentOfCircle(AD,G)",
"IsTangentOfCircle(AF,G)",
"IsTangentOfCircle(BD,G)",
"IsTangentOfCircle(BE... | Value(PerimeterOfTriangle(ABC)) | 36 | [
"tangent_of_circle_property_length_equal(1,AF,AD,G)",
"tangent_of_circle_property_length_equal(1,CE,CF,G)",
"tangent_of_circle_property_length_equal(1,BD,BE,G)",
"line_addition(1,AF,FC)",
"line_addition(1,AD,DB)",
"line_addition(1,BE,EC)",
"triangle_perimeter_formula(1,ABC)"
] | {"START": ["tangent_of_circle_property_length_equal(1,AF,AD,G)", "tangent_of_circle_property_length_equal(1,CE,CF,G)", "tangent_of_circle_property_length_equal(1,BD,BE,G)", "line_addition(1,AF,FC)", "line_addition(1,AD,DB)", "line_addition(1,BE,EC)", "triangle_perimeter_formula(1,ABC)"]} | |
350 | NaZhu_2023-03-12 | Geometry3k-357 | 6 | 如图所示，BC=y，BD=x，CD=2*sqrt(3)，DA=2，AC垂直于BC，BD垂直于CD。求x的值。 | As shown in the diagram, BC=y, BD=x, CD=2*sqrt(3), DA=2, AC⊥BC, BD⊥CD. Find the value of x. | 350.png | [
"Shape(CB,BD,DC)",
"Shape(CD,DA,AC)",
"Collinear(BDA)"
] | [
"Equal(LengthOfLine(BC),y)",
"Equal(LengthOfLine(BD),x)",
"Equal(LengthOfLine(CD),2*sqrt(3))",
"Equal(LengthOfLine(DA),2)",
"PerpendicularBetweenLine(AC,BC)",
"PerpendicularBetweenLine(BD,CD)"
] | [
"Equal(LengthOfLine(BC),y)",
"Equal(LengthOfLine(BD),x)",
"Equal(LengthOfLine(CD),2*sqrt(3))",
"Equal(LengthOfLine(DA),2)",
"PerpendicularBetweenLine(AC,BC)",
"PerpendicularBetweenLine(BD,CD)"
] | Value(x) | 6 | [
"adjacent_complementary_angle(1,BDC,CDA)",
"angle_addition(1,ACD,DCB)",
"triangle_property_angle_sum(1,CBD)",
"similar_triangle_judgment_aa(1,CBD,ACD)",
"similar_triangle_property_line_ratio(1,CBD,ACD)",
"similar_triangle_property_line_ratio(1,BDC,CDA)"
] | {"START": ["adjacent_complementary_angle(1,BDC,CDA)", "angle_addition(1,ACD,DCB)", "triangle_property_angle_sum(1,CBD)"], "adjacent_complementary_angle(1,BDC,CDA)": ["similar_triangle_judgment_aa(1,CBD,ACD)"], "angle_addition(1,ACD,DCB)": ["similar_triangle_judgment_aa(1,CBD,ACD)"], "similar_triangle_judgment_aa(1,CBD,... | |
351 | NaZhu_2023-03-12 | Geometry3k-358 | 2 | 如图所示，∠CEB=40°，∠EFD=2*x°，∠EFD=∠FDE。求x的值。 | As shown in the diagram, ∠CEB=40°, ∠EFD=2*x°, ∠EFD=∠FDE. Find the value of x. | 351.png | [
"Shape(FD,DE,EF)",
"Shape(EB,BC,CE)",
"Collinear(DEC)",
"Collinear(FEB)"
] | [
"Equal(MeasureOfAngle(CEB),40)",
"Equal(MeasureOfAngle(EFD),2*x)",
"Equal(MeasureOfAngle(EFD),MeasureOfAngle(FDE))"
] | [
"Equal(MeasureOfAngle(CEB),40)",
"Equal(MeasureOfAngle(EFD),2*x)",
"Equal(MeasureOfAngle(EFD),MeasureOfAngle(FDE))"
] | Value(x) | 35 | [
"vertical_angle(1,DEF,CEB)",
"triangle_property_angle_sum(1,FDE)"
] | {"START": ["vertical_angle(1,DEF,CEB)", "triangle_property_angle_sum(1,FDE)"]} | |
352 | JiaZou_2023-04-09 | Geometry3k-359 | 1 | 如图所示，∠BGE=75°，∠FCG=100°。求∠EGD的大小。 | As shown in the diagram, ∠BGE=75°, ∠FCG=100°. Find the measure of ∠EGD. | 352.png | [
"Shape(DG,GC,CD)",
"Shape(DC,CA)",
"Shape(AC,CF)",
"Shape(FC,CG)",
"Shape(CG,GB)",
"Shape(BG,GE)",
"Shape(EG,GD)",
"Collinear(DCF)",
"Collinear(DGB)",
"Collinear(ACGE)"
] | [
"Equal(MeasureOfAngle(BGE),75)",
"Equal(MeasureOfAngle(FCG),100)"
] | [
"Equal(MeasureOfAngle(BGE),75)",
"Equal(MeasureOfAngle(FCG),100)"
] | Value(MeasureOfAngle(EGD)) | 105 | [
"adjacent_complementary_angle(1,BGE,EGD)"
] | {"START": ["adjacent_complementary_angle(1,BGE,EGD)"]} | |
353 | JiaZou_2023-04-09 | Geometry3k-360 | 10 | 如图所示，FD=25，圆O的圆心为O，EH是圆O的直径，圆O的切线为DH，圆O的切线为FE，EF垂直于NF，HA垂直于KA，KG⊥EG，ND垂直于HD。求圆O的周长。 | As shown in the diagram, FD=25, the center of ⊙O is O, the diameter of ⊙O is EH, the tangent to circle O is DH, the tangent to ⊙O is FE, EF⊥NF, HA⊥KA, KG is perpendicular to EG, ND is perpendicular to HD. Find the circumference of the circle O. | 353.png | [
"Shape(KG,GE,OKE)",
"Shape(EF,FN,OEN)",
"Shape(ND,DH,ONH)",
"Shape(HA,AK,OHK)",
"Shape(EO,OH,OHK,OKE)",
"Shape(HO,OE,OEN,ONH)",
"Shape(OHK,OKE,OEN,ONH)",
"Collinear(GKA)",
"Collinear(GEF)",
"Collinear(FND)",
"Collinear(AHD)",
"Collinear(EOH)",
"Cocircular(O,KENH)"
] | [
"Equal(LengthOfLine(FD),25)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(EH,O)",
"IsTangentOfCircle(DH,O)",
"IsTangentOfCircle(FE,O)",
"PerpendicularBetweenLine(EF,NF)",
"PerpendicularBetweenLine(HA,KA)",
"PerpendicularBetweenLine(KG,EG)",
"PerpendicularBetweenLine(ND,HD)"
] | [
"Equal(LengthOfLine(FD),25)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(EH,O)",
"IsTangentOfCircle(DH,O)",
"IsTangentOfCircle(FE,O)",
"PerpendicularBetweenLine(EF,NF)",
"PerpendicularBetweenLine(HA,KA)",
"PerpendicularBetweenLine(KG,EG)",
"PerpendicularBetweenLine(ND,HD)"
] | Value(PerimeterOfCircle(O)) | 25*pi | [
"tangent_of_circle_property_perpendicular(1,FE,O,O)",
"tangent_of_circle_property_perpendicular(2,DH,O,O)",
"parallel_judgment_ipsilateral_internal_angle(1,EO,FD)",
"parallel_property_collinear_extend(2,EO,FD,H)",
"parallel_judgment_ipsilateral_internal_angle(1,HD,EF)",
"parallelogram_judgment_parallel_an... | {"START": ["tangent_of_circle_property_perpendicular(1,FE,O,O)", "tangent_of_circle_property_perpendicular(2,DH,O,O)", "diameter_of_circle_property_length_equal(1,EH,O)", "circle_property_length_of_radius_and_diameter(1,O)", "circle_perimeter_formula(1,O)"], "parallel_judgment_ipsilateral_internal_angle(1,EO,FD)": ["pa... | |
354 | NaZhu_2023-03-12 | Geometry3k-361 | 8 | 如图所示，AC=x，AD=8，CB=y，DB=25，DC=z，AC垂直于BC，BD垂直于CD。求x的值。 | As shown in the diagram, AC=x, AD=8, CB=y, DB=25, DC=z, AC is perpendicular to BC, BD is perpendicular to CD. Find the value of x. | 354.png | [
"Shape(AC,CD,DA)",
"Shape(DC,CB,BD)",
"Collinear(ADB)"
] | [
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(AD),8)",
"Equal(LengthOfLine(CB),y)",
"Equal(LengthOfLine(DB),25)",
"Equal(LengthOfLine(DC),z)",
"PerpendicularBetweenLine(AC,BC)",
"PerpendicularBetweenLine(BD,CD)"
] | [
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(AD),8)",
"Equal(LengthOfLine(CB),y)",
"Equal(LengthOfLine(DB),25)",
"Equal(LengthOfLine(DC),z)",
"PerpendicularBetweenLine(AC,BC)",
"PerpendicularBetweenLine(BD,CD)"
] | Value(x) | 2*sqrt(66) | [
"adjacent_complementary_angle(1,BDC,CDA)",
"right_triangle_judgment_angle(1,CDA)",
"triangle_property_angle_sum(1,ACD)",
"triangle_property_angle_sum(1,ACB)",
"similar_triangle_judgment_aa(1,ACD,CBD)",
"similar_triangle_property_line_ratio(1,CDA,BDC)",
"similar_triangle_property_line_ratio(1,ACD,CBD)",
... | {"START": ["adjacent_complementary_angle(1,BDC,CDA)", "triangle_property_angle_sum(1,ACD)", "triangle_property_angle_sum(1,ACB)"], "adjacent_complementary_angle(1,BDC,CDA)": ["right_triangle_judgment_angle(1,CDA)", "similar_triangle_judgment_aa(1,ACD,CBD)"], "right_triangle_judgment_angle(1,CDA)": ["right_triangle_prop... | |
355 | JiaZou_2023-04-09 | Geometry3k-362 | 10 | 如图所示，∠BCJ=105°，∠CIE=x°，∠KEF=125°，BD∥FH。求x的值。 | As shown in the diagram, ∠BCJ=105°, ∠CIE=x°, ∠KEF=125°, BD is parallel to FH. Find the value of x. | 355.png | [
"Shape(BC,CJ)",
"Shape(JC,CD)",
"Shape(CI,IE,EC)",
"Shape(EI,IC)",
"Shape(FE,EI)",
"Shape(CE,EH)",
"Shape(KE,EF)",
"Shape(HE,EK)",
"Shape(IC,CB)",
"Shape(DC,CE)",
"Collinear(BCD)",
"Collinear(JCI)",
"Collinear(IEK)",
"Collinear(FEH)"
] | [
"Equal(MeasureOfAngle(BCJ),105)",
"Equal(MeasureOfAngle(CIE),x)",
"Equal(MeasureOfAngle(KEF),125)",
"ParallelBetweenLine(BD,FH)"
] | [
"Equal(MeasureOfAngle(BCJ),105)",
"Equal(MeasureOfAngle(CIE),x)",
"Equal(MeasureOfAngle(KEF),125)",
"ParallelBetweenLine(BD,FH)"
] | Value(x) | 130 | [
"adjacent_complementary_angle(1,BCJ,JCD)",
"adjacent_complementary_angle(1,HEK,KEF)",
"adjacent_complementary_angle(1,JCD,DCI)",
"adjacent_complementary_angle(1,IEH,HEK)",
"parallel_property_collinear_extend(3,BD,FH,C)",
"parallel_property_collinear_extend(3,HF,DC,E)",
"parallel_property_ipsilateral_int... | {"START": ["adjacent_complementary_angle(1,BCJ,JCD)", "adjacent_complementary_angle(1,HEK,KEF)", "adjacent_complementary_angle(1,JCD,DCI)", "adjacent_complementary_angle(1,IEH,HEK)", "parallel_property_collinear_extend(3,BD,FH,C)", "angle_addition(1,DCE,ECI)", "angle_addition(1,IEC,CEH)", "triangle_property_angle_sum(1... | |
356 | NaZhu_2023-03-12 | Geometry3k-363 | 1 | 如图所示，AB=26，AC=24，CB=10，BC垂直于AC。求sin(AB)的值。 | As shown in the diagram, AB=26, AC=24, CB=10, BC is perpendicular to AC. Find the value of sin(AB). | 356.png | [
"Shape(CA,AB,BC)"
] | [
"Equal(LengthOfLine(AB),26)",
"Equal(LengthOfLine(AC),24)",
"Equal(LengthOfLine(CB),10)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),26)",
"Equal(LengthOfLine(AC),24)",
"Equal(LengthOfLine(CB),10)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(Sin(MeasureOfAngle(ABC))) | 12/13 | [
"cosine_theorem(1,BCA)"
] | {"START": ["cosine_theorem(1,BCA)"]} | |
357 | JiaZou_2023-04-09 | Geometry3k-364 | 5 | 如图所示，AB=15，PB=12，∠DBA=24°，ADCB是菱形。求∠ACB的大小。 | As shown in the diagram, AB=15, PB=12, ∠DBA=24°, ADCB is a rhombus. Find the measure of ∠ACB. | 357.png | [
"Shape(AD,DP,PA)",
"Shape(PD,DC,CP)",
"Shape(PC,CB,BP)",
"Shape(AP,PB,BA)",
"Collinear(DPB)",
"Collinear(APC)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(PB),12)",
"Equal(MeasureOfAngle(DBA),24)",
"Rhombus(ADCB)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(PB),12)",
"Equal(MeasureOfAngle(DBA),24)"
] | Value(MeasureOfAngle(ACB)) | 66 | [
"kite_property_diagonal_perpendicular_bisection(1,BADC,P)",
"isosceles_triangle_judgment_line_equal(1,BAC)",
"altitude_of_triangle_judgment(1,BP,BAC)",
"isosceles_triangle_property_line_coincidence(1,BAC,P)",
"triangle_property_angle_sum(1,BPC)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,BADC,P)", "isosceles_triangle_judgment_line_equal(1,BAC)", "triangle_property_angle_sum(1,BPC)"], "altitude_of_triangle_judgment(1,BP,BAC)": ["isosceles_triangle_property_line_coincidence(1,BAC,P)"], "isosceles_triangle_judgment_line_equal(1,BAC)": ["isoscele... | |
358 | NaZhu_2023-03-12 | Geometry3k-365 | 2 | 如图所示，BA=h，CA=3，∠BAC=45°，AC⊥BC。求h的值。 | As shown in the diagram, BA=h, CA=3, ∠BAC=45°, AC⊥BC. Find the value of h. | 358.png | [
"Shape(CB,BA,AC)"
] | [
"Equal(LengthOfLine(BA),h)",
"Equal(LengthOfLine(CA),3)",
"Equal(MeasureOfAngle(BAC),45)",
"PerpendicularBetweenLine(AC,BC)"
] | [
"Equal(LengthOfLine(BA),h)",
"Equal(LengthOfLine(CA),3)",
"Equal(MeasureOfAngle(BAC),45)",
"PerpendicularBetweenLine(AC,BC)"
] | Value(h) | 3*sqrt(2) | [
"triangle_property_angle_sum(1,ACB)",
"sine_theorem(1,ACB)"
] | {"START": ["triangle_property_angle_sum(1,ACB)", "sine_theorem(1,ACB)"]} | |
359 | NaZhu_2023-03-12 | Geometry3k-366 | 5 | 如图所示，AB=2，AE=x-1，CD=5，ED=x+5，BA平行于DC。求直线AE的长度。 | As shown in the diagram, AB=2, AE=x-1, CD=5, ED=x+5, BA∥DC. Find the length of line AE. | 359.png | [
"Shape(AB,BE,EA)",
"Shape(ED,DC,CE)",
"Collinear(BEC)",
"Collinear(AED)"
] | [
"Equal(LengthOfLine(AB),2)",
"Equal(LengthOfLine(AE),x-1)",
"Equal(LengthOfLine(CD),5)",
"Equal(LengthOfLine(ED),x+5)",
"ParallelBetweenLine(BA,DC)"
] | [
"Equal(LengthOfLine(AB),2)",
"Equal(LengthOfLine(AE),x-1)",
"Equal(LengthOfLine(CD),5)",
"Equal(LengthOfLine(ED),x+5)",
"ParallelBetweenLine(BA,DC)"
] | Value(LengthOfLine(AE)) | 4 | [
"vertical_angle(1,BEA,CED)",
"parallel_property_alternate_interior_angle(1,BA,DC)",
"similar_triangle_judgment_aa(1,ABE,DCE)",
"similar_triangle_property_line_ratio(1,EAB,EDC)",
"similar_triangle_property_line_ratio(1,BEA,CED)"
] | {"START": ["vertical_angle(1,BEA,CED)", "parallel_property_alternate_interior_angle(1,BA,DC)"], "parallel_property_alternate_interior_angle(1,BA,DC)": ["similar_triangle_judgment_aa(1,ABE,DCE)"], "similar_triangle_judgment_aa(1,ABE,DCE)": ["similar_triangle_property_line_ratio(1,BEA,CED)", "similar_triangle_property_li... | |
360 | NaZhu_2023-03-12 | Geometry3k-367 | 1 | 如图所示，AB=26，AC=24，CB=10，BC⊥AC。求tan(AB)的值。 | As shown in the diagram, AB=26, AC=24, CB=10, BC is perpendicular to AC. Find the value of tan(AB). | 360.png | [
"Shape(CA,AB,BC)"
] | [
"Equal(LengthOfLine(AB),26)",
"Equal(LengthOfLine(AC),24)",
"Equal(LengthOfLine(CB),10)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),26)",
"Equal(LengthOfLine(AC),24)",
"Equal(LengthOfLine(CB),10)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(Tan(MeasureOfAngle(ABC))) | 12/5 | [
"cosine_theorem(1,BCA)"
] | {"START": ["cosine_theorem(1,BCA)"]} | |
361 | JiaZou_2023-04-09 | Geometry3k-368 | 8 | 如图所示，CB=23，DE=5，∠EDB=45°，AC和DB是▱CADB的一组对边，AE⊥BE。求CADB的周长。 | As shown in the diagram, CB=23, DE=5, ∠EDB=45°, CADB is a ▱, AE is perpendicular to BE. Find the perimeter of quadrilateral CADB. | 361.png | [
"Shape(CA,AE,EB,BC)",
"Shape(BE,ED,DB)",
"Collinear(AED)"
] | [
"Equal(LengthOfLine(CB),23)",
"Equal(LengthOfLine(DE),5)",
"Equal(MeasureOfAngle(EDB),45)",
"Parallelogram(CADB)",
"PerpendicularBetweenLine(AE,BE)"
] | [
"Equal(LengthOfLine(CB),23)",
"Equal(LengthOfLine(DE),5)",
"Equal(MeasureOfAngle(EDB),45)",
"PerpendicularBetweenLine(AE,BE)"
] | Value(PerimeterOfQuadrilateral(CADB)) | 10*sqrt(2)+46 | [
"altitude_of_quadrilateral_judgment_right_vertex(1,BE,CADB)",
"triangle_property_angle_sum(1,EDB)",
"isosceles_triangle_judgment_angle_equal(1,EDB)",
"right_triangle_judgment_angle(1,BED)",
"right_triangle_property_pythagorean(1,BED)",
"parallelogram_property_opposite_line_equal(1,ADBC)",
"parallelogram... | {"START": ["altitude_of_quadrilateral_judgment_right_vertex(1,BE,CADB)", "triangle_property_angle_sum(1,EDB)", "parallelogram_property_opposite_line_equal(1,ADBC)", "parallelogram_property_opposite_line_equal(1,CADB)", "quadrilateral_perimeter_formula(1,CADB)"], "altitude_of_quadrilateral_judgment_right_vertex(1,BE,CAD... | |
362 | JiaZou_2023-04-09 | Geometry3k-369 | 0 | 如图所示，∠BFC=6*x°，∠CFD=3*x°，∠EFA=12*y-10°，EF⊥AF。求y的值。 | As shown in the diagram, ∠BFC=6*x°, ∠CFD=3*x°, ∠EFA=12*y-10°, EF⊥AF. Find the value of y. | 362.png | [
"Shape(AF,FB)",
"Shape(BF,FC)",
"Shape(CF,FD)",
"Shape(DF,FE)",
"Shape(EF,FA)",
"Collinear(AFD)",
"Collinear(BFE)"
] | [
"Equal(MeasureOfAngle(BFC),6*x)",
"Equal(MeasureOfAngle(CFD),3*x)",
"Equal(MeasureOfAngle(EFA),12*y-10)",
"PerpendicularBetweenLine(EF,AF)"
] | [
"Equal(MeasureOfAngle(BFC),6*x)",
"Equal(MeasureOfAngle(CFD),3*x)",
"Equal(MeasureOfAngle(EFA),12*y-10)",
"PerpendicularBetweenLine(EF,AF)"
] | Value(y) | 25/3 | [] | {"START": []} | |
363 | JiaZou_2023-04-09 | Geometry3k-370 | 1 | 如图所示，MN=3*x-4，NQ=15.4，PN=2*y+5，PQ=11.1，RM=17.9，RP=20，RQ=3*z-3，∠MRQ=38°，∠NQP=83°，∠QNM=33°，RM和PN是▱MRPN的一组对边。求∠RQM的大小。 | As shown in the diagram, MN=3*x-4, NQ=15.4, PN=2*y+5, PQ=11.1, RM=17.9, RP=20, RQ=3*z-3, ∠MRQ=38°, ∠NQP=83°, ∠QNM=33°, quadrilateral MRPN is a parallelogram. Find the measure of ∠RQM. | 363.png | [
"Shape(MQ,QN,NM)",
"Shape(MR,RQ,QM)",
"Shape(QR,RP,PQ)",
"Shape(QP,ON,NQ)",
"Collinear(MQP)",
"Collinear(RQN)"
] | [
"Equal(LengthOfLine(MN),3*x-4)",
"Equal(LengthOfLine(NQ),15.4)",
"Equal(LengthOfLine(PN),2*y+5)",
"Equal(LengthOfLine(PQ),11.1)",
"Equal(LengthOfLine(RM),17.9)",
"Equal(LengthOfLine(RP),20)",
"Equal(LengthOfLine(RQ),3*z-3)",
"Equal(MeasureOfAngle(MRQ),38)",
"Equal(MeasureOfAngle(NQP),83)",
"Equal(... | [
"Equal(LengthOfLine(MN),3*x-4)",
"Equal(LengthOfLine(MN),3*x-4)",
"Equal(LengthOfLine(NQ),15.4)",
"Equal(LengthOfLine(PN),2*y+5)",
"Equal(LengthOfLine(PQ),11.1)",
"Equal(LengthOfLine(RM),17.9)",
"Equal(LengthOfLine(RP),20)",
"Equal(LengthOfLine(RQ),3*z-3)",
"Equal(MeasureOfAngle(MRQ),38)",
"Equal(... | Value(MeasureOfAngle(RQM)) | 83 | [
"vertical_angle(1,RQM,NQP)"
] | {"START": ["vertical_angle(1,RQM,NQP)"]} | |
364 | JiaZou_2023-04-09 | Geometry3k-371 | 6 | 如图所示，AG=1/5*x+3，CJ=2*y+1，CJ=JE，EG=4*x-35，JE=5*y-8，AC∥GJ。求x的值。 | As shown in the diagram, AG=1/5*x+3, CJ=2*y+1, CJ=JE, EG=4*x-35, JE=5*y-8, AC is parallel to GJ. Find the value of x. | 364.png | [
"Shape(AG,GJ,JC,CA)",
"Shape(GE,EJ,JG)",
"Collinear(AGE)",
"Collinear(EJC)"
] | [
"Equal(LengthOfLine(AG),1/5*x+3)",
"Equal(LengthOfLine(CJ),2*y+1)",
"Equal(LengthOfLine(CJ),LengthOfLine(JE))",
"Equal(LengthOfLine(EG),4*x-35)",
"Equal(LengthOfLine(JE),5*y-8)",
"ParallelBetweenLine(AC,GJ)"
] | [
"Equal(LengthOfLine(AG),1/5*x+3)",
"Equal(LengthOfLine(CJ),2*y+1)",
"Equal(LengthOfLine(CJ),LengthOfLine(JE))",
"Equal(LengthOfLine(EG),4*x-35)",
"Equal(LengthOfLine(JE),5*y-8)",
"ParallelBetweenLine(AC,GJ)"
] | Value(x) | 10 | [
"line_addition(1,AG,GE)",
"line_addition(1,EJ,JC)",
"parallel_property_corresponding_angle(2,AC,GJ,E)",
"similar_triangle_judgment_aa(1,JGE,CAE)",
"similar_triangle_property_line_ratio(1,GEJ,AEC)",
"similar_triangle_property_line_ratio(1,JGE,CAE)"
] | {"START": ["line_addition(1,AG,GE)", "line_addition(1,EJ,JC)", "parallel_property_corresponding_angle(2,AC,GJ,E)"], "parallel_property_corresponding_angle(2,AC,GJ,E)": ["similar_triangle_judgment_aa(1,JGE,CAE)"], "similar_triangle_judgment_aa(1,JGE,CAE)": ["similar_triangle_property_line_ratio(1,JGE,CAE)", "similar_tri... | |
365 | NaZhu_2023-03-12 | Geometry3k-372 | 4 | 如图所示，AB=y，AD=z，BD=4，CB=x，CD=10，AB⊥CB，DC垂直于AC。求z的值。 | As shown in the diagram, AB=y, AD=z, BD=4, CB=x, CD=10, AB⊥CB, DC is perpendicular to AC. Find the value of z. | 365.png | [
"Shape(CA,AB,BC)",
"Shape(CB,BD,DC)",
"Collinear(ABD)"
] | [
"Equal(LengthOfLine(AB),y)",
"Equal(LengthOfLine(AD),z)",
"Equal(LengthOfLine(BD),4)",
"Equal(LengthOfLine(CB),x)",
"Equal(LengthOfLine(CD),10)",
"PerpendicularBetweenLine(AB,CB)",
"PerpendicularBetweenLine(DC,AC)"
] | [
"Equal(LengthOfLine(AB),y)",
"Equal(LengthOfLine(AD),z)",
"Equal(LengthOfLine(BD),4)",
"Equal(LengthOfLine(CB),x)",
"Equal(LengthOfLine(CD),10)",
"PerpendicularBetweenLine(AB,CB)",
"PerpendicularBetweenLine(DC,AC)"
] | Value(z) | 25 | [
"adjacent_complementary_angle(1,ABC,CBD)",
"mirror_similar_triangle_judgment_aa(1,CBD,ADC)",
"mirror_similar_triangle_property_line_ratio(1,CBD,ADC)",
"mirror_similar_triangle_property_line_ratio(1,BDC,CAD)"
] | {"START": ["adjacent_complementary_angle(1,ABC,CBD)"], "adjacent_complementary_angle(1,ABC,CBD)": ["mirror_similar_triangle_judgment_aa(1,CBD,ADC)"], "mirror_similar_triangle_judgment_aa(1,CBD,ADC)": ["mirror_similar_triangle_property_line_ratio(1,CBD,ADC)", "mirror_similar_triangle_property_line_ratio(1,BDC,CAD)"]} | |
366 | NaZhu_2023-03-12 | Geometry3k-373 | 2 | 如图所示，AC=20，BA=48，BC=x，CA⊥BA。求x的值。 | As shown in the diagram, AC=20, BA=48, BC=x, CA⊥BA. Find the value of x. | 366.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AC),20)",
"Equal(LengthOfLine(BA),48)",
"Equal(LengthOfLine(BC),x)",
"PerpendicularBetweenLine(CA,BA)"
] | [
"Equal(LengthOfLine(AC),20)",
"Equal(LengthOfLine(BA),48)",
"Equal(LengthOfLine(BC),x)",
"PerpendicularBetweenLine(CA,BA)"
] | Value(x) | 52 | [
"right_triangle_judgment_angle(1,CAB)",
"right_triangle_property_pythagorean(1,CAB)"
] | {"START": ["right_triangle_judgment_angle(1,CAB)"], "right_triangle_judgment_angle(1,CAB)": ["right_triangle_property_pythagorean(1,CAB)"]} | |
367 | NaZhu_2023-03-12 | Geometry3k-374 | 1 | 如图所示，BA=18，BC=x，∠BAC=45°，AC垂直于BC。求x的值。 | As shown in the diagram, BA=18, BC=x, ∠BAC=45°, AC is perpendicular to BC. Find the value of x. | 367.png | [
"Shape(BA,AC,CB)"
] | [
"Equal(LengthOfLine(BA),18)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(BAC),45)",
"PerpendicularBetweenLine(AC,BC)"
] | [
"Equal(LengthOfLine(BA),18)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(BAC),45)",
"PerpendicularBetweenLine(AC,BC)"
] | Value(x) | 9*sqrt(2) | [
"sine_theorem(1,BAC)"
] | {"START": ["sine_theorem(1,BAC)"]} | |
368 | NaZhu_2023-03-12 | Geometry3k-375 | 2 | 如图所示，JM=MK，∠JLM=6*x+8°，∠MLK=9*x-4°，LK垂直于MK，MJ⊥LJ。求∠MLK的大小。 | As shown in the diagram, JM=MK, ∠JLM=6*x+8°, ∠MLK=9*x-4°, LK is perpendicular to MK, MJ⊥LJ. Find the measure of ∠MLK. | 368.png | [
"Shape(JL,LM,MJ)",
"Shape(ML,LK,KM)"
] | [
"Equal(LengthOfLine(JM),LengthOfLine(MK))",
"Equal(MeasureOfAngle(JLM),6*x+8)",
"Equal(MeasureOfAngle(MLK),9*x-4)",
"PerpendicularBetweenLine(LK,MK)",
"PerpendicularBetweenLine(MJ,LJ)"
] | [
"Equal(LengthOfLine(JM),LengthOfLine(MK))",
"Equal(MeasureOfAngle(JLM),6*x+8)",
"Equal(MeasureOfAngle(MLK),9*x-4)",
"PerpendicularBetweenLine(LK,MK)",
"PerpendicularBetweenLine(MJ,LJ)"
] | Value(MeasureOfAngle(MLK)) | 32 | [
"mirror_congruent_triangle_judgment_hl(1,LKM,LMJ)",
"mirror_congruent_triangle_property_angle_equal(1,LKM,LMJ)"
] | {"START": ["mirror_congruent_triangle_judgment_hl(1,LKM,LMJ)"], "mirror_congruent_triangle_judgment_hl(1,LKM,LMJ)": ["mirror_congruent_triangle_property_angle_equal(1,LKM,LMJ)"]} | |
369 | JiaZou_2023-04-09 | Geometry3k-376 | 1 | 如图所示，PQ=6，PS=x，RP=15，TP=4。求x的值。 | As shown in the diagram, PQ=6, PS=x, RP=15, TP=4. Find the value of x. | 369.png | [
"Shape(TP,PQ,AQT)",
"Shape(SP,PT,ATS)",
"Shape(RP,PS,ASR)",
"Shape(QP,PR,ARQ)",
"Collinear(QPS)",
"Collinear(TPR)",
"Cocircular(A,RQTS)"
] | [
"Equal(LengthOfLine(PQ),6)",
"Equal(LengthOfLine(PS),x)",
"Equal(LengthOfLine(RP),15)",
"Equal(LengthOfLine(TP),4)"
] | [
"Equal(LengthOfLine(PQ),6)",
"Equal(LengthOfLine(PS),x)",
"Equal(LengthOfLine(RP),15)",
"Equal(LengthOfLine(TP),4)"
] | Value(x) | 10 | [
"circle_property_circular_power_chord_and_chord(1,QPS,RPT,A)"
] | {"START": ["circle_property_circular_power_chord_and_chord(1,QPS,RPT,A)"]} | |
370 | NaZhu_2023-03-12 | Geometry3k-377 | 0 | 如图所示，WY=6*x+3，XW=9*x，XW=LengthOfLine(WY),LengthOfLine(XY)，XY=4*x+5。求直线XW的长度。 | As shown in the diagram, WY=6*x+3, XW=9*x, XW=LengthOfLine(WY),LengthOfLine(XY), XY=4*x+5. Find the length of line XW. | 370.png | [
"Shape(XW,WY,YX)"
] | [
"Equal(LengthOfLine(WY),6*x+3)",
"Equal(LengthOfLine(XW),9*x)",
"Equal(LengthOfLine(XW),LengthOfLine(WY),LengthOfLine(XY))",
"Equal(LengthOfLine(XY),4*x+5)"
] | [
"Equal(LengthOfLine(WY),6*x+3)",
"Equal(LengthOfLine(XW),9*x)",
"Equal(LengthOfLine(XW),LengthOfLine(WY),LengthOfLine(XY))",
"Equal(LengthOfLine(XY),4*x+5)"
] | Value(LengthOfLine(XW)) | 9 | [] | {"START": []} | |
371 | JiaZou_2023-04-09 | Geometry3k-378 | 2 | 如图所示，AD=7，AE=14，BD=x，AE是圆O的切线。求x的值。 | As shown in the diagram, AD=7, AE=14, BD=x, the tangent to ⊙O is AE. Find the value of x. | 371.png | [
"Shape(EA,AD,OED)",
"Shape(OED,DB,OBE)",
"Shape(BD,ODB)",
"Collinear(ADB)",
"Collinear(AEC)",
"Cocircular(O,EDB)"
] | [
"Equal(LengthOfLine(AD),7)",
"Equal(LengthOfLine(AE),14)",
"Equal(LengthOfLine(BD),x)",
"IsTangentOfCircle(AE,O)"
] | [
"Equal(LengthOfLine(AD),7)",
"Equal(LengthOfLine(AE),14)",
"Equal(LengthOfLine(BD),x)",
"IsTangentOfCircle(AE,O)"
] | Value(x) | 21 | [
"line_addition(1,AD,DB)",
"circle_property_circular_power_tangent_and_segment_line(1,AE,ADB,O)"
] | {"START": ["line_addition(1,AD,DB)", "circle_property_circular_power_tangent_and_segment_line(1,AE,ADB,O)"]} | |
372 | JiaZou_2023-04-09 | Geometry3k-379 | 4 | 如图所示，AB=26，AD=12，DG=9/2，EF=8，GF=14，∠AGF=108°，四边形AGFE相似于四边形ADCB。求AGFE的周长。 | As shown in the diagram, AB=26, AD=12, DG=9/2, EF=8, GF=14, ∠AGF=108°, AGFE is similar to ADCB. Find the perimeter of quadrilateral AGFE. | 372.png | [
"Shape(AG,GF,FE,EA)",
"Shape(GD,DC,CB,BE,EF,FG)",
"Collinear(AEB)",
"Collinear(AGD)"
] | [
"Equal(LengthOfLine(AB),26)",
"Equal(LengthOfLine(AD),12)",
"Equal(LengthOfLine(DG),9/2)",
"Equal(LengthOfLine(EF),8)",
"Equal(LengthOfLine(GF),14)",
"Equal(MeasureOfAngle(AGF),108)",
"SimilarBetweenQuadrilateral(AGFE,ADCB)"
] | [] | Value(PerimeterOfQuadrilateral(AGFE)) | 183/4 | [
"line_addition(1,AG,GD)",
"similar_quadrilateral_property_line_ratio(1,AGFE,ADCB)",
"similar_quadrilateral_property_line_ratio(1,EAGF,BADC)",
"quadrilateral_perimeter_formula(1,AGFE)"
] | {"START": ["line_addition(1,AG,GD)", "similar_quadrilateral_property_line_ratio(1,AGFE,ADCB)", "similar_quadrilateral_property_line_ratio(1,EAGF,BADC)", "quadrilateral_perimeter_formula(1,AGFE)"]} | |
373 | JiaZou_2023-04-09 | Geometry3k-380 | 7 | 如图所示，∠YZX=56°，四边形WYZX是菱形。求∠TWY的大小。 | As shown in the diagram, ∠YZX=56°, quadrilateral WYZX is a rhombus. Find the measure of ∠TWY. | 373.png | [
"Shape(WY,YT,TW)",
"Shape(TY,YZ,ZT)",
"Shape(XW,WT,TX)",
"Shape(TZ,ZX,XT)",
"Collinear(WTZ)",
"Collinear(XTY)"
] | [
"Equal(MeasureOfAngle(YZX),56)",
"Rhombus(WYZX)"
] | [
"Equal(MeasureOfAngle(YZX),56)"
] | Value(MeasureOfAngle(TWY)) | 28 | [
"kite_property_diagonal_perpendicular_bisection(1,ZXWY,T)",
"isosceles_triangle_judgment_line_equal(1,ZXY)",
"altitude_of_triangle_judgment(1,ZT,ZXY)",
"isosceles_triangle_property_line_coincidence(1,ZXY,T)",
"angle_addition(1,YZT,TZX)",
"isosceles_triangle_judgment_line_equal(1,YZW)",
"isosceles_triang... | {"START": ["kite_property_diagonal_perpendicular_bisection(1,ZXWY,T)", "isosceles_triangle_judgment_line_equal(1,ZXY)", "angle_addition(1,YZT,TZX)", "isosceles_triangle_judgment_line_equal(1,YZW)"], "altitude_of_triangle_judgment(1,ZT,ZXY)": ["isosceles_triangle_property_line_coincidence(1,ZXY,T)"], "isosceles_triangle... | |
374 | NaZhu_2023-03-12 | Geometry3k-381 | 6 | 如图所示，BA=5，CB=8，CD=12，NE=5，BA垂直于DA，NE垂直于BE。求△NCB的面积与△BCD的面积之和。 | As shown in the diagram, BA=5, CB=8, CD=12, NE=5, BA is perpendicular to DA, NE⊥BE. Find the sum of the area of △NCB and the area of triangle BCD. | 374.png | [
"Shape(NE,EB,BN)",
"Shape(NC,CE,EN)",
"Shape(BC,CA,AB)",
"Shape(BA,AD,DB)",
"Collinear(CEB)",
"Collinear(CAD)"
] | [
"Equal(LengthOfLine(BA),5)",
"Equal(LengthOfLine(CB),8)",
"Equal(LengthOfLine(CD),12)",
"Equal(LengthOfLine(NE),5)",
"PerpendicularBetweenLine(BA,DA)",
"PerpendicularBetweenLine(NE,BE)"
] | [
"Equal(LengthOfLine(BA),5)",
"Equal(LengthOfLine(CB),8)",
"Equal(LengthOfLine(CD),12)",
"Equal(LengthOfLine(NE),5)",
"PerpendicularBetweenLine(BA,DA)",
"PerpendicularBetweenLine(NE,BE)"
] | Value(Add(AreaOfTriangle(NCB),AreaOfTriangle(BCD))) | 50 | [
"adjacent_complementary_angle(1,CEN,NEB)",
"adjacent_complementary_angle(1,CAB,BAD)",
"altitude_of_triangle_judgment(1,NE,NCB)",
"altitude_of_triangle_judgment(1,BA,BCD)",
"triangle_area_formula_common(1,NCB)",
"triangle_area_formula_common(1,BCD)"
] | {"START": ["adjacent_complementary_angle(1,CEN,NEB)", "adjacent_complementary_angle(1,CAB,BAD)", "triangle_area_formula_common(1,NCB)", "triangle_area_formula_common(1,BCD)"], "adjacent_complementary_angle(1,CAB,BAD)": ["altitude_of_triangle_judgment(1,BA,BCD)"], "adjacent_complementary_angle(1,CEN,NEB)": ["altitude_of... | |
375 | NaZhu_2023-03-12 | Geometry3k-382 | 2 | 如图所示，BA=32，BC=x，CA=y，∠BAC=60°，AC垂直于BC。求y的值。 | As shown in the diagram, BA=32, BC=x, CA=y, ∠BAC=60°, AC is perpendicular to BC. Find the value of y. | 375.png | [
"Shape(BA,AC,CB)"
] | [
"Equal(LengthOfLine(BA),32)",
"Equal(LengthOfLine(BC),x)",
"Equal(LengthOfLine(CA),y)",
"Equal(MeasureOfAngle(BAC),60)",
"PerpendicularBetweenLine(AC,BC)"
] | [
"Equal(LengthOfLine(BA),32)",
"Equal(LengthOfLine(BC),x)",
"Equal(LengthOfLine(CA),y)",
"Equal(MeasureOfAngle(BAC),60)",
"PerpendicularBetweenLine(AC,BC)"
] | Value(y) | 16 | [
"triangle_property_angle_sum(1,ACB)",
"sine_theorem(1,ACB)"
] | {"START": ["triangle_property_angle_sum(1,ACB)", "sine_theorem(1,ACB)"]} | |
376 | JiaZou_2023-04-09 | Geometry3k-383 | 3 | 如图所示，AE=x，BC=4，BE=3，CD=2。求x的值。 | As shown in the diagram, AE=x, BC=4, BE=3, CD=2. Find the value of x. | 376.png | [
"Shape(DC,XCD)",
"Shape(EB,BC,XEC)",
"Shape(AE,XEC,CD,XDA)",
"Shape(EA,XAE)",
"Collinear(DCB)",
"Collinear(AEB)",
"Cocircular(X,AECD)"
] | [
"Equal(LengthOfLine(AE),x)",
"Equal(LengthOfLine(BC),4)",
"Equal(LengthOfLine(BE),3)",
"Equal(LengthOfLine(CD),2)"
] | [
"Equal(LengthOfLine(AE),x)",
"Equal(LengthOfLine(BC),4)",
"Equal(LengthOfLine(BE),3)",
"Equal(LengthOfLine(CD),2)"
] | Value(x) | 5 | [
"line_addition(1,DC,CB)",
"line_addition(1,AE,EB)",
"circle_property_circular_power_segment_and_segment_line(1,BEA,BCD,X)"
] | {"START": ["line_addition(1,DC,CB)", "line_addition(1,AE,EB)", "circle_property_circular_power_segment_and_segment_line(1,BEA,BCD,X)"]} | |
377 | JiaZou_2023-04-09 | Geometry3k-384 | 3 | 如图所示，⊙R的直径为20，⊙S的直径为30，DS=9，R是圆R的圆心，⊙S的圆心为S。求直线CD的长度。 | As shown in the diagram, the diameter of ⊙R is 20, the diameter of ⊙S is 30, DS=9, R is the center of circle R, the center of ⊙S is S. Find the length of line CD. | 377.png | [
"Shape(SEC,CD,RDE)",
"Shape(SCF,RFD,DC)",
"Shape(REF,SEC,SCF)",
"Shape(SFB,BS,SD,RFD)",
"Shape(DS,SB,SBE,RDE)",
"Collinear(RCDSB)",
"Cocircular(R,FDE)",
"Cocircular(S,ECFB)"
] | [
"Equal(DiameterOfCircle(R),20)",
"Equal(DiameterOfCircle(S),30)",
"Equal(LengthOfLine(DS),9)",
"IsCentreOfCircle(R,R)",
"IsCentreOfCircle(S,S)"
] | [
"Equal(DiameterOfCircle(R),20)",
"Equal(DiameterOfCircle(S),30)",
"Equal(LengthOfLine(DS),9)",
"IsCentreOfCircle(R,R)",
"IsCentreOfCircle(S,S)"
] | Value(LengthOfLine(CD)) | 6 | [
"circle_property_length_of_radius_and_diameter(1,S)",
"radius_of_circle_property_length_equal(1,SC,S)",
"line_addition(1,CD,DS)"
] | {"START": ["circle_property_length_of_radius_and_diameter(1,S)", "radius_of_circle_property_length_equal(1,SC,S)", "line_addition(1,CD,DS)"]} | |
378 | JiaZou_2023-04-09 | Geometry3k-385 | 2 | 如图所示，FA=10，FW=x，WA=4，BFWC相似于NFAD。求四边形BFWC和四边形NFAD的相似比。 | As shown in the diagram, FA=10, FW=x, WA=4, quadrilateral BFWC is similar to quadrilateral NFAD. Find The ratio of similar quadrilaterals BFWC and NFAD. | 378.png | [
"Shape(NB,BC,CW,WA,AD,DN)",
"Shape(BF,FW,WC,CB)",
"Collinear(NBF)",
"Collinear(FWA)"
] | [
"Equal(LengthOfLine(FA),10)",
"Equal(LengthOfLine(FW),x)",
"Equal(LengthOfLine(WA),4)",
"SimilarBetweenQuadrilateral(BFWC,NFAD)"
] | [
"Equal(LengthOfLine(FA),10)",
"Equal(LengthOfLine(FW),x)",
"Equal(LengthOfLine(WA),4)"
] | Value(RatioOfSimilarQuadrilateral(BFWC,NFAD)) | 3/5 | [
"line_addition(1,FW,WA)",
"similar_quadrilateral_property_line_ratio(1,FWCB,FADN)"
] | {"START": ["line_addition(1,FW,WA)", "similar_quadrilateral_property_line_ratio(1,FWCB,FADN)"]} | |
379 | NaZhu_2023-03-12 | Geometry3k-386 | 7 | 如图所示，BA=c，BC=a，BE=x，CA=b，EA=y，∠CAE=30°，∠EBC=60°，x=7*sqrt(3)，AE垂直于CE，BC垂直于AC。求y的值。 | As shown in the diagram, BA=c, BC=a, BE=x, CA=b, EA=y, ∠CAE=30°, ∠EBC=60°, x=7*sqrt(3), AE is perpendicular to CE, BC is perpendicular to AC. Find the value of y. | 379.png | [
"Shape(BC,CE,EB)",
"Shape(EC,CA,AE)",
"Collinear(BEA)"
] | [
"Equal(LengthOfLine(BA),c)",
"Equal(LengthOfLine(BC),a)",
"Equal(LengthOfLine(BE),x)",
"Equal(LengthOfLine(CA),b)",
"Equal(LengthOfLine(EA),y)",
"Equal(MeasureOfAngle(CAE),30)",
"Equal(MeasureOfAngle(EBC),60)",
"Equal(x,7*sqrt(3))",
"PerpendicularBetweenLine(AE,CE)",
"PerpendicularBetweenLine(BC,A... | [
"Equal(LengthOfLine(BA),c)",
"Equal(LengthOfLine(BC),a)",
"Equal(LengthOfLine(BE),x)",
"Equal(LengthOfLine(CA),b)",
"Equal(LengthOfLine(EA),y)",
"Equal(MeasureOfAngle(CAE),30)",
"Equal(MeasureOfAngle(EBC),60)",
"PerpendicularBetweenLine(AE,CE)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(y) | 21*sqrt(3) | [
"adjacent_complementary_angle(1,AEC,CEB)",
"triangle_property_angle_sum(1,AEC)",
"sine_theorem(1,AEC)",
"sine_theorem(1,CAE)",
"triangle_property_angle_sum(1,CEB)",
"sine_theorem(1,CEB)",
"sine_theorem(1,BCE)"
] | {"START": ["adjacent_complementary_angle(1,AEC,CEB)", "triangle_property_angle_sum(1,AEC)", "sine_theorem(1,AEC)", "sine_theorem(1,CAE)", "triangle_property_angle_sum(1,CEB)", "sine_theorem(1,CEB)", "sine_theorem(1,BCE)"]} | |
380 | NaZhu_2023-03-12 | Geometry3k-387 | 1 | 如图所示，AB=26，AD=10，CB=16，DC=8，AD是△ACB的高。求三角形ACB的面积。 | As shown in the diagram, AB=26, AD=10, CB=16, DC=8, AD is the altitude of triangle ACB. Find the area of △ACB. | 380.png | [
"Shape(AD,DC,CA)",
"Shape(AC,CB,BA)",
"Collinear(DCB)"
] | [
"Equal(LengthOfLine(AB),26)",
"Equal(LengthOfLine(AD),10)",
"Equal(LengthOfLine(CB),16)",
"Equal(LengthOfLine(DC),8)",
"IsAltitudeOfTriangle(AD,ACB)"
] | [
"Equal(LengthOfLine(AB),26)",
"Equal(LengthOfLine(AD),10)",
"Equal(LengthOfLine(CB),16)",
"Equal(LengthOfLine(DC),8)",
"IsAltitudeOfTriangle(AD,ACB)"
] | Value(AreaOfTriangle(ACB)) | 80 | [
"triangle_area_formula_common(1,ACB)"
] | {"START": ["triangle_area_formula_common(1,ACB)"]} | |
381 | JiaZou_2023-04-09 | Geometry3k-388 | 6 | 如图所示，PT=8，SR=3，TQ=3，ST∥RQ。求直线PS的长度。 | As shown in the diagram, PT=8, SR=3, TQ=3, ST∥RQ. Find the length of line PS. | 381.png | [
"Shape(PS,ST,TP)",
"Shape(SR,RQ,QT,TS)",
"Collinear(PSR)",
"Collinear(PTQ)"
] | [
"Equal(LengthOfLine(PT),8)",
"Equal(LengthOfLine(SR),3)",
"Equal(LengthOfLine(TQ),3)",
"ParallelBetweenLine(ST,RQ)"
] | [
"Equal(LengthOfLine(PT),8)",
"Equal(LengthOfLine(SR),3)",
"Equal(LengthOfLine(TQ),3)",
"ParallelBetweenLine(ST,RQ)"
] | Value(LengthOfLine(PS)) | 8 | [
"line_addition(1,PS,SR)",
"line_addition(1,PT,TQ)",
"parallel_property_corresponding_angle(1,ST,RQ,P)",
"similar_triangle_judgment_aa(1,TPS,QPR)",
"similar_triangle_property_line_ratio(1,TPS,QPR)",
"similar_triangle_property_line_ratio(1,STP,RQP)"
] | {"START": ["line_addition(1,PS,SR)", "line_addition(1,PT,TQ)", "parallel_property_corresponding_angle(1,ST,RQ,P)"], "parallel_property_corresponding_angle(1,ST,RQ,P)": ["similar_triangle_judgment_aa(1,TPS,QPR)"], "similar_triangle_judgment_aa(1,TPS,QPR)": ["similar_triangle_property_line_ratio(1,TPS,QPR)", "similar_tri... | |
382 | NaZhu_2023-03-12 | Geometry3k-389 | 5 | 如图所示，ST=TU，SY=YZ，UJ=9，VJ=3，ZT=18，ZV=VU。求直线JT的长度。 | As shown in the diagram, ST=TU, SY=YZ, UJ=9, VJ=3, ZT=18, ZV=VU. Find the length of line JT. | 382.png | [
"Shape(SY,YJ,JS)",
"Shape(SJ,JT,TS)",
"Shape(YZ,ZJ,JY)",
"Shape(TJ,JU,UT)",
"Shape(JZ,ZV,VJ)",
"Shape(JV,VU,UJ)",
"Collinear(SYZ)",
"Collinear(STU)",
"Collinear(ZVU)",
"Collinear(SJV)",
"Collinear(YJU)",
"Collinear(ZJT)"
] | [
"Equal(LengthOfLine(ST),LengthOfLine(TU))",
"Equal(LengthOfLine(SY),LengthOfLine(YZ))",
"Equal(LengthOfLine(UJ),9)",
"Equal(LengthOfLine(VJ),3)",
"Equal(LengthOfLine(ZT),18)",
"Equal(LengthOfLine(ZV),LengthOfLine(VU))"
] | [
"Equal(LengthOfLine(ST),LengthOfLine(TU))",
"Equal(LengthOfLine(SY),LengthOfLine(YZ))",
"Equal(LengthOfLine(ZV),LengthOfLine(VU))"
] | Value(LengthOfLine(JT)) | 6 | [
"median_of_triangle_judgment(1,ZT,ZUS)",
"median_of_triangle_judgment(1,UY,USZ)",
"line_addition(1,ZJ,JT)",
"centroid_of_triangle_judgment_intersection(1,J,SZU,Y,T)",
"centroid_of_triangle_property_line_ratio(1,J,ZUS,T)"
] | {"START": ["median_of_triangle_judgment(1,ZT,ZUS)", "median_of_triangle_judgment(1,UY,USZ)", "line_addition(1,ZJ,JT)"], "centroid_of_triangle_judgment_intersection(1,J,SZU,Y,T)": ["centroid_of_triangle_property_line_ratio(1,J,ZUS,T)"], "median_of_triangle_judgment(1,UY,USZ)": ["centroid_of_triangle_judgment_intersectio... | |
383 | JiaZou_2023-04-09 | Geometry3k-390 | 1 | 如图所示，∠EAH=38°，∠GEB=52°，AH⊥EH。求∠CHA的大小。 | As shown in the diagram, ∠EAH=38°, ∠GEB=52°, AH is perpendicular to EH. Find the measure of ∠CHA. | 383.png | [
"Shape(IA,AF)",
"Shape(FA,AE)",
"Shape(AE,EG)",
"Shape(GE,EB)",
"Shape(CH,HA)",
"Shape(HA,AI)",
"Shape(DH,HC)",
"Shape(BE,EH)",
"Shape(EH,HD)",
"Shape(AH,HE,EA)",
"Collinear(IAEB)",
"Collinear(FAHD)",
"Collinear(CHEG)"
] | [
"Equal(MeasureOfAngle(EAH),38)",
"Equal(MeasureOfAngle(GEB),52)",
"PerpendicularBetweenLine(AH,EH)"
] | [
"Equal(MeasureOfAngle(EAH),38)",
"Equal(MeasureOfAngle(GEB),52)",
"PerpendicularBetweenLine(AH,EH)"
] | Value(MeasureOfAngle(CHA)) | 90 | [
"adjacent_complementary_angle(1,CHA,AHE)"
] | {"START": ["adjacent_complementary_angle(1,CHA,AHE)"]} | |
384 | NaZhu_2023-03-12 | Geometry3k-391 | 4 | 如图所示，AD=10，BC=15，BE=6，EC=12，ED=3*x-2，∠ADE=∠CBE。求x的值。 | As shown in the diagram, AD=10, BC=15, BE=6, EC=12, ED=3*x-2, ∠ADE=∠CBE. Find the value of x. | 384.png | [
"Shape(AD,DE,EA)",
"Shape(BE,EC,CB)",
"Collinear(AEC)",
"Collinear(BED)"
] | [
"Equal(LengthOfLine(AD),10)",
"Equal(LengthOfLine(BC),15)",
"Equal(LengthOfLine(BE),6)",
"Equal(LengthOfLine(EC),12)",
"Equal(LengthOfLine(ED),3*x-2)",
"Equal(MeasureOfAngle(ADE),MeasureOfAngle(CBE))"
] | [
"Equal(LengthOfLine(AD),10)",
"Equal(LengthOfLine(BC),15)",
"Equal(LengthOfLine(BE),6)",
"Equal(LengthOfLine(EC),12)",
"Equal(LengthOfLine(ED),3*x-2)",
"Equal(MeasureOfAngle(ADE),MeasureOfAngle(CBE))"
] | Value(x) | 2 | [
"vertical_angle(1,BEC,DEA)",
"similar_triangle_judgment_aa(1,ADE,CBE)",
"similar_triangle_property_line_ratio(1,ADE,CBE)",
"similar_triangle_property_line_ratio(1,EAD,ECB)"
] | {"START": ["vertical_angle(1,BEC,DEA)"], "similar_triangle_judgment_aa(1,ADE,CBE)": ["similar_triangle_property_line_ratio(1,ADE,CBE)", "similar_triangle_property_line_ratio(1,EAD,ECB)"], "vertical_angle(1,BEC,DEA)": ["similar_triangle_judgment_aa(1,ADE,CBE)"]} | |
385 | JiaZou_2023-04-09 | Geometry3k-392 | 5 | 如图所示，圆A的直径为8，⊙B的直径为18，圆C的直径为11，A是⊙A的圆心，⊙B的圆心为B，圆C的圆心为C。求直线FB的长度。 | As shown in the diagram, the diameter of circle A is 8, the diameter of circle B is 18, the diameter of ⊙C is 11, the center of ⊙A is A, the center of ⊙B is B, C is the center of circle C. Find the length of line FB. | 385.png | [
"Shape(AF,AFI,BIA)",
"Shape(FA,BAJ,AJF)",
"Shape(BIJ,AIJ)",
"Shape(FB,BM,CKM,BKI,AFI)",
"Shape(MB,BF,AJF,BJL,CML)",
"Shape(MG,BGK,CKM)",
"Shape(GM,CML,BLG)",
"Shape(BLG,BGK,CLK)",
"Collinear(AFBMGC)",
"Cocircular(A,JFI)",
"Cocircular(B,IAJLGK)",
"Cocircular(C,KML)"
] | [
"Equal(DiameterOfCircle(A),8)",
"Equal(DiameterOfCircle(B),18)",
"Equal(DiameterOfCircle(C),11)",
"IsCentreOfCircle(A,A)",
"IsCentreOfCircle(B,B)",
"IsCentreOfCircle(C,C)"
] | [
"Equal(DiameterOfCircle(A),8)",
"Equal(DiameterOfCircle(B),18)",
"Equal(DiameterOfCircle(C),11)",
"IsCentreOfCircle(A,A)",
"IsCentreOfCircle(B,B)",
"IsCentreOfCircle(C,C)"
] | Value(LengthOfLine(FB)) | 5 | [
"circle_property_length_of_radius_and_diameter(1,A)",
"circle_property_length_of_radius_and_diameter(1,B)",
"radius_of_circle_property_length_equal(1,AF,A)",
"radius_of_circle_property_length_equal(1,BA,B)",
"line_addition(1,AF,FB)"
] | {"START": ["circle_property_length_of_radius_and_diameter(1,A)", "circle_property_length_of_radius_and_diameter(1,B)", "radius_of_circle_property_length_equal(1,AF,A)", "radius_of_circle_property_length_equal(1,BA,B)", "line_addition(1,AF,FB)"]} | |
386 | NaZhu_2023-03-12 | Geometry3k-393 | 0 | 如图所示，AP=3*y+11，HP=7*y-5，∠HWQ=4*x-16°，∠HWX=x+12°，∠QAP=3*x-2°，WX平分∠HWQ，WP是△WAH的中线。求y的值。 | As shown in the diagram, AP=3*y+11, HP=7*y-5, ∠HWQ=4*x-16°, ∠HWX=x+12°, ∠QAP=3*x-2°, WX is the angle bisector of ∠HWQ, WP is the median of △ WAH. Find the value of y. | 386.png | [
"Shape(HW,WX,XH)",
"Shape(HX,XP,PH)",
"Shape(XW,WQ,QX)",
"Shape(PX,XQ,QA,AP)",
"Collinear(HXQ)",
"Collinear(WXP)",
"Collinear(HPA)",
"Collinear(WQA)"
] | [
"Equal(LengthOfLine(AP),3*y+11)",
"Equal(LengthOfLine(HP),7*y-5)",
"Equal(MeasureOfAngle(HWQ),4*x-16)",
"Equal(MeasureOfAngle(HWX),x+12)",
"Equal(MeasureOfAngle(QAP),3*x-2)",
"IsBisectorOfAngle(WX,HWQ)",
"IsMedianOfTriangle(WP,WAH)"
] | [] | Value(y) | 4 | [] | {"START": []} | |
387 | JiaZou_2023-04-09 | Geometry3k-394 | 3 | 如图所示，∠ADK=96°，∠HGJ=42°，DH∥AG。求∠EAD的大小。 | As shown in the diagram, ∠ADK=96°, ∠HGJ=42°, DH∥AG. Find the measure of ∠EAD. | 387.png | [
"Shape(KD,DL)",
"Shape(LD,DH)",
"Shape(DH,HI)",
"Shape(IH,HC)",
"Shape(AD,DK)",
"Shape(HD,DA)",
"Shape(GH,HD)",
"Shape(CH,HG)",
"Shape(EA,AD)",
"Shape(DA,AG)",
"Shape(AG,GH)",
"Shape(HG,GJ)",
"Shape(MA,AE)",
"Shape(GA,AM)",
"Shape(FG,GA)",
"Shape(JG,GF)",
"Collinear(KDHC)",
"Collin... | [
"Equal(MeasureOfAngle(ADK),96)",
"Equal(MeasureOfAngle(HGJ),42)",
"ParallelBetweenLine(DH,AG)"
] | [
"Equal(MeasureOfAngle(ADK),96)",
"Equal(MeasureOfAngle(HGJ),42)",
"ParallelBetweenLine(DH,AG)"
] | Value(MeasureOfAngle(EAD)) | 84 | [
"parallel_property_collinear_extend(1,DH,AG,K)",
"parallel_property_collinear_extend(2,GA,DK,E)",
"parallel_property_ipsilateral_internal_angle(1,AE,DK)"
] | {"START": ["parallel_property_collinear_extend(1,DH,AG,K)"], "parallel_property_collinear_extend(1,DH,AG,K)": ["parallel_property_collinear_extend(2,GA,DK,E)"], "parallel_property_collinear_extend(2,GA,DK,E)": ["parallel_property_ipsilateral_internal_angle(1,AE,DK)"]} | |
388 | NaZhu_2023-03-12 | Geometry3k-395 | 4 | 如图所示，AC=13，CD=6，DB=29，CD垂直于AD。求三角形ADB的面积。 | As shown in the diagram, AC=13, CD=6, DB=29, CD⊥AD. Find the area of triangle ADB. | 388.png | [
"Shape(AC,CD,DA)",
"Shape(AD,DB,BA)",
"Collinear(CDB)"
] | [
"Equal(LengthOfLine(AC),13)",
"Equal(LengthOfLine(CD),6)",
"Equal(LengthOfLine(DB),29)",
"PerpendicularBetweenLine(CD,AD)"
] | [
"Equal(LengthOfLine(AC),13)",
"Equal(LengthOfLine(CD),6)",
"Equal(LengthOfLine(DB),29)",
"PerpendicularBetweenLine(CD,AD)"
] | Value(AreaOfTriangle(ADB)) | 29*sqrt(133)/2 | [
"adjacent_complementary_angle(1,CDA,ADB)",
"right_triangle_judgment_angle(1,CDA)",
"right_triangle_property_pythagorean(1,CDA)",
"triangle_area_formula_sine(1,DBA)"
] | {"START": ["adjacent_complementary_angle(1,CDA,ADB)", "right_triangle_judgment_angle(1,CDA)", "triangle_area_formula_sine(1,DBA)"], "right_triangle_judgment_angle(1,CDA)": ["right_triangle_property_pythagorean(1,CDA)"]} | |
389 | JiaZou_2023-04-09 | Geometry3k-396 | 6 | 如图所示，AB=15，DA=9，DB=12，DE=x，ADCB是平行四边形，BE⊥DE，CB垂直于DB。求x的值。 | As shown in the diagram, AB=15, DA=9, DB=12, DE=x, quadrilateral ADCB is a ▱, BE is perpendicular to DE, CB⊥DB. Find the value of x. | 389.png | [
"Shape(AD,DE,EA)",
"Shape(ED,DB,BE)",
"Shape(BD,DC,CB)",
"Collinear(AEB)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(DA),9)",
"Equal(LengthOfLine(DB),12)",
"Equal(LengthOfLine(DE),x)",
"Parallelogram(ADCB)",
"PerpendicularBetweenLine(BE,DE)",
"PerpendicularBetweenLine(CB,DB)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(DA),9)",
"Equal(LengthOfLine(DB),12)",
"Equal(LengthOfLine(DE),x)",
"Parallelogram(ADCB)",
"PerpendicularBetweenLine(BE,DE)",
"PerpendicularBetweenLine(CB,DB)"
] | Value(x) | 36/5 | [
"line_addition(1,AE,EB)",
"adjacent_complementary_angle(1,BED,DEA)",
"right_triangle_judgment_angle(1,BED)",
"right_triangle_judgment_angle(1,DEA)",
"right_triangle_property_pythagorean(1,BED)",
"right_triangle_property_pythagorean(1,DEA)"
] | {"START": ["line_addition(1,AE,EB)", "adjacent_complementary_angle(1,BED,DEA)", "right_triangle_judgment_angle(1,BED)"], "adjacent_complementary_angle(1,BED,DEA)": ["right_triangle_judgment_angle(1,DEA)"], "right_triangle_judgment_angle(1,BED)": ["right_triangle_property_pythagorean(1,BED)"], "right_triangle_judgment_a... | |
390 | JiaZou_2023-04-09 | Geometry3k-397 | 3 | 如图所示，AB=15，PB=12，∠PBA=24°，ADCB是菱形。求直线AP的长度。 | As shown in the diagram, AB=15, PB=12, ∠PBA=24°, quadrilateral ADCB is a rhombus. Find the length of line AP. | 390.png | [
"Shape(AD,DP,PA)",
"Shape(PD,DC,CP)",
"Shape(PC,CB,BP)",
"Shape(PB,BA,AP)",
"Collinear(APC)",
"Collinear(DPB)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(PB),12)",
"Equal(MeasureOfAngle(PBA),24)",
"Rhombus(ADCB)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(PB),12)",
"Equal(MeasureOfAngle(PBA),24)"
] | Value(LengthOfLine(AP)) | 9 | [
"kite_property_diagonal_perpendicular_bisection(1,BADC,P)",
"right_triangle_judgment_angle(1,APB)",
"right_triangle_property_pythagorean(1,APB)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,BADC,P)"], "kite_property_diagonal_perpendicular_bisection(1,BADC,P)": ["right_triangle_judgment_angle(1,APB)"], "right_triangle_judgment_angle(1,APB)": ["right_triangle_property_pythagorean(1,APB)"]} | |
391 | NaZhu_2023-03-12 | Geometry3k-398 | 6 | 如图所示，PQ=UQ，PR=RT，∠UQP=40°。求∠SRQ的大小。 | As shown in the diagram, PQ=UQ, PR=RT, ∠UQP=40°. Find the measure of ∠SRQ. | 391.png | [
"Shape(RQ,QS,SR)",
"Shape(QP,PU,UQ)",
"Shape(QU,US,SQ)",
"Shape(SU,UT,TS)",
"Collinear(RQP)",
"Collinear(RST)",
"Collinear(PUT)"
] | [
"Equal(LengthOfLine(PQ),LengthOfLine(UQ))",
"Equal(LengthOfLine(PR),LengthOfLine(RT))",
"Equal(MeasureOfAngle(UQP),40)"
] | [] | Value(MeasureOfAngle(SRQ)) | 40 | [
"isosceles_triangle_judgment_line_equal(1,QPU)",
"isosceles_triangle_judgment_line_equal(1,RPT)",
"isosceles_triangle_property_angle_equal(1,QPU)",
"isosceles_triangle_property_angle_equal(1,RPT)",
"triangle_property_angle_sum(1,QPU)",
"triangle_property_angle_sum(1,TRP)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,QPU)", "isosceles_triangle_judgment_line_equal(1,RPT)", "triangle_property_angle_sum(1,QPU)", "triangle_property_angle_sum(1,TRP)"], "isosceles_triangle_judgment_line_equal(1,QPU)": ["isosceles_triangle_property_angle_equal(1,QPU)"], "isosceles_triangle_judgment_line... | |
392 | JiaZou_2023-04-09 | Geometry3k-399 | 10 | 如图所示，GK=14，MF=8，⌒FKG的角度为142，F是圆F的圆心，HJ⊥KJ。求弧FMK的角度。 | As shown in the diagram, GK=14, MF=8, the measure of arc FKG is 142, F is the center of circle F, HJ is perpendicular to KJ. Find the measure of arc FMK. | 392.png | [
"Shape(GJ,JH,FHG)",
"Shape(HJ,JK,FKH)",
"Shape(FJ,JG,GF)",
"Shape(MF,FG,FGM)",
"Shape(JF,FK,KJ)",
"Shape(KF,FM,FMK)",
"Collinear(GJK)",
"Collinear(MFJH)",
"Cocircular(F,GMKH)"
] | [
"Equal(LengthOfLine(GK),14)",
"Equal(LengthOfLine(MF),8)",
"Equal(MeasureOfArc(FKG),142)",
"IsCentreOfCircle(F,F)",
"PerpendicularBetweenLine(HJ,KJ)"
] | [
"Equal(LengthOfLine(GK),14)",
"Equal(LengthOfLine(MF),8)",
"Equal(MeasureOfArc(FKG),142)",
"IsCentreOfCircle(F,F)",
"PerpendicularBetweenLine(HJ,KJ)"
] | Value(MeasureOfArc(FMK)) | 109 | [
"radius_of_circle_property_length_equal(1,FK,F)",
"radius_of_circle_property_length_equal(1,FG,F)",
"adjacent_complementary_angle(1,JFK,KFM)",
"adjacent_complementary_angle(1,HJK,KJM)",
"arc_property_center_angle(1,FKG,F)",
"isosceles_triangle_judgment_line_equal(1,FKG)",
"altitude_of_triangle_judgment(... | {"START": ["radius_of_circle_property_length_equal(1,FK,F)", "radius_of_circle_property_length_equal(1,FG,F)", "adjacent_complementary_angle(1,JFK,KFM)", "adjacent_complementary_angle(1,HJK,KJM)", "arc_property_center_angle(1,FKG,F)", "angle_addition(1,GFJ,JFK)", "arc_property_center_angle(1,FMK,F)"], "adjacent_complem... | |
393 | NaZhu_2023-03-12 | Geometry3k-400 | 2 | 如图所示，AB=2*sqrt(3)，AY=x，YC=y，∠BAY=30°，∠YCB=60°，AY⊥BY，CB⊥AB。求x的值。 | As shown in the diagram, AB=2*sqrt(3), AY=x, YC=y, ∠BAY=30°, ∠YCB=60°, AY⊥BY, CB is perpendicular to AB. Find the value of x. | 393.png | [
"Shape(AY,YB,BA)",
"Shape(BY,YC,CB)",
"Collinear(AYC)"
] | [
"Equal(LengthOfLine(AB),2*sqrt(3))",
"Equal(LengthOfLine(AY),x)",
"Equal(LengthOfLine(YC),y)",
"Equal(MeasureOfAngle(BAY),30)",
"Equal(MeasureOfAngle(YCB),60)",
"PerpendicularBetweenLine(AY,BY)",
"PerpendicularBetweenLine(CB,AB)"
] | [
"Equal(LengthOfLine(AB),2*sqrt(3))",
"Equal(LengthOfLine(AY),x)",
"Equal(LengthOfLine(YC),y)",
"Equal(MeasureOfAngle(BAY),30)",
"Equal(MeasureOfAngle(YCB),60)",
"PerpendicularBetweenLine(AY,BY)",
"PerpendicularBetweenLine(CB,AB)"
] | Value(x) | 3 | [
"triangle_property_angle_sum(1,AYB)",
"sine_theorem(1,AYB)"
] | {"START": ["triangle_property_angle_sum(1,AYB)", "sine_theorem(1,AYB)"]} | |
394 | JiaZou_2023-04-09 | Geometry3k-401 | 2 | 如图所示，AB=x+1，CD=x-1，FE=8，HG=5，∠ADC=∠EHG，∠BAD=∠FEH，∠CBA=∠GFE，∠DCB=∠HGF，四边形BADC与四边形FEHG相似。求直线CD的长度。 | As shown in the diagram, AB=x+1, CD=x-1, FE=8, HG=5, ∠ADC=∠EHG, ∠BAD=∠FEH, ∠CBA=∠GFE, ∠DCB=∠HGF, BADC is similar to FEHG. Find the length of line CD. | 394.png | [
"Shape(BA,AD,DC,CB)",
"Shape(EH,HG,GF,FE)"
] | [
"Equal(LengthOfLine(AB),x+1)",
"Equal(LengthOfLine(CD),x-1)",
"Equal(LengthOfLine(FE),8)",
"Equal(LengthOfLine(HG),5)",
"Equal(MeasureOfAngle(ADC),MeasureOfAngle(EHG))",
"Equal(MeasureOfAngle(BAD),MeasureOfAngle(FEH))",
"Equal(MeasureOfAngle(CBA),MeasureOfAngle(GFE))",
"Equal(MeasureOfAngle(DCB),Measu... | [
"Equal(LengthOfLine(AB),x+1)",
"Equal(LengthOfLine(CD),x-1)",
"Equal(LengthOfLine(FE),8)",
"Equal(LengthOfLine(HG),5)",
"Equal(MeasureOfAngle(ADC),MeasureOfAngle(EHG))",
"Equal(MeasureOfAngle(BAD),MeasureOfAngle(FEH))",
"Equal(MeasureOfAngle(CBA),MeasureOfAngle(GFE))",
"Equal(MeasureOfAngle(DCB),Measu... | Value(LengthOfLine(CD)) | 10/3 | [
"similar_quadrilateral_property_line_ratio(1,BADC,FEHG)",
"similar_quadrilateral_property_line_ratio(1,DCBA,HGFE)"
] | {"START": ["similar_quadrilateral_property_line_ratio(1,BADC,FEHG)", "similar_quadrilateral_property_line_ratio(1,DCBA,HGFE)"]} | |
395 | NaZhu_2023-03-12 | Geometry3k-402 | 9 | 如图所示，CBTA的面积为104，AB=16，CE=ET，ET=x，CE⊥AE。求x的值。 | As shown in the diagram, the area of CBTA is 104, AB=16, CE=ET, ET=x, CE is perpendicular to AE. Find the value of x. | 395.png | [
"Shape(CB,BE,EC)",
"Shape(CE,EA,AC)",
"Shape(BT,TE,EB)",
"Shape(ET,TA,AE)",
"Collinear(CET)",
"Collinear(BEA)"
] | [
"Equal(AreaOfQuadrilateral(CBTA),104)",
"Equal(LengthOfLine(AB),16)",
"Equal(LengthOfLine(CE),LengthOfLine(ET))",
"Equal(LengthOfLine(ET),x)",
"PerpendicularBetweenLine(CE,AE)"
] | [
"Equal(LengthOfLine(AB),16)",
"Equal(LengthOfLine(CE),LengthOfLine(ET))",
"Equal(LengthOfLine(ET),x)"
] | Value(x) | 13/2 | [
"adjacent_complementary_angle(1,CEA,AET)",
"adjacent_complementary_angle(1,AET,TEB)",
"perpendicular_bisector_judgment_per_and_mid(1,BE,TC)",
"perpendicular_bisector_judgment_per_and_mid(1,AE,CT)",
"perpendicular_bisector_property_distance_equal(1,BE,TC)",
"perpendicular_bisector_property_distance_equal(1... | {"START": ["adjacent_complementary_angle(1,CEA,AET)", "adjacent_complementary_angle(1,AET,TEB)", "perpendicular_bisector_judgment_per_and_mid(1,AE,CT)", "line_addition(1,CE,ET)"], "adjacent_complementary_angle(1,AET,TEB)": ["perpendicular_bisector_judgment_per_and_mid(1,BE,TC)"], "adjacent_complementary_angle(1,CEA,AET... | |
396 | NaZhu_2023-03-12 | Geometry3k-404 | 2 | 如图所示，KJ=11，KL=11，ML=5.5，∠KJM=60°，KM垂直于LM。求∠LKJ的大小。 | As shown in the diagram, KJ=11, KL=11, ML=5.5, ∠KJM=60°, KM is perpendicular to LM. Find the measure of ∠LKJ. | 396.png | [
"Shape(KJ,JM,MK)",
"Shape(KM,ML,LK)",
"Collinear(JML)"
] | [
"Equal(LengthOfLine(KJ),11)",
"Equal(LengthOfLine(KL),11)",
"Equal(LengthOfLine(ML),5.5)",
"Equal(MeasureOfAngle(KJM),60)",
"PerpendicularBetweenLine(KM,LM)"
] | [
"Equal(LengthOfLine(KJ),11)",
"Equal(LengthOfLine(KL),11)",
"Equal(LengthOfLine(ML),5.5)",
"Equal(MeasureOfAngle(KJM),60)",
"PerpendicularBetweenLine(KM,LM)"
] | Value(MeasureOfAngle(LKJ)) | 60 | [
"cosine_theorem(1,JLK)",
"cosine_theorem(1,KJL)"
] | {"START": ["cosine_theorem(1,JLK)", "cosine_theorem(1,KJL)"]} | |
397 | JiaZou_2023-04-09 | Geometry3k-405 | 3 | 如图所示，ML=7，∠MLK=92°，圆L的圆心为L。求扇形LKM的面积。 | As shown in the diagram, ML=7, ∠MLK=92°, L is the center of circle L. Find the area of the sector LKM. | 397.png | [
"Shape(ML,LK,LKM)",
"Cocircular(L,KM)"
] | [
"Equal(LengthOfLine(ML),7)",
"Equal(MeasureOfAngle(MLK),92)",
"IsCentreOfCircle(L,L)"
] | [
"Equal(LengthOfLine(ML),7)",
"Equal(MeasureOfAngle(MLK),92)",
"IsCentreOfCircle(L,L)"
] | Value(AreaOfSector(LKM)) | 1127*pi/90 | [
"radius_of_circle_property_length_equal(1,LM,L)",
"arc_property_center_angle(1,LKM,L)",
"sector_area_formula(1,LKM)"
] | {"START": ["radius_of_circle_property_length_equal(1,LM,L)", "arc_property_center_angle(1,LKM,L)", "sector_area_formula(1,LKM)"]} | |
398 | JiaZou_2023-04-09 | Geometry3k-406 | 1 | 如图所示，∠ILR=2*y+8°，∠JME=z°，∠KIL=4*x+6°，∠RLP=142°，HK∥BR，JN∥HK。求y的值。 | As shown in the diagram, ∠ILR=2*y+8°, ∠JME=z°, ∠KIL=4*x+6°, ∠RLP=142°, HK is parallel to BR, JN∥HK. Find the value of y. | 398.png | [
"Shape(JM,ME)",
"Shape(EM,MN)",
"Shape(IM,MJ)",
"Shape(NM,MI)",
"Shape(HI,IM)",
"Shape(MI,IK)",
"Shape(LI,IH)",
"Shape(KI,IL)",
"Shape(BL,LI)",
"Shape(IL,LR)",
"Shape(PL,LB)",
"Shape(RL,LP)",
"Collinear(JMN)",
"Collinear(HIK)",
"Collinear(BLR)",
"Collinear(PLIME)"
] | [
"Equal(MeasureOfAngle(ILR),2*y+8)",
"Equal(MeasureOfAngle(JME),z)",
"Equal(MeasureOfAngle(KIL),4*x+6)",
"Equal(MeasureOfAngle(RLP),142)",
"ParallelBetweenLine(HK,BR)",
"ParallelBetweenLine(JN,HK)"
] | [
"Equal(MeasureOfAngle(ILR),2*y+8)",
"Equal(MeasureOfAngle(JME),z)",
"Equal(MeasureOfAngle(KIL),4*x+6)",
"Equal(MeasureOfAngle(RLP),142)",
"ParallelBetweenLine(HK,BR)",
"ParallelBetweenLine(JN,HK)"
] | Value(y) | 15 | [
"adjacent_complementary_angle(1,ILR,RLP)"
] | {"START": ["adjacent_complementary_angle(1,ILR,RLP)"]} | |
399 | NaZhu_2023-03-12 | Geometry3k-407 | 1 | 如图所示，∠ACE=25°，∠AEG=51°，∠DAB=35°，∠GBA=28°，AB垂直于FB，AG垂直于BG，BD垂直于FD。求∠BAG的大小。 | As shown in the diagram, ∠ACE=25°, ∠AEG=51°, ∠DAB=35°, ∠GBA=28°, AB is perpendicular to FB, AG is perpendicular to BG, BD is perpendicular to FD. Find the measure of ∠BAG. | 399.png | [
"Shape(AC,CE,EA)",
"Shape(AE,EG,GA)",
"Shape(AG,GB,BA)",
"Shape(AB,BD,DA)",
"Shape(DB,BF,FD)",
"Collinear(CEGB)",
"Collinear(ADF)"
] | [
"Equal(MeasureOfAngle(ACE),25)",
"Equal(MeasureOfAngle(AEG),51)",
"Equal(MeasureOfAngle(DAB),35)",
"Equal(MeasureOfAngle(GBA),28)",
"PerpendicularBetweenLine(AB,FB)",
"PerpendicularBetweenLine(AG,BG)",
"PerpendicularBetweenLine(BD,FD)"
] | [
"Equal(MeasureOfAngle(ACE),25)",
"Equal(MeasureOfAngle(AEG),51)",
"Equal(MeasureOfAngle(DAB),35)",
"Equal(MeasureOfAngle(GBA),28)",
"PerpendicularBetweenLine(AB,FB)",
"PerpendicularBetweenLine(AG,BG)",
"PerpendicularBetweenLine(BD,FD)"
] | Value(MeasureOfAngle(BAG)) | 62 | [
"triangle_property_angle_sum(1,AGB)"
] | {"START": ["triangle_property_angle_sum(1,AGB)"]} | |
400 | NaZhu_2023-03-12 | Geometry3k-408 | 2 | 如图所示，PQ=UQ，∠QPU=32°。求∠PUQ的大小。 | As shown in the diagram, PQ=UQ, ∠QPU=32°. Find the measure of ∠PUQ. | 400.png | [
"Shape(RQ,QS,SR)",
"Shape(QP,PU,UQ)",
"Shape(QU,US,SQ)",
"Shape(SU,UT,TS)",
"Collinear(RQP)",
"Collinear(RST)",
"Collinear(PUT)"
] | [
"Equal(LengthOfLine(PQ),LengthOfLine(UQ))",
"Equal(MeasureOfAngle(QPU),32)"
] | [
"Equal(LengthOfLine(PQ),LengthOfLine(UQ))",
"Equal(MeasureOfAngle(QPU),32)"
] | Value(MeasureOfAngle(PUQ)) | 32 | [
"isosceles_triangle_judgment_line_equal(1,QPU)",
"isosceles_triangle_property_angle_equal(1,QPU)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,QPU)"], "isosceles_triangle_judgment_line_equal(1,QPU)": ["isosceles_triangle_property_angle_equal(1,QPU)"]} |
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