SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, UV=7, UW=12, WV=x, the center of ⊙V is V, the tangent to ⊙V is WU. Find the value of x.
问题：如图所示，UV=7，UW=12，WV=x，V是⊙V的圆心，WU是圆O的切线。求x的值。
Construction (CDL):
Shape(AV,VU,VUA)
Shape(UW,WA,VUA)
Shape(VA,VAU,UV)
Collinear(WAV)
Cocircular(V,AU)
Conditions (CDL):
Equal(LengthOfLine(UV),7)
Equal(LengthOfLine(UW),12)
Equal(LengthOfLine(WV),x)
IsCentreOfCircle(V,V)
IsTangentOfCircle(WU,V)
Goal: Value(x)
Answer: sqrt(193)
Theorem sequence:
tangent_of_circle_property_perpendicular(1,WU,V,V)
right_triangle_judgment_angle(1,VUW)
right_triangle_property_pythagorean(1,VUW)
Problem: As shown in the diagram, BF=2*y+4, DG=3/2*x+8, EB=3*y-6, FG=1/2*X+12, FG=GD, GB is parallel to DE. Find the value of y.
问题：如图所示，BF=2*y+4，DG=3/2*x+8，EB=3*y-6，FG=1/2*X+12，FG=GD，GB∥DE。求y的值。
Construction (CDL):
Shape(FG,GB,BF)
Shape(BG,GD,DE,EB)
Collinear(FBE)
Collinear(FGD)
Conditions (CDL):
Equal(LengthOfLine(BF),2*y+4)
Equal(LengthOfLine(DG),3/2*x+8)
Equal(LengthOfLine(EB),3*y-6)
Equal(LengthOfLine(FG),1/2*X+12)
Equal(LengthOfLine(FG),LengthOfLine(GD))
ParallelBetweenLine(GB,DE)
Goal: Value(y)
Answer: 10
Theorem sequence:
parallel_property_corresponding_angle(1,GB,DE,F)
parallel_property_corresponding_angle(2,ED,BG,F)
similar_triangle_judgment_aa(1,FGB,FDE)
similar_triangle_property_line_ratio(1,GBF,DEF)
similar_triangle_property_line_ratio(1,BFG,EFD)
line_addition(1,FB,BE)
line_addition(1,FG,GD)
Problem: As shown in the diagram, JN=x-2, MJ=2*x-5, MJ=NM, MN=3*x-9, ∠JMN and ∠MNJ are the base angles of the isosceles triangle JMN. Find the length of line JM.
问题：如图所示，JN=x-2，MJ=2*x-5，MJ=NM，MN=3*x-9，△JMN为等腰△。求直线JM的长度。
Construction (CDL):
Shape(MJ,JN,NM)
Conditions (CDL):
Equal(LengthOfLine(JN),x-2)
Equal(LengthOfLine(MJ),2*x-5)
Equal(LengthOfLine(MJ),LengthOfLine(NM))
Equal(LengthOfLine(MN),3*x-9)
IsoscelesTriangle(JMN)
Goal: Value(LengthOfLine(JM))
Answer: 3
Theorem sequence:
Problem: As shown in the diagram, DE=4, DF=x, EF=3, the center of circle F is F, DE is the tangent to ⊙F. Find the value of x.
问题：如图所示，DE=4，DF=x，EF=3，⊙F的圆心为F，⊙O的切线为DE。求x的值。
Construction (CDL):
Shape(FEC,ED,DC)
Shape(CF,FE,FEC)
Shape(FC,FCE,EF)
Collinear(DCF)
Cocircular(F,CE)
Conditions (CDL):
Equal(LengthOfLine(DE),4)
Equal(LengthOfLine(DF),x)
Equal(LengthOfLine(EF),3)
IsCentreOfCircle(F,F)
IsTangentOfCircle(DE,F)
Goal: Value(x)
Answer: 5
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DE,F,F)
right_triangle_judgment_angle(1,FED)
right_triangle_property_pythagorean(1,FED)
Problem: As shown in the diagram, AB=19, BC=x, ∠ABC=27°, BC is perpendicular to AC. Find the value of x.
问题：如图所示，AB=19，BC=x，∠ABC=27°，BC垂直于AC。求x的值。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),19)
Equal(LengthOfLine(BC),x)
Equal(MeasureOfAngle(ABC),27)
PerpendicularBetweenLine(BC,AC)
Goal: Value(x)
Answer: -19*sqrt(2)/8+19*sqrt(10)/8+19*sqrt(sqrt(5)+5)/4
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
cosine_theorem(1,BCA)
Problem: As shown in the diagram, FJ=FH, GF=GH, ∠HGF=32°, ∠JHG=152°. Find the measure of ∠HFJ.
问题：如图所示，FJ=FH，GF=GH，∠HGF=32°，∠JHG=152°。求∠HFJ的大小。
Construction (CDL):
Shape(FJ,JH,HF)
Shape(FH,HG,GF)
Conditions (CDL):
Equal(LengthOfLine(FJ),LengthOfLine(FH))
Equal(LengthOfLine(GF),LengthOfLine(GH))
Equal(MeasureOfAngle(HGF),32)
Equal(MeasureOfAngle(JHG),152)
Goal: Value(MeasureOfAngle(HFJ))
Answer: 24
Theorem sequence:
triangle_property_angle_sum(1,FHG)
triangle_property_angle_sum(1,FJH)
isosceles_triangle_judgment_line_equal(1,GFH)
isosceles_triangle_judgment_line_equal(1,FJH)
isosceles_triangle_property_angle_equal(1,GFH)
angle_addition(1,JHF,FHG)
isosceles_triangle_property_angle_equal(1,FJH)
Problem: As shown in the diagram, the measure of ⌒GBD is 154, G is the center of circle G, the tangent to circle G is AB. Find the measure of ∠ABD.
问题：如图所示，弧GBD的角度为154，圆G的圆心为G，AB是圆O的切线。求∠ABD的大小。
Construction (CDL):
Shape(GB,BD,DG)
Shape(GD,GDB,BG)
Shape(DB,GBD)
Shape(AB,BG)
Cocircular(G,BD)
Conditions (CDL):
Equal(MeasureOfArc(GBD),154)
IsCentreOfCircle(G,G)
IsTangentOfCircle(AB,G)
Goal: Value(MeasureOfAngle(ABD))
Answer: 103
Theorem sequence:
arc_property_center_angle(1,GBD,G)
tangent_of_circle_property_perpendicular(2,AB,G,G)
radius_of_circle_property_length_equal(1,GD,G)
radius_of_circle_property_length_equal(1,GB,G)
isosceles_triangle_judgment_line_equal(1,GBD)
isosceles_triangle_property_angle_equal(1,GBD)
triangle_property_angle_sum(1,GBD)
angle_addition(1,ABG,GBD)
Problem: As shown in the diagram, AC=b, BA=c, BC=a, a=14, b=48, c=50, BC is perpendicular to AC. Find the value of sin(AB).
问题：如图所示，AC=b，BA=c，BC=a，a=14，b=48，c=50，BC⊥AC。求sin(AB)的值。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AC),b)
Equal(LengthOfLine(BA),c)
Equal(LengthOfLine(BC),a)
Equal(a,14)
Equal(b,48)
Equal(c,50)
PerpendicularBetweenLine(BC,AC)
Goal: Value(Sin(MeasureOfAngle(ABC)))
Answer: 24/25
Theorem sequence:
cosine_theorem(1,BCA)
Problem: As shown in the diagram, DF=6, FC=8, the perimeter of triangle CFB is 27, triangle DFE is similar to triangle CFB.. Find the perimeter of △DFE.
问题：如图所示，DF=6，FC=8，△CFB的周长为27，△DFE与△CFB是相似三角形。求△DFE的周长。
Construction (CDL):
Shape(EA,AD,DE)
Shape(ED,DF,FE)
Shape(FD,DB,BF)
Shape(FB,BC,CF)
Shape(EF,FC,CE)
Collinear(E,F,B)
Collinear(D,F,C)
Conditions (CDL):
Equal(LengthOfLine(DF),6)
Equal(LengthOfLine(FC),8)
Equal(PerimeterOfTriangle(CFB),27)
SimilarBetweenTriangle(DFE,CFB)
Goal: Value(PerimeterOfTriangle(DFE))
Answer: 81/4
Theorem sequence:
similar_triangle_property_line_ratio(1,EDF,BCF)
similar_triangle_property_perimeter_ratio(1,EDF,BCF)
Problem: As shown in the diagram, BH=74, HF=21, DEBF is a rhombus. Find the area of DEBF.
问题：如图所示，BH=74，HF=21，DEBF是菱形。求DEBF的面积。
Construction (CDL):
Shape(DH,HF,FD)
Shape(FH,HB,BF)
Shape(DE,EH,HD)
Shape(HE,EB,BH)
Collinear(FHE)
Collinear(DHB)
Conditions (CDL):
Equal(LengthOfLine(BH),74)
Equal(LengthOfLine(HF),21)
Rhombus(DEBF)
Goal: Value(AreaOfQuadrilateral(DEBF))
Answer: 3108
Theorem sequence:
parallelogram_property_diagonal_bisection(1,DEBF,H)
parallelogram_property_diagonal_bisection(1,EBFD,H)
line_addition(1,FH,HE)
line_addition(1,DH,HB)
kite_area_formula_diagonal(1,DEBF)
Problem: As shown in the diagram, HP=5*x-16, PJ=3*x+8, ∠GJN=6*y-3°, ∠GMH=4*z+14°, ∠NJH=4*y+23°, HM is the altitude of △ HGJ. Find the value of z.
问题：如图所示，HP=5*x-16，PJ=3*x+8，∠GJN=6*y-3°，∠GMH=4*z+14°，∠NJH=4*y+23°，HM是△HGJ的高。求z的值。
Construction (CDL):
Shape(GA,AN,NG)
Shape(GM,MB,BA,AG)
Shape(BM,MJ,JB)
Shape(BJ,JP,PZ,ZB)
Shape(ZP,PH,HZ)
Shape(NA,AZ,ZH,HN)
Shape(AB,BZ,ZA)
Collinear(GAZP)
Collinear(NABJ)
Collinear(HZBM)
Collinear(HNG)
Collinear(HPJ)
Collinear(GMJ)
Conditions (CDL):
Equal(LengthOfLine(HP),5*x-16)
Equal(LengthOfLine(PJ),3*x+8)
Equal(MeasureOfAngle(GJN),6*y-3)
Equal(MeasureOfAngle(GMH),4*z+14)
Equal(MeasureOfAngle(NJH),4*y+23)
IsAltitudeOfTriangle(HM,HGJ)
Goal: Value(z)
Answer: 19
Theorem sequence:
Problem: As shown in the diagram, AB=y, AC=x, BC=6*sqrt(3), ∠BAC=60°, AC is perpendicular to BC. Find the value of x.
问题：如图所示，AB=y，AC=x，BC=6*sqrt(3)，∠BAC=60°，AC⊥BC。求x的值。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BC),6*sqrt(3))
Equal(MeasureOfAngle(BAC),60)
PerpendicularBetweenLine(AC,BC)
Goal: Value(x)
Answer: 6
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
cosine_theorem(1,ACB)
Problem: As shown in the diagram, DE=EC, the measure of ⌒DFC is 60, DG⊥BG. Find the measure of ∠CAF.
问题：如图所示，DE=EC，弧DFC的角度为60，DG垂直于BG。求∠CAF的大小。
Construction (CDL):
Shape(DBA,AB)
Shape(BA,AD,DG,GB)
Shape(GD,DC,CG)
Shape(DEB,BG,GE)
Shape(EG,GC,CE)
Shape(EC,DCE)
Shape(CD,DA,AF,FC)
Shape(FA,DAF)
Shape(CF,DFC)
Collinear(BGC)
Collinear(ADC)
Collinear(DGE)
Cocircular(D,AFCEB)
Conditions (CDL):
Equal(LengthOfLine(DE),LengthOfLine(EC))
Equal(MeasureOfArc(DFC),60)
PerpendicularBetweenLine(DG,BG)
Goal: Value(MeasureOfAngle(CAF))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,DFC,A)
Problem: As shown in the diagram, AB=6, the center of ⊙B is B. Find the area of the ⊙B.
问题：如图所示，AB=6，圆B的圆心为B。求⊙B的面积。
Construction (CDL):
Shape(BAC,BCA)
Shape(AB)
Cocircular(B,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
IsCentreOfCircle(B,B)
Goal: Value(AreaOfCircle(B))
Answer: 36*pi
Theorem sequence:
radius_of_circle_property_length_equal(1,BA,B)
circle_area_formula(1,B)
Problem: As shown in the diagram, JK=2*b+3, JM=3*a, LK=21, LM=45, ∠MLR=30°, ∠RLK=70°, JK and ML are opposite sides of the ▱ JMLK. Find the measure of ∠JML.
问题：如图所示，JK=2*b+3，JM=3*a，LK=21，LM=45，∠MLR=30°，∠RLK=70°，MJ和LK是▱JMLK的一组对边。求∠JML的大小。
Construction (CDL):
Shape(JM,MR,RJ)
Shape(RM,ML,LR)
Shape(RL,LK,KR)
Shape(JR,RK,KJ)
Collinear(JRL)
Collinear(MRK)
Conditions (CDL):
Equal(LengthOfLine(JK),2*b+3)
Equal(LengthOfLine(JM),3*a)
Equal(LengthOfLine(LK),21)
Equal(LengthOfLine(LM),45)
Equal(MeasureOfAngle(MLR),30)
Equal(MeasureOfAngle(RLK),70)
Parallelogram(JMLK)
Goal: Value(MeasureOfAngle(JML))
Answer: 80
Theorem sequence:
angle_addition(1,MLR,RLK)
parallel_property_ipsilateral_internal_angle(1,MJ,LK)
Problem: As shown in the diagram, XY=8, XZ=2*c, ZY=6, XWZY is a rectangle. Find the length of line WY.
问题：如图所示，XY=8，XZ=2*c，ZY=6，XWZY是矩形。求直线WY的长度。
Construction (CDL):
Shape(XW,WA,AX)
Shape(AW,WZ,ZA)
Shape(AZ,ZY,YA)
Shape(XA,AY,YX)
Collinear(XAZ)
Collinear(WAY)
Conditions (CDL):
Equal(LengthOfLine(XY),8)
Equal(LengthOfLine(XZ),2*c)
Equal(LengthOfLine(ZY),6)
Rectangle(XWZY)
Goal: Value(LengthOfLine(WY))
Answer: 10
Theorem sequence:
right_triangle_judgment_angle(1,ZYX)
right_triangle_property_pythagorean(1,ZYX)
rectangle_property_diagonal_equal(1,XWZY)
Problem: As shown in the diagram, ∠DOB=∠AOF, ∠DOB=∠EOD, ∠DOB=∠FOE, the center of ⊙O is O, CO⊥AO. Find the measure of arc OAB.
问题：如图所示，∠DOB=∠AOF，∠DOB=∠EOD，∠DOB=∠FOE，O是⊙O的圆心，CO垂直于AO。求弧OAB的角度。
Construction (CDL):
Shape(OC,OCB,BO)
Shape(DO,OB,OBD)
Shape(EO,OD,ODE)
Shape(FO,OE,OEF)
Shape(AO,OF,OFA)
Shape(CO,OA,OAC)
Collinear(AOB)
Collinear(EOC)
Cocircular(O,ACBDEF)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),MeasureOfAngle(AOF))
Equal(MeasureOfAngle(DOB),MeasureOfAngle(EOD))
Equal(MeasureOfAngle(DOB),MeasureOfAngle(FOE))
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CO,AO)
Goal: Value(MeasureOfArc(OAB))
Answer: 180
Theorem sequence:
arc_property_center_angle(1,OAB,O)
flat_angle(1,BOA)
Problem: As shown in the diagram, PA=7, A is the center of circle A, the diameter of circle A is PG. Find the length of line PG.
问题：如图所示，PA=7，⊙A的圆心为A，圆A的直径为PG。求直线PG的长度。
Construction (CDL):
Shape(DA,AP,APD)
Shape(LA,AD,ADL)
Shape(GA,AL,ALG)
Shape(FA,AG,AGF)
Shape(PA,AF,AFP)
Collinear(DAF)
Collinear(PAG)
Cocircular(A,PDLGF)
Conditions (CDL):
Equal(LengthOfLine(PA),7)
IsCentreOfCircle(A,A)
IsDiameterOfCircle(PG,A)
Goal: Value(LengthOfLine(PG))
Answer: 14
Theorem sequence:
diameter_of_circle_property_length_equal(1,PG,A)
radius_of_circle_property_length_equal(1,AP,A)
circle_property_length_of_radius_and_diameter(1,A)
Problem: As shown in the diagram, AB=222, the center of ⊙C is C, the diameter of circle C is AB. Find the area of the ⊙C.
问题：如图所示，AB=222，⊙C的圆心为C，⊙C的直径为AB。求圆C的面积。
Construction (CDL):
Shape(CAB,BC,CA)
Shape(AC,CB,CBA)
Collinear(BCA)
Cocircular(C,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),222)
IsCentreOfCircle(C,C)
IsDiameterOfCircle(AB,C)
Goal: Value(AreaOfCircle(C))
Answer: 12321*pi
Theorem sequence:
diameter_of_circle_property_length_equal(1,AB,C)
circle_property_length_of_radius_and_diameter(1,C)
circle_area_formula(1,C)
Problem: As shown in the diagram, JN=3*y-5, LN=y+5, NK=2*z+7, NM=z+9, MJ and LK are opposite sides of the ▱ JMLK. Find the value of z.
问题：如图所示，JN=3*y-5，LN=y+5，NK=2*z+7，NM=z+9，JMLK是▱。求z的值。
Construction (CDL):
Shape(JM,MN,NJ)
Shape(NM,ML,LN)
Shape(NL,LK,KN)
Shape(JN,NK,KJ)
Collinear(JNL)
Collinear(MNK)
Conditions (CDL):
Equal(LengthOfLine(JN),3*y-5)
Equal(LengthOfLine(LN),y+5)
Equal(LengthOfLine(NK),2*z+7)
Equal(LengthOfLine(NM),z+9)
Parallelogram(JMLK)
Goal: Value(z)
Answer: 2
Theorem sequence:
parallelogram_property_diagonal_bisection(1,MLKJ,N)
Problem: As shown in the diagram, ∠ABF=115°, ∠CED=x+24°, ∠DEA=2*y-56°, ∠FDE=x°, ED is parallel to AB. Find the value of y.
问题：如图所示，∠ABF=115°，∠CED=x+24°，∠DEA=2*y-56°，∠FDE=x°，ED平行于AB。求y的值。
Construction (CDL):
Shape(EA,AB,BF,FD,DE)
Shape(CE)
Shape(CE,ED)
Collinear(CEA)
Collinear(BFD)
Conditions (CDL):
Equal(MeasureOfAngle(ABF),115)
Equal(MeasureOfAngle(CED),x+24)
Equal(MeasureOfAngle(DEA),2*y-56)
Equal(MeasureOfAngle(FDE),x)
ParallelBetweenLine(ED,AB)
Goal: Value(y)
Answer: 147/2
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,BA,DE)
adjacent_complementary_angle(1,CED,DEA)
Problem: As shown in the diagram, the area of triangle DEF is 25, AB=18, FD=10, △DEF is similar to △BCA.. Find the area of △BCA.
问题：如图所示，三角形DEF的面积为25，AB=18，FD=10，△DEF与△BCA是相似三角形。求三角形BCA的面积。
Construction (CDL):
Shape(BC,CA,AB)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(AreaOfTriangle(DEF),25)
Equal(LengthOfLine(AB),18)
Equal(LengthOfLine(FD),10)
SimilarBetweenTriangle(DEF,BCA)
Goal: Value(AreaOfTriangle(BCA))
Answer: 81
Theorem sequence:
similar_triangle_property_line_ratio(1,EFD,CAB)
similar_triangle_property_area_square_ratio(1,EFD,CAB)
Problem: As shown in the diagram, QR=2, QW=15, ST=5, XW=12, WS∥VT, XR is parallel to WS. Find the length of line RS.
问题：如图所示，QR=2，QW=15，ST=5，XW=12，WS平行于VT，XR平行于WS。求直线RS的长度。
Construction (CDL):
Shape(QX,XR,RQ)
Shape(RX,XW,WS,SR)
Shape(SW,WV,VT,TS)
Collinear(QXWV)
Collinear(QRST)
Conditions (CDL):
Equal(LengthOfLine(QR),2)
Equal(LengthOfLine(QW),15)
Equal(LengthOfLine(ST),5)
Equal(LengthOfLine(XW),12)
ParallelBetweenLine(WS,VT)
ParallelBetweenLine(XR,WS)
Goal: Value(LengthOfLine(RS))
Answer: 8
Theorem sequence:
parallel_property_corresponding_angle(1,XR,WS,Q)
parallel_property_corresponding_angle(2,SW,RX,Q)
similar_triangle_judgment_aa(1,QXR,QWS)
line_addition(1,QX,XW)
line_addition(1,QR,RS)
similar_triangle_property_line_ratio(1,XRQ,WSQ)
similar_triangle_property_line_ratio(1,RQX,SQW)
Problem: As shown in the diagram, AB=y, AC=z, AD=8, BD=x, ∠ABC=45°, ∠ADB=60°, BC⊥AC, DB⊥AB. Find the value of x.
问题：如图所示，AB=y，AC=z，AD=8，BD=x，∠ABC=45°，∠ADB=60°，BC⊥AC，DB垂直于AB。求x的值。
Construction (CDL):
Shape(AD,DB,BA)
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AC),z)
Equal(LengthOfLine(AD),8)
Equal(LengthOfLine(BD),x)
Equal(MeasureOfAngle(ABC),45)
Equal(MeasureOfAngle(ADB),60)
PerpendicularBetweenLine(BC,AC)
PerpendicularBetweenLine(DB,AB)
Goal: Value(x)
Answer: 4
Theorem sequence:
right_triangle_judgment_angle(1,DBA)
cosine_theorem(1,DBA)
right_triangle_property_pythagorean(1,DBA)
Problem: As shown in the diagram, AC=2, triangle ABC is an equilateral triangle, the center of ⊙D is D, E bisects segment CA. Find the area of the circle D minus the area of △ABC.
问题：如图所示，AC=2，△ABC为等边△，D是圆D的圆心，E是线段CA的中点。求⊙D的面积减去△ABC的面积。
Construction (CDL):
Shape(DBC,CB)
Shape(BC,CD,DB)
Shape(CE,ED,DC)
Shape(DE,EA,AD)
Shape(BD,DA,AB)
Shape(CA,DAC)
Shape(BA,DAB)
Collinear(BDE)
Collinear(CEA)
Cocircular(D,ABC)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
EquilateralTriangle(ABC)
IsCentreOfCircle(D,D)
IsMidpointOfLine(E,CA)
Goal: Value(Sub(AreaOfCircle(D),AreaOfTriangle(ABC)))
Answer: -sqrt(3)+4*pi/3
Theorem sequence:
line_addition(1,BD,DE)
line_addition(1,CE,EA)
radius_of_circle_property_length_equal(1,DB,D)
radius_of_circle_property_length_equal(1,DC,D)
equilateral_triangle_property_angle(1,ABC)
median_of_triangle_judgment(1,BE,BCA)
isosceles_triangle_property_line_coincidence(2,BCA,E)
right_triangle_judgment_angle(1,CEB)
right_triangle_judgment_angle(1,CED)
right_triangle_property_pythagorean(1,CEB)
right_triangle_property_pythagorean(1,CED)
triangle_area_formula_sine(1,ABC)
circle_area_formula(1,D)
Problem: As shown in the diagram, ∠ACB=38°, ∠ADB=72°, ∠CBA=42°. Find the measure of ∠CDA.
问题：如图所示，∠ACB=38°，∠ADB=72°，∠CBA=42°。求∠CDA的大小。
Construction (CDL):
Shape(BA,AD,DB)
Shape(DA,AC,CD)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),38)
Equal(MeasureOfAngle(ADB),72)
Equal(MeasureOfAngle(CBA),42)
Goal: Value(MeasureOfAngle(CDA))
Answer: 108
Theorem sequence:
adjacent_complementary_angle(1,CDA,ADB)
Problem: As shown in the diagram, the center of circle C is C, the diameter of circle C is SQ, the incenter of quadrilateral PSRQ is C. Find the measure of ∠CRQ.
问题：如图所示，C是圆C的圆心，SQ是圆C的直径，四边形PSRQ的内心为C。求∠CRQ的大小。
Construction (CDL):
Shape(CPS,SP)
Shape(PS,SC,CP)
Shape(CSR,RS)
Shape(SR,RC,CS)
Shape(CR,RQ,QC)
Shape(CRQ,QR)
Shape(CQ,QP,PC)
Shape(PQ,CQP)
Collinear(PCR)
Collinear(SCQ)
Cocircular(C,PSRQ)
Conditions (CDL):
IsCentreOfCircle(C,C)
IsDiameterOfCircle(SQ,C)
IsIncenterOfQuadrilateral(C,PSRQ)
Goal: Value(MeasureOfAngle(CRQ))
Answer: 45
Theorem sequence:
diameter_of_circle_property_right_angle(1,SRQ,C)
angle_addition(1,SRC,CRQ)
Problem: As shown in the diagram, ZX=28, ZY=x, ∠YZX=34°, XY is perpendicular to ZY. Find the value of x.
问题：如图所示，ZX=28，ZY=x，∠YZX=34°，XY垂直于ZY。求x的值。
Construction (CDL):
Shape(ZX,XY,YZ)
Conditions (CDL):
Equal(LengthOfLine(ZX),28)
Equal(LengthOfLine(ZY),x)
Equal(MeasureOfAngle(YZX),34)
PerpendicularBetweenLine(XY,ZY)
Goal: Value(x)
Answer: 28*cos(17*pi/90)
Theorem sequence:
right_triangle_judgment_angle(1,XYZ)
cosine_theorem(1,ZXY)
right_triangle_property_pythagorean(1,XYZ)
Problem: As shown in the diagram, the area of triangle BAC is 150, the area of triangle DEF is 96, AB=x, DE=14, △DEF is similar to △BAC.. Find the ratio of similar triangles DEF and BAC.
问题：如图所示，三角形BAC的面积为150，三角形DEF的面积为96，AB=x，DE=14，△DEF相似于△BAC。求△DEF和△BAC的相似比。
Construction (CDL):
Shape(BA,AC,CB)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(AreaOfTriangle(BAC),150)
Equal(AreaOfTriangle(DEF),96)
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(DE),14)
SimilarBetweenTriangle(DEF,BAC)
Goal: Value(RatioOfSimilarTriangle(DEF,BAC))
Answer: 4/5
Theorem sequence:
similar_triangle_property_area_square_ratio(1,DEF,BAC)
Problem: As shown in the diagram, BC=7, BD=x, CX=2*x+1, DX=15, ∠BXC=∠DXB. Find the value of x.
问题：如图所示，BC=7，BD=x，CX=2*x+1，DX=15，∠BXC=∠DXB。求x的值。
Construction (CDL):
Shape(CB,BX,XC)
Shape(XB,BD,DX)
Collinear(CBD)
Conditions (CDL):
Equal(LengthOfLine(BC),7)
Equal(LengthOfLine(BD),x)
Equal(LengthOfLine(CX),2*x+1)
Equal(LengthOfLine(DX),15)
Equal(MeasureOfAngle(BXC),MeasureOfAngle(DXB))
Goal: Value(x)
Answer: 7
Theorem sequence:
bisector_of_angle_judgment_angle_equal(1,XB,DXC)
bisector_of_angle_property_line_ratio(1,XB,DXC)
Problem: As shown in the diagram, ∠AGK=60°, ∠MJB=50°, HA∥CG, MH∥EN, WD is parallel to BT. Find the measure of ∠ORP.
问题：如图所示，∠AGK=60°，∠MJB=50°，HA∥CG，MH∥EN，WD∥BT。求∠ORP的大小。
Construction (CDL):
Shape(HC,CP,PN,NH)
Shape(SJ,JF,FO,OS)
Shape(RG,GA,AQ,QR)
Shape(WS)
Shape(BJ)
Shape(MJ)
Shape(EF)
Shape(IC)
Shape(NL)
Shape(CS,SO,OP,PC)
Shape(PO,OR,RP)
Shape(NP,PR,RQ,QN)
Shape(OF,FG,GR,RO)
Shape(QD)
Shape(AT)
Shape(AU)
Shape(GK)
Shape(MJ,JB)
Shape(EF,FJ)
Shape(AG,GK)
Shape(NQ,QD)
Shape(LN,NQ)
Collinear(WSORQD)
Collinear(BJFGAT)
Collinear(HCSJM)
Collinear(LNPOFE)
Collinear(ICPRGK)
Collinear(HNQAU)
Conditions (CDL):
Equal(MeasureOfAngle(AGK),60)
Equal(MeasureOfAngle(MJB),50)
ParallelBetweenLine(HA,CG)
ParallelBetweenLine(MH,EN)
ParallelBetweenLine(WD,BT)
Goal: Value(MeasureOfAngle(ORP))
Answer: 60
Theorem sequence:
parallel_property_collinear_extend(3,WD,BT,O)
parallel_property_collinear_extend(3,OD,BT,R)
parallel_property_collinear_extend(3,TB,RO,F)
parallel_property_collinear_extend(3,TF,RO,G)
vertical_angle(1,AGK,FGR)
parallel_property_corresponding_angle(2,GF,RO,P)
Problem: As shown in the diagram, ∠RWZ=45°, ∠TRW=4*x°, ∠WZT=2*x+30°. Find the measure of ∠ZTR.
问题：如图所示，∠RWZ=45°，∠TRW=4*x°，∠WZT=2*x+30°。求∠ZTR的大小。
Construction (CDL):
Shape(ORW,WR)
Shape(ZW,OWZ)
Shape(TZ,OZT)
Shape(RT,OTR)
Shape(RW,WZ,ZT,TR)
Cocircular(O,RWZT)
Conditions (CDL):
Equal(MeasureOfAngle(RWZ),45)
Equal(MeasureOfAngle(TRW),4*x)
Equal(MeasureOfAngle(WZT),2*x+30)
Goal: Value(MeasureOfAngle(ZTR))
Answer: 135
Theorem sequence:
arc_property_circumference_angle_external(1,OZR,W)
arc_property_circumference_angle_internal(1,OZR,T)
Problem: As shown in the diagram, DE=DF, FE=2*x-3, ∠EFD=4*x+11°, ∠FDE=42°. Find the length of line EF.
问题：如图所示，DE=DF，FE=2*x-3，∠EFD=4*x+11°，∠FDE=42°。求直线EF的长度。
Construction (CDL):
Shape(DE,EF,FD)
Conditions (CDL):
Equal(LengthOfLine(DE),LengthOfLine(DF))
Equal(LengthOfLine(FE),2*x-3)
Equal(MeasureOfAngle(EFD),4*x+11)
Equal(MeasureOfAngle(FDE),42)
Goal: Value(LengthOfLine(EF))
Answer: 26
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DEF)
isosceles_triangle_property_angle_equal(1,DEF)
triangle_property_angle_sum(1,DEF)
Problem: As shown in the diagram, JK=ML, KP=4, PM=7, KL is parallel to JM. Find the length of line JL.
问题：如图所示，JK=ML，KP=4，PM=7，KL∥JM。求直线JL的长度。
Construction (CDL):
Shape(KJ,JP,PK)
Shape(PJ,JM,MP)
Shape(PM,ML,LP)
Shape(KP,PL,LK)
Collinear(KPM)
Collinear(JPL)
Conditions (CDL):
Equal(LengthOfLine(JK),LengthOfLine(ML))
Equal(LengthOfLine(KP),4)
Equal(LengthOfLine(PM),7)
ParallelBetweenLine(KL,JM)
Goal: Value(LengthOfLine(JL))
Answer: 11
Theorem sequence:
trapezoid_judgment_parallel(1,KJML)
isosceles_trapezoid_judgment_line_equal(1,KJML)
line_addition(1,KP,PM)
isosceles_trapezoid_property_diagonal_equal(1,KJML)
Problem: As shown in the diagram, RW=14, RZ=18, UZ=5, the centroid of △RST is Z. Find the length of line ZV.
问题：如图所示，RW=14，RZ=18，UZ=5，△RST的重心为O。求直线ZV的长度。
Construction (CDL):
Shape(RW,WZ,ZR)
Shape(WS,SZ,ZW)
Shape(ZS,SV,VZ)
Shape(ZV,VT,TZ)
Shape(ZT,TU,UZ)
Shape(RZ,ZU,UR)
Collinear(RZV)
Collinear(WZT)
Collinear(SZU)
Collinear(RWS)
Collinear(SVT)
Collinear(TUR)
Conditions (CDL):
Equal(LengthOfLine(RW),14)
Equal(LengthOfLine(RZ),18)
Equal(LengthOfLine(UZ),5)
IsCentroidOfTriangle(Z,RST)
Goal: Value(LengthOfLine(ZV))
Answer: 9
Theorem sequence:
centroid_of_triangle_property_line_ratio(1,Z,RST,V)
Problem: As shown in the diagram, AB=6, triangle ABC is an equilateral triangle. Find the area of △CAB.
问题：如图所示，AB=6，△ABC为等边△。求三角形CAB的面积。
Construction (CDL):
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
EquilateralTriangle(ABC)
Goal: Value(AreaOfTriangle(CAB))
Answer: 9*sqrt(3)
Theorem sequence:
equilateral_triangle_property_angle(1,ABC)
triangle_area_formula_sine(1,ABC)
Problem: As shown in the diagram, Add(LengthOfLine(JM)=LengthOfLine(MG), JK=Mul(LengthOfLine(KL),2), KL=Mul(LengthOfLine(LM),2), quadrilateral AKLD is a square, quadrilateral BLMG is a square, quadrilateral EJKF is a square. Find the sum of the area of quadrilateral EJKF, the area of quadrilateral AKLD and the area of quadrilateral BLMG.
问题：如图所示，Add(LengthOfLine(JM)=LengthOfLine(MG)，JK=Mul(LengthOfLine(KL),2)，KL=Mul(LengthOfLine(LM),2)，AKLD是正方形，四边形BLMG是正方形，四边形EJKF是正方形。求四边形EJKF的面积、四边形AKLD的面积与BLMG的面积之和。
Construction (CDL):
Shape(EJ,JK,KA,AF,FE)
Shape(AK,KL,LB,BD,DA)
Shape(BL,LM,MG,GB)
Collinear(FAK)
Collinear(DBL)
Collinear(JKLM)
Conditions (CDL):
Equal(Add(LengthOfLine(JM),LengthOfLine(MG),LengthOfLine(BG),LengthOfLine(BD),LengthOfLine(DA),LengthOfLine(AF),LengthOfLine(FE),LengthOfLine(EJ)),66)
Equal(LengthOfLine(JK),Mul(LengthOfLine(KL),2))
Equal(LengthOfLine(KL),Mul(LengthOfLine(LM),2))
Square(AKLD)
Square(BLMG)
Square(EJKF)
Goal: Value(Add(AreaOfQuadrilateral(EJKF),AreaOfQuadrilateral(AKLD),AreaOfQuadrilateral(BLMG)))
Answer: 189
Theorem sequence:
line_addition(1,FA,AK)
line_addition(1,DB,BL)
line_addition(1,JK,KL)
line_addition(1,JL,LM)
parallelogram_area_formula_sine(1,EJKF)
parallelogram_area_formula_sine(1,AKLD)
parallelogram_area_formula_sine(1,BLMG)
Problem: As shown in the diagram, ∠ABD=3*x-60°, ∠DBC=y°, ∠ECF=2*x+15°, BD is parallel to CE. Find the value of y.
问题：如图所示，∠ABD=3*x-60°，∠DBC=y°，∠ECF=2*x+15°，BD∥CE。求y的值。
Construction (CDL):
Shape(AB)
Shape(BC)
Shape(CF)
Shape(BD)
Shape(CE)
Shape(AB,BD)
Shape(DB,BC)
Shape(BC,CE)
Shape(EC,CF)
Collinear(ABCF)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),3*x-60)
Equal(MeasureOfAngle(DBC),y)
Equal(MeasureOfAngle(ECF),2*x+15)
ParallelBetweenLine(BD,CE)
Goal: Value(y)
Answer: 105
Theorem sequence:
adjacent_complementary_angle(1,ABD,DBC)
parallel_property_corresponding_angle(2,BD,CE,F)
Problem: As shown in the diagram, AB=5, AC=4, ∠CAB=37°. Find the area of triangle CAB.
问题：如图所示，AB=5，AC=4，∠CAB=37°。求△CAB的面积。
Construction (CDL):
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(AC),4)
Equal(MeasureOfAngle(CAB),37)
Goal: Value(AreaOfTriangle(CAB))
Answer: 10*sin(37*pi/180)
Theorem sequence:
triangle_area_formula_sine(1,ABC)