SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠GFC=14*x+9°, ∠GHD=5*x+90°, AD∥BC. Find the value of x.
问题：如图所示，∠GFC=14*x+9°，∠GHD=5*x+90°，AD平行于BC。求x的值。
Construction (CDL):
Shape(GH,HD)
Shape(HF,FC)
Collinear(GHFE)
Collinear(BFC)
Collinear(AHD)
Conditions (CDL):
Equal(MeasureOfAngle(GFC),14*x+9)
Equal(MeasureOfAngle(GHD),5*x+90)
ParallelBetweenLine(AD,BC)
Goal: Value(x)
Answer: 9
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,H)
parallel_property_collinear_extend(3,CB,DH,F)
parallel_property_corresponding_angle(1,HD,FC,G)
Problem: As shown in the diagram, HG=10, G is the center of circle G, the center of circle J is J, K is the center of ⊙K. Find the length of line GJ.
问题：如图所示，HG=10，⊙G的圆心为G，⊙J的圆心为J，圆K的圆心为K。求直线GJ的长度。
Construction (CDL):
Shape(GHB,BG,GH)
Shape(GBF,FG,GB)
Shape(GFL,JGL,GF)
Shape(JGL,KJL,JG)
Shape(KJL,LJ)
Shape(KLJ,JL)
Shape(JLG,GJ,KLJ)
Shape(GLH,HG,JLG)
Collinear(HGF)
Collinear(BGJKL)
Cocircular(G,HBFL)
Cocircular(J,GL)
Cocircular(K,LDJIC)
Conditions (CDL):
Equal(LengthOfLine(HG),10)
IsCentreOfCircle(G,G)
IsCentreOfCircle(J,J)
IsCentreOfCircle(K,K)
Goal: Value(LengthOfLine(GJ))
Answer: 5
Theorem sequence:
radius_of_circle_property_length_equal(1,GH,G)
radius_of_circle_property_length_equal(1,GL,G)
radius_of_circle_property_length_equal(1,JG,J)
radius_of_circle_property_length_equal(1,JL,J)
line_addition(1,BG,GJ)
line_addition(1,BG,GL)
line_addition(1,BJ,JL)
Problem: As shown in the diagram, the area of the sector KJL is 128, JK=x, ∠LKJ=152°, K is the center of ⊙K. Find the value of x.
问题：如图所示，扇形KJL的面积为128，JK=x，∠LKJ=152°，⊙K的圆心为K。求x的值。
Construction (CDL):
Shape(KJL,LK,KJ)
Shape(KLJ,JK,KL)
Cocircular(K,JL)
Conditions (CDL):
Equal(AreaOfSector(KJL),128)
Equal(LengthOfLine(JK),x)
Equal(MeasureOfAngle(LKJ),152)
IsCentreOfCircle(K,K)
Goal: Value(x)
Answer: 24*sqrt(190)/(19*sqrt(pi))
Theorem sequence:
arc_property_center_angle(1,KJL,K)
radius_of_circle_property_length_equal(1,KJ,K)
sector_area_formula(1,KJL)
Problem: As shown in the diagram, ∠HUN=123°, KA is parallel to UQ, QI∥UH. Find the measure of ∠UQA.
问题：如图所示，∠HUN=123°，KA∥UQ，QI∥UH。求∠UQA的大小。
Construction (CDL):
Shape(GK,KF)
Shape(FK,KA)
Shape(KA,AJ)
Shape(JA,AM)
Shape(KU,UQ,QS,SK)
Shape(UK,KG)
Shape(MS,SQ)
Shape(NU,UK)
Shape(SQ,QE)
Shape(HU,UN)
Shape(QU,UH)
Shape(IQ,QU)
Shape(EQ,QI)
Collinear(FKUH)
Collinear(JAQI)
Collinear(GKAM)
Collinear(NUQE)
Conditions (CDL):
Equal(MeasureOfAngle(HUN),123)
ParallelBetweenLine(KA,UQ)
ParallelBetweenLine(QI,UH)
Goal: Value(MeasureOfAngle(UQA))
Answer: 57
Theorem sequence:
parallel_property_corresponding_angle(2,QI,UH,N)
adjacent_complementary_angle(1,IQU,UQA)
Problem: As shown in the diagram, ∠ABJ=3*y-11°, ∠BAE=y+19°, ∠EBA=4*z+2°, ∠JBL=x°, EI∥BJ, AE is perpendicular to BE. Find the value of y.
问题：如图所示，∠ABJ=3*y-11°，∠BAE=y+19°，∠EBA=4*z+2°，∠JBL=x°，EI平行于BJ，AE垂直于BE。求y的值。
Construction (CDL):
Shape(LB,BF)
Shape(FB,BM)
Shape(MB,BG)
Shape(JB,BL)
Shape(AB,BJ)
Shape(BA,AE,EB)
Shape(GE,EA)
Shape(EA,AH)
Shape(IA,AB)
Shape(HA,AI)
Collinear(MBJ)
Collinear(LBEG)
Collinear(FBAH)
Collinear(EAI)
Conditions (CDL):
Equal(MeasureOfAngle(ABJ),3*y-11)
Equal(MeasureOfAngle(BAE),y+19)
Equal(MeasureOfAngle(EBA),4*z+2)
Equal(MeasureOfAngle(JBL),x)
ParallelBetweenLine(EI,BJ)
PerpendicularBetweenLine(AE,BE)
Goal: Value(y)
Answer: 15
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,EI,BJ)
angle_addition(1,EBA,ABJ)
triangle_property_angle_sum(1,AEB)
Problem: As shown in the diagram, ∠DEA=95°, the measure of ⌒FAD is 120. Find the measure of ∠BAC.
问题：如图所示，∠DEA=95°，⌒FAD的角度为120。求∠BAC的大小。
Construction (CDL):
Shape(BE,EC,ECB)
Shape(EB,BA,AE)
Shape(AB,EBA)
Shape(EA,EAD,DE)
Shape(ED,EDC,CE)
Collinear(BED)
Collinear(CEA)
Collinear(AB)
Cocircular(F,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(DEA),95)
Equal(MeasureOfArc(FAD),120)
Goal: Value(MeasureOfAngle(BAC))
Answer: 35
Theorem sequence:
arc_property_circumference_angle_external(1,FAD,B)
adjacent_complementary_angle(1,DEA,AEB)
triangle_property_angle_sum(1,AEB)
Problem: As shown in the diagram, EC=x+4, ED=3*x-6, ∠BAC=55°, ∠CBA=5*y°, ∠DAE=60°, CA and BD are opposite sides of the parallelogram ACBD. Find the value of y.
问题：如图所示，EC=x+4，ED=3*x-6，∠BAC=55°，∠CBA=5*y°，∠DAE=60°，AD和CB是平行四边形ACBD的一组对边。求y的值。
Construction (CDL):
Shape(CE,EA,AC)
Shape(BE,EC,CB)
Shape(EB,BD,DE)
Shape(AE,ED,DA)
Collinear(CED)
Collinear(BEA)
Conditions (CDL):
Equal(LengthOfLine(EC),x+4)
Equal(LengthOfLine(ED),3*x-6)
Equal(MeasureOfAngle(BAC),55)
Equal(MeasureOfAngle(CBA),5*y)
Equal(MeasureOfAngle(DAE),60)
Parallelogram(ACBD)
Goal: Value(y)
Answer: 12
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,CB)
Problem: As shown in the diagram, KN=9, LN=16, PM=Mul(LengthOfLine(KP),2), LK∥RP, KN is perpendicular to MN, LM⊥KM. Find the length of line MN.
问题：如图所示，KN=9，LN=16，PM=Mul(LengthOfLine(KP),2)，LK平行于RP，KN⊥MN，LM⊥KM。求直线MN的长度。
Construction (CDL):
Shape(LR,RQ,QN,NL)
Shape(RM,MQ,QR)
Shape(NQ,QP,PK,KN)
Shape(QM,MP,PQ)
Collinear(LNK)
Collinear(RQP)
Collinear(NQM)
Collinear(LRM)
Collinear(MPK)
Conditions (CDL):
Equal(LengthOfLine(KN),9)
Equal(LengthOfLine(LN),16)
Equal(LengthOfLine(PM),Mul(LengthOfLine(KP),2))
ParallelBetweenLine(LK,RP)
PerpendicularBetweenLine(KN,MN)
PerpendicularBetweenLine(LM,KM)
Goal: Value(LengthOfLine(MN))
Answer: 12
Theorem sequence:
adjacent_complementary_angle(1,KNQ,QNL)
triangle_property_angle_sum(1,KNM)
triangle_property_angle_sum(1,KLM)
similar_triangle_judgment_aa(1,KNM,MNL)
similar_triangle_property_line_ratio(1,KNM,MNL)
similar_triangle_property_line_ratio(1,MKN,LMN)
Problem: As shown in the diagram, XZ=8, YX=8, ∠YXZ=60°. Find the length of line YZ.
问题：如图所示，XZ=8，YX=8，∠YXZ=60°。求直线YZ的长度。
Construction (CDL):
Shape(XZ,ZY,YX)
Conditions (CDL):
Equal(LengthOfLine(XZ),8)
Equal(LengthOfLine(YX),8)
Equal(MeasureOfAngle(YXZ),60)
Goal: Value(LengthOfLine(YZ))
Answer: 8
Theorem sequence:
cosine_theorem(1,XZY)
Problem: As shown in the diagram, △LMN is congruent to △QRS, ML=2*x+11, QR=3*x-9, ∠LMN=y+10°, ∠QRS=2*y-40°. Find the value of y.
问题：如图所示，三角形LMN与三角形QRS是全等三角形，ML=2*x+11，QR=3*x-9，∠LMN=y+10°，∠QRS=2*y-40°。求y的值。
Construction (CDL):
Shape(LM,MN,NL)
Shape(SQ,QR,RS)
Conditions (CDL):
CongruentBetweenTriangle(LMN,QRS)
Equal(LengthOfLine(ML),2*x+11)
Equal(LengthOfLine(QR),3*x-9)
Equal(MeasureOfAngle(LMN),y+10)
Equal(MeasureOfAngle(QRS),2*y-40)
Goal: Value(y)
Answer: 50
Theorem sequence:
congruent_triangle_property_angle_equal(1,MNL,RSQ)
Problem: As shown in the diagram, the measure of arc EBA is 180, the measure of ⌒EBG is 45, the measure of arc EGC is 110. Find the measure of ∠GDB.
问题：如图所示，⌒EBA的角度为180，弧EBG的角度为45，⌒EGC的角度为110。求∠GDB的大小。
Construction (CDL):
Shape(CG,EGC)
Shape(CD,DA,ECA)
Shape(GC,ECA,AB,EBG)
Shape(BA,EAB)
Collinear(DAEB)
Collinear(DCG)
Cocircular(E,ABGC)
Conditions (CDL):
Equal(MeasureOfArc(EBA),180)
Equal(MeasureOfArc(EBG),45)
Equal(MeasureOfArc(EGC),110)
Goal: Value(MeasureOfAngle(GDB))
Answer: 10
Theorem sequence:
arc_addition_measure(1,EBG,EGC)
arc_addition_measure(1,EBC,ECA)
circle_property_circular_power_segment_and_segment_angle(1,DCG,DAB,E)
Problem: As shown in the diagram, AB=10, AE=25/4, BC=x+2, DE=x-1, △ABE is similar to △ACD.. Find the length of line ED.
问题：如图所示，AB=10，AE=25/4，BC=x+2，DE=x-1，三角形ABE与三角形ACD是相似三角形。求直线ED的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(BC,CD,DE,EB)
Collinear(ABC)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AE),25/4)
Equal(LengthOfLine(BC),x+2)
Equal(LengthOfLine(DE),x-1)
SimilarBetweenTriangle(ABE,ACD)
Goal: Value(LengthOfLine(ED))
Answer: 5
Theorem sequence:
line_addition(1,AB,BC)
line_addition(1,AE,ED)
similar_triangle_property_line_ratio(1,EAB,DAC)
similar_triangle_property_line_ratio(1,BEA,CDA)
Problem: As shown in the diagram, AB=2*x+7, CY=x+9, ∠ABY=2*y-5°, ∠BYC=2*y+21°, BA and YC are opposite sides of the ▱ ABYC. Find the value of y.
问题：如图所示，AB=2*x+7，CY=x+9，∠ABY=2*y-5°，∠BYC=2*y+21°，ABYC是▱。求y的值。
Construction (CDL):
Shape(BY,YC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),2*x+7)
Equal(LengthOfLine(CY),x+9)
Equal(MeasureOfAngle(ABY),2*y-5)
Equal(MeasureOfAngle(BYC),2*y+21)
Parallelogram(ABYC)
Goal: Value(y)
Answer: 41
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,BA,YC)
Problem: As shown in the diagram, ∠AGB=30°, the center of ⊙G is G, CG⊥DG, GC is perpendicular to GC. Find the measure of ⌒GCB.
问题：如图所示，∠AGB=30°，G是⊙G的圆心，CG⊥DG，GC垂直于GC。求弧GCB的角度。
Construction (CDL):
Shape(AG,GB,GBA)
Shape(GA,GAF,FG)
Shape(GF,GFD,DG)
Shape(GD,GDC,CG)
Shape(BG,GC,GCB)
Collinear(AGD)
Collinear(BGF)
Collinear(GC)
Cocircular(G,FDCBA)
Conditions (CDL):
Equal(MeasureOfAngle(AGB),30)
IsCentreOfCircle(G,G)
PerpendicularBetweenLine(CG,DG)
PerpendicularBetweenLine(GC,GD)
Goal: Value(MeasureOfArc(GCB))
Answer: 60
Theorem sequence:
vertical_angle(1,AGB,DGF)
angle_addition(1,CGD,DGF)
adjacent_complementary_angle(1,BGC,CGF)
arc_property_center_angle(1,GCB,G)
Problem: As shown in the diagram, PT=16, QS=x, QT=20, RS=12, RS is parallel to PT. Find the length of line ST.
问题：如图所示，PT=16，QS=x，QT=20，RS=12，RS∥PT。求直线ST的长度。
Construction (CDL):
Shape(QR,RS,SQ)
Shape(RP,PT,TS,SR)
Collinear(QRP)
Collinear(QST)
Conditions (CDL):
Equal(LengthOfLine(PT),16)
Equal(LengthOfLine(QS),x)
Equal(LengthOfLine(QT),20)
Equal(LengthOfLine(RS),12)
ParallelBetweenLine(RS,PT)
Goal: Value(LengthOfLine(ST))
Answer: 5
Theorem sequence:
parallel_property_corresponding_angle(1,RS,PT,Q)
parallel_property_corresponding_angle(2,TP,SR,Q)
similar_triangle_judgment_aa(1,QRS,QPT)
similar_triangle_property_line_ratio(1,QRS,QPT)
similar_triangle_property_line_ratio(1,RSQ,PTQ)
line_addition(1,QS,ST)
Problem: As shown in the diagram, ∠FCE=3*x-14°, ∠NXA=2*x+25°, CT is parallel to XN. Find the value of x.
问题：如图所示，∠FCE=3*x-14°，∠NXA=2*x+25°，CT平行于XN。求x的值。
Construction (CDL):
Shape(FC,CE)
Shape(EC,CT)
Shape(XC,CF)
Shape(TC,CX)
Shape(DX,XC)
Shape(CX,XN)
Shape(AX,XD)
Shape(NX,XA)
Collinear(FCT)
Collinear(ECXA)
Collinear(DXN)
Conditions (CDL):
Equal(MeasureOfAngle(FCE),3*x-14)
Equal(MeasureOfAngle(NXA),2*x+25)
ParallelBetweenLine(CT,XN)
Goal: Value(x)
Answer: 39
Theorem sequence:
adjacent_complementary_angle(1,CXN,NXA)
parallel_property_corresponding_angle(1,CT,XN,E)
adjacent_complementary_angle(1,FCE,ECT)
Problem: As shown in the diagram, AB=y, AD=z, BC=x, BD=4, CD=10, AB is perpendicular to CB, DC⊥AC. Find the value of x.
问题：如图所示，AB=y，AD=z，BC=x，BD=4，CD=10，AB⊥CB，DC⊥AC。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Shape(CB,BD,DC)
Collinear(ABD)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AD),z)
Equal(LengthOfLine(BC),x)
Equal(LengthOfLine(BD),4)
Equal(LengthOfLine(CD),10)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(DC,AC)
Goal: Value(x)
Answer: 2*sqrt(21)
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBD)
right_triangle_judgment_angle(1,CBD)
right_triangle_property_pythagorean(1,CBD)
Problem: As shown in the diagram, AB=y, BD=8, CD=x, ∠ACD=43°, BA is perpendicular to CA, CD⊥AD. Find the value of x.
问题：如图所示，AB=y，BD=8，CD=x，∠ACD=43°，BA垂直于CA，CD垂直于AD。求x的值。
Construction (CDL):
Shape(BA,AD,DB)
Shape(DA,AC,CD)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(BD),8)
Equal(LengthOfLine(CD),x)
Equal(MeasureOfAngle(ACD),43)
PerpendicularBetweenLine(BA,CA)
PerpendicularBetweenLine(CD,AD)
Goal: Value(x)
Answer: 8/tan(43*pi/180)**2
Theorem sequence:
adjacent_complementary_angle(1,CDA,ADB)
triangle_property_angle_sum(1,ACD)
angle_addition(1,BAD,DAC)
sine_theorem(1,BAD)
sine_theorem(1,BAC)
line_addition(1,BD,DC)
Problem: As shown in the diagram, ∠EDB=6*x°, ∠YDE=3*x°, AD is perpendicular to YD. Find the value of x.
问题：如图所示，∠EDB=6*x°，∠YDE=3*x°，AD⊥YD。求x的值。
Construction (CDL):
Shape(ED,DB)
Shape(BD,DA)
Shape(YD,DE)
Shape(AD,DY)
Collinear(BDY)
Conditions (CDL):
Equal(MeasureOfAngle(EDB),6*x)
Equal(MeasureOfAngle(YDE),3*x)
PerpendicularBetweenLine(AD,YD)
Goal: Value(x)
Answer: 20
Theorem sequence:
adjacent_complementary_angle(1,YDE,EDB)
Problem: As shown in the diagram, ∠AEI=62°, ∠GEL=5*x+2°. Find the value of x.
问题：如图所示，∠AEI=62°，∠GEL=5*x+2°。求x的值。
Construction (CDL):
Shape(IE,EG)
Shape(AE,EI)
Shape(LE,EA)
Shape(GE,EL)
Collinear(IEL)
Collinear(AEG)
Conditions (CDL):
Equal(MeasureOfAngle(AEI),62)
Equal(MeasureOfAngle(GEL),5*x+2)
Goal: Value(x)
Answer: 12
Theorem sequence:
vertical_angle(1,AEI,GEL)
Problem: As shown in the diagram, triangle RST is congruent to triangle ABC, CB=y+13, ST=2*y-8, ∠ABC=3*x+40°, ∠RST=5*x+20°. Find the value of y.
问题：如图所示，三角形RST与三角形ABC是全等三角形，CB=y+13，ST=2*y-8，∠ABC=3*x+40°，∠RST=5*x+20°。求y的值。
Construction (CDL):
Shape(RS,ST,TR)
Shape(AB,BC,CA)
Conditions (CDL):
CongruentBetweenTriangle(RST,ABC)
Equal(LengthOfLine(CB),y+13)
Equal(LengthOfLine(ST),2*y-8)
Equal(MeasureOfAngle(ABC),3*x+40)
Equal(MeasureOfAngle(RST),5*x+20)
Goal: Value(y)
Answer: 21
Theorem sequence:
congruent_triangle_property_line_equal(1,RST,ABC)
Problem: As shown in the diagram, AC=x, BA=17, BC=14, the center of ⊙A is A, BC is the tangent to circle A. Find the value of x.
问题：如图所示，AC=x，BA=17，BC=14，A是圆A的圆心，圆O的切线为BC。求x的值。
Construction (CDL):
Shape(FB,BC,AFC)
Shape(AF,AFC,CA)
Shape(AC,ACF,FA)
Collinear(BFA)
Cocircular(A,FC)
Conditions (CDL):
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BA),17)
Equal(LengthOfLine(BC),14)
IsCentreOfCircle(A,A)
IsTangentOfCircle(BC,A)
Goal: Value(x)
Answer: sqrt(93)
Theorem sequence:
tangent_of_circle_property_perpendicular(2,BC,A,A)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, AB=12, EB=9, ∠DBA=55°, quadrilateral ADCB is a rhombus. Find the length of line CE.
问题：如图所示，AB=12，EB=9，∠DBA=55°，四边形ADCB是菱形。求直线CE的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(AE,EB,BA)
Shape(ED,DC,CE)
Shape(BE,EC,CB)
Collinear(AEC)
Collinear(DEB)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(EB),9)
Equal(MeasureOfAngle(DBA),55)
Rhombus(ADCB)
Goal: Value(LengthOfLine(CE))
Answer: 3*sqrt(7)
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ADCB,E)
right_triangle_judgment_angle(1,AEB)
right_triangle_property_pythagorean(1,AEB)
parallelogram_property_diagonal_bisection(1,ADCB,E)
Problem: As shown in the diagram, XY=6*x-3, YZ=4*x+5, ∠YWX=33°, ∠ZWY=33°, WX is perpendicular to YX, YZ is perpendicular to WZ. Find the length of line XY.
问题：如图所示，XY=6*x-3，YZ=4*x+5，∠YWX=33°，∠ZWY=33°，WX垂直于YX，YZ垂直于WZ。求直线XY的长度。
Construction (CDL):
Shape(ZW,WY,YZ)
Shape(WX,XY,YW)
Conditions (CDL):
Equal(LengthOfLine(XY),6*x-3)
Equal(LengthOfLine(YZ),4*x+5)
Equal(MeasureOfAngle(YWX),33)
Equal(MeasureOfAngle(ZWY),33)
PerpendicularBetweenLine(WX,YX)
PerpendicularBetweenLine(YZ,WZ)
Goal: Value(LengthOfLine(XY))
Answer: 21
Theorem sequence:
mirror_congruent_triangle_judgment_aas(1,YWX,YZW)
mirror_congruent_triangle_property_line_equal(1,WXY,WYZ)
Problem: As shown in the diagram, the measure of arc GBD is 50, the measure of ⌒GEA is 120. Find the measure of ∠BCD.
问题：如图所示，弧GBD的角度为50，弧GEA的角度为120。求∠BCD的大小。
Construction (CDL):
Shape(ED,GDE)
Shape(DE,GEA,AB,GBD)
Shape(BA,GAB)
Shape(GBD,BC,CD)
Collinear(EDC)
Collinear(ABC)
Cocircular(G,ABDE)
Conditions (CDL):
Equal(MeasureOfArc(GBD),50)
Equal(MeasureOfArc(GEA),120)
Goal: Value(MeasureOfAngle(BCD))
Answer: 35
Theorem sequence:
circle_property_circular_power_segment_and_segment_angle(1,CBA,CDE,G)
Problem: As shown in the diagram, KL=NL, NM=ML, ∠JLK=25°, ∠KLN=18°, ∠NKJ=130°, ∠NLM=20°. Find the measure of ∠MNL.
问题：如图所示，KL=NL，NM=ML，∠JLK=25°，∠KLN=18°，∠NKJ=130°，∠NLM=20°。求∠MNL的大小。
Construction (CDL):
Shape(JL,LK,KJ)
Shape(KL,LN,NK)
Shape(NL,LM,MN)
Conditions (CDL):
Equal(LengthOfLine(KL),LengthOfLine(NL))
Equal(LengthOfLine(NM),LengthOfLine(ML))
Equal(MeasureOfAngle(JLK),25)
Equal(MeasureOfAngle(KLN),18)
Equal(MeasureOfAngle(NKJ),130)
Equal(MeasureOfAngle(NLM),20)
Goal: Value(MeasureOfAngle(MNL))
Answer: 20
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,MNL)
isosceles_triangle_property_angle_equal(1,MNL)
Problem: As shown in the diagram, RS=6*x+2, RT=4*x+3, RW=12, SW=16, TW=9, RW⊥TW. Find the length of line RS.
问题：如图所示，RS=6*x+2，RT=4*x+3，RW=12，SW=16，TW=9，RW⊥TW。求直线RS的长度。
Construction (CDL):
Shape(SW,WR,RS)
Shape(RW,WT,TR)
Collinear(SWT)
Conditions (CDL):
Equal(LengthOfLine(RS),6*x+2)
Equal(LengthOfLine(RT),4*x+3)
Equal(LengthOfLine(RW),12)
Equal(LengthOfLine(SW),16)
Equal(LengthOfLine(TW),9)
PerpendicularBetweenLine(RW,TW)
Goal: Value(LengthOfLine(RS))
Answer: 20
Theorem sequence:
adjacent_complementary_angle(1,SWR,RWT)
right_triangle_judgment_angle(1,SWR)
right_triangle_property_pythagorean(1,SWR)
Problem: As shown in the diagram, ∠AXW=30°, WYZX is a rectangle. Find the measure of ∠XWA.
问题：如图所示，∠AXW=30°，四边形WYZX是矩形。求∠XWA的大小。
Construction (CDL):
Shape(WY,YA,AW)
Shape(AY,YZ,ZA)
Shape(AZ,ZX,XA)
Shape(WA,AX,XW)
Collinear(WAZ)
Collinear(YAX)
Conditions (CDL):
Equal(MeasureOfAngle(AXW),30)
Rectangle(WYZX)
Goal: Value(MeasureOfAngle(XWA))
Answer: 30
Theorem sequence:
rectangle_property_diagonal_equal(1,WYZX)
line_addition(1,WA,AZ)
line_addition(1,XA,AY)
parallelogram_property_diagonal_bisection(1,WYZX,A)
parallelogram_property_diagonal_bisection(1,XWYZ,A)
isosceles_triangle_judgment_line_equal(1,AXW)
isosceles_triangle_property_angle_equal(1,AXW)
Problem: As shown in the diagram, RS=21, RT=44, TS⊥RS. Find the measure of ∠RTS.
问题：如图所示，RS=21，RT=44，TS垂直于RS。求∠RTS的大小。
Construction (CDL):
Shape(RT,TS,SR)
Conditions (CDL):
Equal(LengthOfLine(RS),21)
Equal(LengthOfLine(RT),44)
PerpendicularBetweenLine(TS,RS)
Goal: Value(MeasureOfAngle(RTS))
Answer: 180*acos(sqrt(1495)/44)/pi
Theorem sequence:
right_triangle_judgment_angle(1,TSR)
right_triangle_property_pythagorean(1,TSR)
cosine_theorem(1,TSR)
Problem: As shown in the diagram, YP=YZ, ∠PYZ=26°, YZ⊥XZ. Find the measure of ∠PZX.
问题：如图所示，YP=YZ，∠PYZ=26°，YZ⊥XZ。求∠PZX的大小。
Construction (CDL):
Shape(YZ,ZP,PY)
Shape(PZ,ZX,XP)
Collinear(YPX)
Conditions (CDL):
Equal(LengthOfLine(YP),LengthOfLine(YZ))
Equal(MeasureOfAngle(PYZ),26)
PerpendicularBetweenLine(YZ,XZ)
Goal: Value(MeasureOfAngle(PZX))
Answer: 13
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,YZP)
isosceles_triangle_property_angle_equal(1,YZP)
triangle_property_angle_sum(1,YZP)
angle_addition(1,YZP,PZX)
Problem: As shown in the diagram, AE=8, BE=5, BE=EC, SE=14, quadrilateral SBAC is a kite. Find the area of SBAC.
问题：如图所示，AE=8，BE=5，BE=EC，SE=14，SB和SC是风筝形SBAC的一组临边。求SBAC的面积。
Construction (CDL):
Shape(SB,BE,ES)
Shape(SE,EC,CS)
Shape(EB,BA,AE)
Shape(CE,EA,AC)
Collinear(SEA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AE),8)
Equal(LengthOfLine(BE),5)
Equal(LengthOfLine(BE),LengthOfLine(EC))
Equal(LengthOfLine(SE),14)
Kite(SBAC)
Goal: Value(AreaOfQuadrilateral(SBAC))
Answer: 110
Theorem sequence:
line_addition(1,SE,EA)
line_addition(1,BE,EC)
kite_area_formula_diagonal(1,SBAC)
Problem: As shown in the diagram, ∠WZY=4*x°, ∠XWZ=3*x°, ∠YXW=x°, ∠ZYX=2*x°. Find the measure of ∠YXW.
问题：如图所示，∠WZY=4*x°，∠XWZ=3*x°，∠YXW=x°，∠ZYX=2*x°。求∠YXW的大小。
Construction (CDL):
Shape(XW,WZ,ZY,YX)
Conditions (CDL):
Equal(MeasureOfAngle(WZY),4*x)
Equal(MeasureOfAngle(XWZ),3*x)
Equal(MeasureOfAngle(YXW),x)
Equal(MeasureOfAngle(ZYX),2*x)
Goal: Value(MeasureOfAngle(YXW))
Answer: 36
Theorem sequence:
quadrilateral_property_angle_sum(1,XWZY)
Problem: As shown in the diagram, AB=4, AD=3, CB=8, ED=x. Find the value of x.
问题：如图所示，AB=4，AD=3，CB=8，ED=x。求x的值。
Construction (CDL):
Shape(BA,AD,OBD)
Shape(ED,ODE)
Shape(CB,OBD,DE,OEC)
Shape(BC,OCB)
Collinear(ABC)
Collinear(ADE)
Cocircular(O,DECB)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(CB),8)
Equal(LengthOfLine(ED),x)
Goal: Value(x)
Answer: 13
Theorem sequence:
line_addition(1,AB,BC)
line_addition(1,AD,DE)
circle_property_circular_power_segment_and_segment_line(1,ABC,ADE,O)
Problem: As shown in the diagram, AB=12, EB=9, ∠DBA=55°, ADCB is a rhombus. Find the measure of ∠ADB.
问题：如图所示，AB=12，EB=9，∠DBA=55°，ADCB是菱形。求∠ADB的大小。
Construction (CDL):
Shape(AE,EB,BA)
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Shape(EC,CB,BE)
Collinear(AEC)
Collinear(BED)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(EB),9)
Equal(MeasureOfAngle(DBA),55)
Rhombus(ADCB)
Goal: Value(MeasureOfAngle(ADB))
Answer: 55
Theorem sequence:
sine_theorem(1,ADB)
Problem: As shown in the diagram, AC=12, AD=x, BF=2, CE=6, EF=y, FH=3, CA is the tangent to ⊙D. Find the value of y.
问题：如图所示，AC=12，AD=x，BF=2，CE=6，EF=y，FH=3，圆O的切线为CA。求y的值。
Construction (CDL):
Shape(FA,DAH,HF)
Shape(FH,DHB,BF)
Shape(FB,DBE,EF)
Shape(FE,DEA,AF)
Shape(EC,CA,DEA)
Collinear(ADFB)
Collinear(CEFH)
Cocircular(D,EAHB)
Conditions (CDL):
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(AD),x)
Equal(LengthOfLine(BF),2)
Equal(LengthOfLine(CE),6)
Equal(LengthOfLine(EF),y)
Equal(LengthOfLine(FH),3)
IsTangentOfCircle(CA,D)
Goal: Value(y)
Answer: 15
Theorem sequence:
line_addition(1,EF,FH)
line_addition(1,CE,EH)
circle_property_circular_power_tangent_and_segment_line(1,CA,CEH,D)
Problem: As shown in the diagram, KJ=x+10, LK=x, MK=x+8, PK=x+1. Find the value of x.
问题：如图所示，KJ=x+10，LK=x，MK=x+8，PK=x+1。求x的值。
Construction (CDL):
Shape(JK,KP,OPJ)
Shape(KJ,OJM,MK)
Shape(KM,OML,LK)
Shape(KL,OLP,PK)
Collinear(PKM)
Collinear(JKL)
Cocircular(O,JMLP)
Conditions (CDL):
Equal(LengthOfLine(KJ),x+10)
Equal(LengthOfLine(LK),x)
Equal(LengthOfLine(MK),x+8)
Equal(LengthOfLine(PK),x+1)
Goal: Value(x)
Answer: 8
Theorem sequence:
circle_property_circular_power_chord_and_chord(1,PKM,LKJ,O)
Problem: As shown in the diagram, AB=x, AC=14, BC=y, ∠CBA=30°, BA is perpendicular to CA. Find the value of y.
问题：如图所示，AB=x，AC=14，BC=y，∠CBA=30°，BA⊥CA。求y的值。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(AC),14)
Equal(LengthOfLine(BC),y)
Equal(MeasureOfAngle(CBA),30)
PerpendicularBetweenLine(BA,CA)
Goal: Value(y)
Answer: 28
Theorem sequence:
sine_theorem(1,CBA)
Problem: As shown in the diagram, AB=20, AC=16, BC=19, ∠CBA=x°. Find the value of x.
问题：如图所示，AB=20，AC=16，BC=19，∠CBA=x°。求x的值。
Construction (CDL):
Shape(CB,BA,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),20)
Equal(LengthOfLine(AC),16)
Equal(LengthOfLine(BC),19)
Equal(MeasureOfAngle(CBA),x)
Goal: Value(x)
Answer: 180*acos(101/152)/pi
Theorem sequence:
cosine_theorem(1,BAC)
Problem: As shown in the diagram, ∠MLP=31°, ∠QPN=22°, NM is perpendicular to LM. Find the measure of ∠NQP.
问题：如图所示，∠MLP=31°，∠QPN=22°，NM垂直于LM。求∠NQP的大小。
Construction (CDL):
Shape(LN,NM,ML)
Shape(NQ,QP,PN)
Collinear(LNP)
Collinear(MNQ)
Conditions (CDL):
Equal(MeasureOfAngle(MLP),31)
Equal(MeasureOfAngle(QPN),22)
PerpendicularBetweenLine(NM,LM)
Goal: Value(MeasureOfAngle(NQP))
Answer: 99
Theorem sequence:
triangle_property_angle_sum(1,LNM)
vertical_angle(1,LNM,PNQ)
triangle_property_angle_sum(1,NQP)
Problem: As shown in the diagram, ∠ADF=35°, ∠CFB=70°, AD is perpendicular to BD, CE is perpendicular to AE. Find the measure of ∠FCE.
问题：如图所示，∠ADF=35°，∠CFB=70°，AD垂直于BD，CE垂直于AE。求∠FCE的大小。
Construction (CDL):
Shape(EA,AF,FC,CE)
Shape(AD,DF,FA)
Shape(FD,DB,BF)
Collinear(EAD)
Collinear(AFB)
Collinear(CFD)
Conditions (CDL):
Equal(MeasureOfAngle(ADF),35)
Equal(MeasureOfAngle(CFB),70)
PerpendicularBetweenLine(AD,BD)
PerpendicularBetweenLine(CE,AE)
Goal: Value(MeasureOfAngle(FCE))
Answer: 55
Theorem sequence:
triangle_property_angle_sum(1,EDC)
Problem: As shown in the diagram, AB=21, CA=16, CB=15, AC⊥BC. Find the value of sin(BA).
问题：如图所示，AB=21，CA=16，CB=15，AC垂直于BC。求sin(BA)的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),21)
Equal(LengthOfLine(CA),16)
Equal(LengthOfLine(CB),15)
PerpendicularBetweenLine(AC,BC)
Goal: Value(Sin(MeasureOfAngle(BAC)))
Answer: 5/7
Theorem sequence:
sine_theorem(1,BAC)