SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, CD=2, the measure of ⌒ODC is 134, the radius of circle O is 13, O is the center of circle O, BX is perpendicular to DX. Find the measure of arc OBC.
问题：如图所示，CD=2，⌒ODC的角度为134，⊙O的半径为13，圆O的圆心为O，BX垂直于DX。求弧OBC的角度。
Construction (CDL):
Shape(OBC,CX,XB)
Shape(CO,OX,XC)
Shape(OCD,DO,OC)
Shape(DX,XO,OD)
Shape(ODB,BX,XD)
Collinear(CXD)
Collinear(OXB)
Cocircular(O,CDB)
Conditions (CDL):
Equal(LengthOfLine(CD),2)
Equal(MeasureOfArc(ODC),134)
Equal(RadiusOfCircle(O),13)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BX,DX)
Goal: Value(MeasureOfArc(OBC))
Answer: 67
Theorem sequence:
arc_property_center_angle(1,ODC,O)
arc_property_center_angle(1,OBC,O)
adjacent_complementary_angle(1,CXB,BXD)
adjacent_complementary_angle(1,OXC,CXB)
adjacent_complementary_angle(1,BXD,DXO)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OD,O)
mirror_congruent_triangle_judgment_hl(2,OXC,ODX)
mirror_congruent_triangle_property_angle_equal(1,OXC,ODX)
angle_addition(1,COX,XOD)
Problem: As shown in the diagram, GH=10, the center of circle G is G, the center of ⊙J is J, K is the center of ⊙K. Find the length of line FH.
问题：如图所示，GH=10，G是⊙G的圆心，圆J的圆心为J，K是圆K的圆心。求直线FH的长度。
Construction (CDL):
Shape(GHB,BG,GH)
Shape(GBF,FG,GB)
Shape(GFL,JGL,GF)
Shape(JGL,KJL,JG)
Shape(KJL,LK,KJ)
Shape(KLJ,JK,KL)
Shape(JLG,GJ,KLJ)
Shape(GLH,HG,JLG)
Collinear(BGJKL)
Collinear(HGF)
Cocircular(G,BFLH)
Cocircular(J,GL)
Cocircular(K,JL)
Conditions (CDL):
Equal(LengthOfLine(GH),10)
IsCentreOfCircle(G,G)
IsCentreOfCircle(J,J)
IsCentreOfCircle(K,K)
Goal: Value(LengthOfLine(FH))
Answer: 20
Theorem sequence:
line_addition(1,HG,GF)
radius_of_circle_property_length_equal(1,GH,G)
radius_of_circle_property_length_equal(1,GF,G)
Problem: As shown in the diagram, FC=12, FG=x, GC=8, quadrilateral DFGE is similar to quadrilateral AFCB. Find The ratio of similar quadrilaterals DFGE and AFCB.
问题：如图所示，FC=12，FG=x，GC=8，DFGE相似于AFCB。求四边形DFGE和四边形AFCB的相似比。
Construction (CDL):
Shape(AD,DE,EG,GC,CB,BA)
Shape(DF,FG,GE,ED)
Collinear(ADF)
Collinear(FGC)
Conditions (CDL):
Equal(LengthOfLine(FC),12)
Equal(LengthOfLine(FG),x)
Equal(LengthOfLine(GC),8)
SimilarBetweenQuadrilateral(DFGE,AFCB)
Goal: Value(RatioOfSimilarQuadrilateral(DFGE,AFCB))
Answer: 1/3
Theorem sequence:
line_addition(1,FG,GC)
similar_quadrilateral_property_line_ratio(1,FGED,FCBA)
Problem: As shown in the diagram, ∠QRS=x°, ∠RST=2*x+7°, ∠STQ=x°, ∠TQR=2*x+5°. Find the measure of ∠QRS.
问题：如图所示，∠QRS=x°，∠RST=2*x+7°，∠STQ=x°，∠TQR=2*x+5°。求∠QRS的大小。
Construction (CDL):
Shape(QR,RS,ST,TQ)
Conditions (CDL):
Equal(MeasureOfAngle(QRS),x)
Equal(MeasureOfAngle(RST),2*x+7)
Equal(MeasureOfAngle(STQ),x)
Equal(MeasureOfAngle(TQR),2*x+5)
Goal: Value(MeasureOfAngle(QRS))
Answer: 58
Theorem sequence:
quadrilateral_property_angle_sum(1,QRST)
Problem: As shown in the diagram, AE=3*x-6, ED=x+4, ∠ACE=60°, ∠DBE=5*y°, ∠ECD=55°, BD and AC are opposite sides of the ▱ DBAC. Find the value of x.
问题：如图所示，AE=3*x-6，ED=x+4，∠ACE=60°，∠DBE=5*y°，∠ECD=55°，DC和BA是平行四边形DBAC的一组对边。求x的值。
Construction (CDL):
Shape(DB,BE,ED)
Shape(EB,BA,AE)
Shape(EA,AC,CE)
Shape(DE,EC,CD)
Collinear(DEA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AE),3*x-6)
Equal(LengthOfLine(ED),x+4)
Equal(MeasureOfAngle(ACE),60)
Equal(MeasureOfAngle(DBE),5*y)
Equal(MeasureOfAngle(ECD),55)
Parallelogram(DBAC)
Goal: Value(x)
Answer: 5
Theorem sequence:
parallelogram_property_diagonal_bisection(1,DBAC,E)
Problem: As shown in the diagram, AC=24, BA=26, BC=10, BC is perpendicular to AC. Find the value of cos(AB).
问题：如图所示，AC=24，BA=26，BC=10，BC⊥AC。求cos(AB)的值。
Construction (CDL):
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(AC),24)
Equal(LengthOfLine(BA),26)
Equal(LengthOfLine(BC),10)
PerpendicularBetweenLine(BC,AC)
Goal: Value(Cos(MeasureOfAngle(ABC)))
Answer: 5/13
Theorem sequence:
cosine_theorem(1,BCA)
Problem: As shown in the diagram, AB=DF, CB=9*x, ED=2*x+14, L is the center of circle L, FE is perpendicular to LE, LC⊥BC. Find the value of x.
问题：如图所示，AB=DF，CB=9*x，ED=2*x+14，L是⊙L的圆心，FE垂直于LE，LC⊥BC。求x的值。
Construction (CDL):
Shape(LDA,AC,CL,LE,ED)
Shape(LBF,FE,EL,LC,CB)
Shape(LAG,GC,CA)
Shape(LGB,BC,CG)
Shape(LHD,DE,EH)
Shape(LFH,HE,EF)
Collinear(GCEH)
Collinear(ACB)
Collinear(DEF)
Cocircular(L,AGBFHD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(DF))
Equal(LengthOfLine(CB),9*x)
Equal(LengthOfLine(ED),2*x+14)
IsCentreOfCircle(L,L)
PerpendicularBetweenLine(FE,LE)
PerpendicularBetweenLine(LC,BC)
Goal: Value(x)
Answer: 2
Theorem sequence:
line_addition(1,AC,CB)
line_addition(1,DE,EF)
adjacent_complementary_angle(1,ACL,LCB)
circle_property_chord_perpendicular_bisect_chord(1,L,LC,AB)
circle_property_chord_perpendicular_bisect_chord(1,L,LE,FD)
Problem: As shown in the diagram, FL=24, HJ=48, the measure of arc MHP is 65, M is the center of ⊙M, JN is perpendicular to PN, KL is perpendicular to GL. Find the length of line FG.
问题：如图所示，FL=24，HJ=48，弧MHP的角度为65，M是圆M的圆心，JN⊥PN，KL⊥GL。求直线FG的长度。
Construction (CDL):
Shape(MKF,FL,LK)
Shape(MFH,HN,NM,ML,LF)
Shape(MHP,PN,NH)
Shape(MPJ,JN,NP)
Shape(MJG,GL,LM,MN,NJ)
Shape(MGK,KL,LG)
Collinear(FLG)
Collinear(HNJ)
Collinear(KLM)
Collinear(MNP)
Cocircular(M,FHPJGK)
Conditions (CDL):
Equal(LengthOfLine(FL),24)
Equal(LengthOfLine(HJ),48)
Equal(MeasureOfArc(MHP),65)
IsCentreOfCircle(M,M)
PerpendicularBetweenLine(JN,PN)
PerpendicularBetweenLine(KL,GL)
Goal: Value(LengthOfLine(FG))
Answer: 48
Theorem sequence:
line_addition(1,FL,LG)
adjacent_complementary_angle(1,KLG,GLM)
circle_property_chord_perpendicular_bisect_chord(1,M,ML,GF)
Problem: As shown in the diagram, HI=PH, IH=4/3*y+1, JB=6-x, JL=BJ, LJ=2*x+3, PH=2*y, JH∥BP, LI∥BP, LI∥JH. Find the value of x.
问题：如图所示，HI=PH，IH=4/3*y+1，JB=6-x，JL=BJ，LJ=2*x+3，PH=2*y，JH∥BP，LI平行于BP，LI∥JH。求x的值。
Construction (CDL):
Shape(GL,LA)
Shape(MJ,JL)
Shape(DB,NJ)
Shape(OB,BD)
Shape(FP,PB)
Shape(NP,PF)
Shape(CH,HB)
Shape(KI,IH)
Shape(EI,IK)
Shape(AJ,JI)
Shape(LJ,JH,HI,IL)
Shape(JB,BP,PH,HJ)
Collinear(GLIK)
Collinear(MJHC)
Collinear(DBPN)
Collinear(ALJBO)
Collinear(EIHPF)
Conditions (CDL):
Equal(LengthOfLine(HI),LengthOfLine(PH))
Equal(LengthOfLine(IH),4/3*y+1)
Equal(LengthOfLine(JB),6-x)
Equal(LengthOfLine(JL),LengthOfLine(BJ))
Equal(LengthOfLine(LJ),2*x+3)
Equal(LengthOfLine(PH),2*y)
ParallelBetweenLine(JH,BP)
ParallelBetweenLine(LI,BP)
ParallelBetweenLine(LI,JH)
Goal: Value(x)
Answer: 1
Theorem sequence:
Problem: As shown in the diagram, AD=x, BA=z, BC=8, CD=6, DB=y, AC is perpendicular to BC, DB⊥AB. Find the value of z.
问题：如图所示，AD=x，BA=z，BC=8，CD=6，DB=y，AC⊥BC，DB垂直于AB。求z的值。
Construction (CDL):
Shape(DB,BC,CD)
Shape(CB,BA,AC)
Collinear(DCA)
Conditions (CDL):
Equal(LengthOfLine(AD),x)
Equal(LengthOfLine(BA),z)
Equal(LengthOfLine(BC),8)
Equal(LengthOfLine(CD),6)
Equal(LengthOfLine(DB),y)
PerpendicularBetweenLine(AC,BC)
PerpendicularBetweenLine(DB,AB)
Goal: Value(z)
Answer: 40/3
Theorem sequence:
adjacent_complementary_angle(1,ACB,BCD)
right_triangle_judgment_angle(1,BCD)
right_triangle_judgment_angle(1,ACB)
right_triangle_judgment_angle(1,DBA)
right_triangle_property_pythagorean(1,BCD)
right_triangle_property_pythagorean(1,ACB)
right_triangle_property_pythagorean(1,DBA)
line_addition(1,DC,CA)
Problem: As shown in the diagram, AE=x, BE=x, EC=9, ED=4. Find the value of x.
问题：如图所示，AE=x，BE=x，EC=9，ED=4。求x的值。
Construction (CDL):
Shape(GAD,DE,EA)
Shape(GDB,BE,ED)
Shape(GBC,CE,EB)
Shape(GCA,AE,EC)
Collinear(DEC)
Collinear(AEB)
Cocircular(G,ADBC)
Conditions (CDL):
Equal(LengthOfLine(AE),x)
Equal(LengthOfLine(BE),x)
Equal(LengthOfLine(EC),9)
Equal(LengthOfLine(ED),4)
Goal: Value(x)
Answer: 6
Theorem sequence:
circle_property_circular_power_chord_and_chord(1,AEB,DEC,G)
Problem: As shown in the diagram, the diameter of circle H is 18, LM=12, the measure of arc HML is 84, the center of circle H is H, MP is perpendicular to HP. Find the length of line HP.
问题：如图所示，⊙H的直径为18，LM=12，⌒HML的角度为84，⊙H的圆心为H，MP⊥HP。求直线HP的长度。
Construction (CDL):
Shape(HLJ,JH,HP,PL)
Shape(HJM,MP,PH,HJ)
Shape(HMK,KP,PM)
Shape(HKL,LP,PK)
Collinear(JHPK)
Collinear(LPM)
Cocircular(H,JMKL)
Conditions (CDL):
Equal(DiameterOfCircle(H),18)
Equal(LengthOfLine(LM),12)
Equal(MeasureOfArc(HML),84)
IsCentreOfCircle(H,H)
PerpendicularBetweenLine(MP,HP)
Goal: Value(LengthOfLine(HP))
Answer: 3*sqrt(5)
Theorem sequence:
line_addition(1,LP,PM)
line_addition(1,JH,HP)
line_addition(1,HP,PK)
circle_property_length_of_radius_and_diameter(1,H)
circle_property_circular_power_chord_and_chord(1,JPK,LPM,H)
radius_of_circle_property_length_equal(1,HJ,H)
radius_of_circle_property_length_equal(1,HK,H)
circle_property_chord_perpendicular_bisect_chord(1,H,HP,ML)
Problem: As shown in the diagram, ∠ABD=27°, ∠DAB=52°. Find the measure of ∠ADC.
问题：如图所示，∠ABD=27°，∠DAB=52°。求∠ADC的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),27)
Equal(MeasureOfAngle(DAB),52)
Goal: Value(MeasureOfAngle(ADC))
Answer: 79
Theorem sequence:
triangle_property_angle_sum(1,ABD)
adjacent_complementary_angle(1,BDA,ADC)
Problem: As shown in the diagram, AB=y, BC=x, CA=7, ∠ABC=30°, CA is perpendicular to BA. Find the value of x.
问题：如图所示，AB=y，BC=x，CA=7，∠ABC=30°，CA垂直于BA。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(BC),x)
Equal(LengthOfLine(CA),7)
Equal(MeasureOfAngle(ABC),30)
PerpendicularBetweenLine(CA,BA)
Goal: Value(x)
Answer: 14
Theorem sequence:
sine_theorem(1,CAB)
Problem: As shown in the diagram, ∠EDF=39°, ∠FBA=48°, FC is parallel to ED, AF⊥BF, DC⊥FC, FE⊥DE. Find the measure of ∠CFD.
问题：如图所示，∠EDF=39°，∠FBA=48°，FC平行于ED，AF⊥BF，DC垂直于FC，FE⊥DE。求∠CFD的大小。
Construction (CDL):
Shape(AF,FB,BA)
Shape(FD,DC,CF)
Shape(FE,ED,DF)
Collinear(AFE)
Conditions (CDL):
Equal(MeasureOfAngle(EDF),39)
Equal(MeasureOfAngle(FBA),48)
ParallelBetweenLine(FC,ED)
PerpendicularBetweenLine(AF,BF)
PerpendicularBetweenLine(DC,FC)
PerpendicularBetweenLine(FE,DE)
Goal: Value(MeasureOfAngle(CFD))
Answer: 39
Theorem sequence:
parallel_property_alternate_interior_angle(1,FC,ED)
Problem: As shown in the diagram, PS=25/2, PT=15, SR=5, ST∥RQ, TS is parallel to QR. Find the length of line TQ.
问题：如图所示，PS=25/2，PT=15，SR=5，ST平行于RQ，TS平行于QR。求直线TQ的长度。
Construction (CDL):
Shape(PS,ST,TP)
Shape(SR,RQ,QT,TS)
Collinear(PSR)
Collinear(PTQ)
Conditions (CDL):
Equal(LengthOfLine(PS),25/2)
Equal(LengthOfLine(PT),15)
Equal(LengthOfLine(SR),5)
ParallelBetweenLine(ST,RQ)
ParallelBetweenLine(TS,QR)
Goal: Value(LengthOfLine(TQ))
Answer: 6
Theorem sequence:
line_addition(1,PS,SR)
line_addition(1,PT,TQ)
parallel_property_corresponding_angle(1,ST,RQ,P)
parallel_property_corresponding_angle(2,QR,TS,P)
similar_triangle_judgment_aa(1,PST,PRQ)
similar_triangle_property_line_ratio(1,STP,RQP)
similar_triangle_property_line_ratio(1,TPS,QPR)
Problem: As shown in the diagram, ∠DAB=123°, DC is perpendicular to BC. Find the measure of ∠CBA.
问题：如图所示，∠DAB=123°，DC垂直于BC。求∠CBA的大小。
Construction (CDL):
Shape(AC,CB,BA)
Shape(DA,AB)
Collinear(DAC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),123)
PerpendicularBetweenLine(DC,BC)
Goal: Value(MeasureOfAngle(CBA))
Answer: 33
Theorem sequence:
adjacent_complementary_angle(1,DAB,BAC)
triangle_property_angle_sum(1,ACB)
Problem: As shown in the diagram, EC=5, ∠DBE=30°, D is the center of ⊙D, the diameter of ⊙D is BC, the diameter of circle D is CB. Find the area of the circle D minus the area of triangle ECB.
问题：如图所示，EC=5，∠DBE=30°，D是⊙D的圆心，BC是圆D的直径，CB是⊙D的直径。求圆D的面积减去三角形ECB的面积。
Construction (CDL):
Shape(DEC,CE)
Shape(DCB,BD,DC)
Shape(DBE,EB)
Shape(EC,CD,DB,BE)
Collinear(CDB)
Cocircular(D,ECB)
Conditions (CDL):
Equal(LengthOfLine(EC),5)
Equal(MeasureOfAngle(DBE),30)
IsCentreOfCircle(D,D)
IsDiameterOfCircle(BC,D)
IsDiameterOfCircle(CB,D)
Goal: Value(Sub(AreaOfCircle(D),AreaOfTriangle(ECB)))
Answer: -25*sqrt(3)/2+25*pi
Theorem sequence:
diameter_of_circle_property_right_angle(1,BEC,D)
sine_theorem(1,CBE)
diameter_of_circle_property_length_equal(1,CB,D)
circle_property_length_of_radius_and_diameter(1,D)
circle_area_formula(1,D)
triangle_area_formula_sine(1,CBE)
triangle_property_angle_sum(1,ECB)
Problem: As shown in the diagram, BA=x, CB=15, ∠CBA=70°, AC is perpendicular to BC. Find the value of x.
问题：如图所示，BA=x，CB=15，∠CBA=70°，AC⊥BC。求x的值。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(BA),x)
Equal(LengthOfLine(CB),15)
Equal(MeasureOfAngle(CBA),70)
PerpendicularBetweenLine(AC,BC)
Goal: Value(x)
Answer: 15/sin(pi/9)
Theorem sequence:
cosine_theorem(1,BAC)
cosine_theorem(1,CBA)
Problem: As shown in the diagram, AE=7, AE=BE, CE=DE, DE=12, quadrilateral BDAC is a rhombus. Find the area of BDAC.
问题：如图所示，AE=7，AE=BE，CE=DE，DE=12，BDAC是菱形。求BDAC的面积。
Construction (CDL):
Shape(BD,DE,EB)
Shape(DA,AE,ED)
Shape(AC,CE,EA)
Shape(CB,BE,EC)
Collinear(BEA)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(AE),7)
Equal(LengthOfLine(AE),LengthOfLine(BE))
Equal(LengthOfLine(CE),LengthOfLine(DE))
Equal(LengthOfLine(DE),12)
Rhombus(BDAC)
Goal: Value(AreaOfQuadrilateral(BDAC))
Answer: 168
Theorem sequence:
line_addition(1,BE,EA)
line_addition(1,DE,EC)
kite_area_formula_diagonal(1,BDAC)
Problem: As shown in the diagram, JK=2*b+3, JM=3*a, LK=21, LM=45, ∠MLR=30°, ∠RLK=70°, JMLK is a parallelogram. Find the measure of ∠KJR.
问题：如图所示，JK=2*b+3，JM=3*a，LK=21，LM=45，∠MLR=30°，∠RLK=70°，四边形JMLK是平行四边形。求∠KJR的大小。
Construction (CDL):
Shape(JM,MR,RJ)
Shape(ML,LR,RM)
Shape(LK,KR,RL)
Shape(KJ,JR,RK)
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(KJR))
Answer: 30
Theorem sequence:
parallel_property_alternate_interior_angle(1,JK,ML)
Problem: As shown in the diagram, JL=3*y-2, LM=6*x-3, MJ=4, QS=5, ST=3, TQ=2, ∠LMJ=∠STQ, ∠MJL=∠TQS, △JLM is similar to △QST.. Find the value of x.
问题：如图所示，JL=3*y-2，LM=6*x-3，MJ=4，QS=5，ST=3，TQ=2，∠LMJ=∠STQ，∠MJL=∠TQS，△JLM相似于△QST。求x的值。
Construction (CDL):
Shape(JL,LM,MJ)
Shape(ST,TQ,QS)
Conditions (CDL):
Equal(LengthOfLine(JL),3*y-2)
Equal(LengthOfLine(LM),6*x-3)
Equal(LengthOfLine(MJ),4)
Equal(LengthOfLine(QS),5)
Equal(LengthOfLine(ST),3)
Equal(LengthOfLine(TQ),2)
Equal(MeasureOfAngle(LMJ),MeasureOfAngle(STQ))
Equal(MeasureOfAngle(MJL),MeasureOfAngle(TQS))
SimilarBetweenTriangle(JLM,QST)
Goal: Value(x)
Answer: 3/2
Theorem sequence:
similar_triangle_property_line_ratio(1,JLM,QST)
similar_triangle_property_line_ratio(1,LMJ,STQ)
Problem: As shown in the diagram, AB=2*x+11, CA=6*x-9, ∠ABC=∠BCA, ∠BCA=∠CAB. Find the value of x.
问题：如图所示，AB=2*x+11，CA=6*x-9，∠ABC=∠BCA，∠BCA=∠CAB。求x的值。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),2*x+11)
Equal(LengthOfLine(CA),6*x-9)
Equal(MeasureOfAngle(ABC),MeasureOfAngle(BCA))
Equal(MeasureOfAngle(BCA),MeasureOfAngle(CAB))
Goal: Value(x)
Answer: 5
Theorem sequence:
isosceles_triangle_judgment_angle_equal(1,BCA)
isosceles_triangle_judgment_angle_equal(1,CAB)
Problem: As shown in the diagram, ∠HUN=123°, GM is parallel to NE, JI∥FH. Find the measure of ∠KAJ.
问题：如图所示，∠HUN=123°，GM∥NE，JI平行于FH。求∠KAJ的大小。
Construction (CDL):
Shape(GK,Kf)
Shape(NU,UK)
Shape(HU,UN)
Shape(IQ,QU)
Shape(EQ,QI)
Shape(MA,AQ)
Shape(JA,AM)
Shape(FK,KA)
Shape(KU,UQ,QA,AK)
Collinear(GKAM)
Collinear(NUQE)
Collinear(FKUH)
Collinear(JAQI)
Conditions (CDL):
Equal(MeasureOfAngle(HUN),123)
ParallelBetweenLine(GM,NE)
ParallelBetweenLine(JI,FH)
Goal: Value(MeasureOfAngle(KAJ))
Answer: 57
Theorem sequence:
parallel_property_collinear_extend(3,GM,NE,K)
parallel_property_collinear_extend(3,EN,KG,U)
parallel_property_collinear_extend(3,JI,FH,A)
parallel_property_collinear_extend(3,HF,AJ,K)
parallel_property_corresponding_angle(2,UN,KG,F)
parallel_property_corresponding_angle(1,KF,AJ,G)
adjacent_complementary_angle(1,HUN,NUK)
Problem: As shown in the diagram, BC=5, the radius of ⊙J is 10, the radius of circle K is 8, the center of circle J is J, K is the center of ⊙K. Find the length of line JK.
问题：如图所示，BC=5，⊙J的半径为10，圆K的半径为8，J是⊙J的圆心，⊙K的圆心为K。求直线JK的长度。
Construction (CDL):
Shape(JAC,CJ,JA)
Shape(JCA,AJ,JC)
Shape(KDB,BK,KD)
Shape(KBD,DK,KB)
Collinear(AJBCKD)
Cocircular(J,AC)
Cocircular(K,BD)
Conditions (CDL):
Equal(LengthOfLine(BC),5)
Equal(RadiusOfCircle(J),10)
Equal(RadiusOfCircle(K),8)
IsCentreOfCircle(J,J)
IsCentreOfCircle(K,K)
Goal: Value(LengthOfLine(JK))
Answer: 13
Theorem sequence:
line_addition(1,BC,CK)
line_addition(1,JC,CK)
radius_of_circle_property_length_equal(1,JC,J)
radius_of_circle_property_length_equal(1,KB,K)
Problem: As shown in the diagram, AC=x, BA=20, CB=x, AC is perpendicular to BC. Find the value of x.
问题：如图所示，AC=x，BA=20，CB=x，AC⊥BC。求x的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BA),20)
Equal(LengthOfLine(CB),x)
PerpendicularBetweenLine(AC,BC)
Goal: Value(x)
Answer: 10*sqrt(2)
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
Problem: As shown in the diagram, AB=12, BC=y, BD=x, CD=z, DA=10, AB is perpendicular to CB, BD is perpendicular to AD. Find the value of x.
问题：如图所示，AB=12，BC=y，BD=x，CD=z，DA=10，AB⊥CB，BD⊥AD。求x的值。
Construction (CDL):
Shape(BC,CD,DB)
Shape(BD,DA,AB)
Collinear(CDA)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(BC),y)
Equal(LengthOfLine(BD),x)
Equal(LengthOfLine(CD),z)
Equal(LengthOfLine(DA),10)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(BD,AD)
Goal: Value(x)
Answer: 2*sqrt(11)
Theorem sequence:
right_triangle_judgment_angle(1,BDA)
right_triangle_property_pythagorean(1,BDA)
Problem: As shown in the diagram, BA=x, CB=16, ∠ACB=21°, ∠BAC=97°. Find the value of x.
问题：如图所示，BA=x，CB=16，∠ACB=21°，∠BAC=97°。求x的值。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(BA),x)
Equal(LengthOfLine(CB),16)
Equal(MeasureOfAngle(ACB),21)
Equal(MeasureOfAngle(BAC),97)
Goal: Value(x)
Answer: (-sqrt(30)-sqrt(3*sqrt(5)+15)-sqrt(sqrt(5)+5)-sqrt(6)+sqrt(2)+sqrt(10)+sqrt(5*sqrt(5)+25)+sqrt(15*sqrt(5)+75))/sin(83*pi/180)
Theorem sequence:
sine_theorem(1,BAC)
Problem: As shown in the diagram, GJ=8, HG=12, IH=4, KH∥JI. Find the length of line KJ.
问题：如图所示，GJ=8，HG=12，IH=4，KH∥JI。求直线KJ的长度。
Construction (CDL):
Shape(GK,KH,HG)
Shape(KJ,JI,IH,HK)
Collinear(GKJ)
Collinear(IHG)
Conditions (CDL):
Equal(LengthOfLine(GJ),8)
Equal(LengthOfLine(HG),12)
Equal(LengthOfLine(IH),4)
ParallelBetweenLine(KH,JI)
Goal: Value(LengthOfLine(KJ))
Answer: 2
Theorem sequence:
parallel_property_corresponding_angle(1,KH,JI,G)
parallel_property_corresponding_angle(2,IJ,HK,G)
similar_triangle_judgment_aa(1,GKH,GJI)
similar_triangle_property_line_ratio(1,KHG,JIG)
similar_triangle_property_line_ratio(1,HGK,IGJ)
line_addition(1,GK,KJ)
line_addition(1,IH,HG)
Problem: As shown in the diagram, triangle RST is congruent to triangle ABC, BC=y+13, ST=2*y-8, ∠ABC=3*x+40°, ∠RST=5*x+20°. Find the value of x.
问题：如图所示，△RST全等于△ABC，BC=y+13，ST=2*y-8，∠ABC=3*x+40°，∠RST=5*x+20°。求x的值。
Construction (CDL):
Shape(RS,ST,TR)
Shape(CA,AB,BC)
Conditions (CDL):
CongruentBetweenTriangle(RST,ABC)
Equal(LengthOfLine(BC),y+13)
Equal(LengthOfLine(ST),2*y-8)
Equal(MeasureOfAngle(ABC),3*x+40)
Equal(MeasureOfAngle(RST),5*x+20)
Goal: Value(x)
Answer: 10
Theorem sequence:
congruent_triangle_property_angle_equal(1,STR,BCA)
Problem: As shown in the diagram, triangle SRV is congruent to triangle VTS, SR=2*y-1, TS=24.5, VT=24, ∠RVS=78°, ∠VTS=x°, VS⊥RS. Find the value of x.
问题：如图所示，△SRV与△VTS是全等三角形，SR=2*y-1，TS=24.5，VT=24，∠RVS=78°，∠VTS=x°，VS垂直于RS。求x的值。
Construction (CDL):
Shape(SR,RV,VS)
Shape(SV,VT,TS)
Conditions (CDL):
CongruentBetweenTriangle(SRV,VTS)
Equal(LengthOfLine(SR),2*y-1)
Equal(LengthOfLine(TS),24.5)
Equal(LengthOfLine(VT),24)
Equal(MeasureOfAngle(RVS),78)
Equal(MeasureOfAngle(VTS),x)
PerpendicularBetweenLine(VS,RS)
Goal: Value(x)
Answer: 12
Theorem sequence:
triangle_property_angle_sum(1,SRV)
congruent_triangle_property_angle_equal(1,RVS,TSV)
Problem: As shown in the diagram, ∠ABF=10*x-1°, ∠FBC=2*x+25°, BD bisects ∠ABF. Find the measure of ∠DBF.
问题：如图所示，∠ABF=10*x-1°，∠FBC=2*x+25°，BD平分∠ABF。求∠DBF的大小。
Construction (CDL):
Shape(AB,BD)
Shape(DB,BF)
Shape(FB,BC)
Collinear(ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABF),10*x-1)
Equal(MeasureOfAngle(FBC),2*x+25)
IsBisectorOfAngle(BD,ABF)
Goal: Value(MeasureOfAngle(DBF))
Answer: 129/2
Theorem sequence:
angle_addition(1,ABD,DBF)
adjacent_complementary_angle(1,ABF,FBC)
Problem: As shown in the diagram, VW=72, WX=21, XV=75, vw is perpendicular to xw. Find the value of cos(WX).
问题：如图所示，VW=72，WX=21，XV=75，vw垂直于xw。求cos(WX)的值。
Construction (CDL):
Shape(VW,WX,XV)
Conditions (CDL):
Equal(LengthOfLine(VW),72)
Equal(LengthOfLine(WX),21)
Equal(LengthOfLine(XV),75)
PerpendicularBetweenLine(vw,xw)
Goal: Value(Cos(MeasureOfAngle(WXV)))
Answer: 7/25
Theorem sequence:
cosine_theorem(1,XVW)
Problem: As shown in the diagram, ∠LCR=3*x+5°, ∠MCN=60°, ∠RCM=x-1°. Find the measure of ∠RCM.
问题：如图所示，∠LCR=3*x+5°，∠MCN=60°，∠RCM=x-1°。求∠RCM的大小。
Construction (CDL):
Shape(CMR,RC,CM)
Shape(CRL,LC,CR)
Shape(CLN,NC,CL)
Shape(CNM,MC,CN)
Collinear(MCL)
Cocircular(C,RLNM)
Conditions (CDL):
Equal(MeasureOfAngle(LCR),3*x+5)
Equal(MeasureOfAngle(MCN),60)
Equal(MeasureOfAngle(RCM),x-1)
Goal: Value(MeasureOfAngle(RCM))
Answer: 43
Theorem sequence:
adjacent_complementary_angle(1,LCR,RCM)
Problem: As shown in the diagram, CA=4, EG=6, GH=3, AB⊥CB, GH⊥EH, △ADC is similar to △GFE.. Find the length of line AB.
问题：如图所示，CA=4，EG=6，GH=3，AB⊥CB，GH垂直于EH，三角形ADC相似于三角形GFE。求直线AB的长度。
Construction (CDL):
Shape(AD,DB,BA)
Shape(AB,BC,CA)
Shape(GF,FH,HG)
Shape(GH,HE,EG)
Collinear(DBC)
Collinear(FHE)
Conditions (CDL):
Equal(LengthOfLine(CA),4)
Equal(LengthOfLine(EG),6)
Equal(LengthOfLine(GH),3)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(GH,EH)
SimilarBetweenTriangle(ADC,GFE)
Goal: Value(LengthOfLine(AB))
Answer: 2
Theorem sequence:
similar_triangle_property_angle_equal(1,CAD,EGF)
similar_triangle_judgment_aa(1,ABC,GHE)
similar_triangle_property_line_ratio(1,BCA,HEG)
similar_triangle_property_line_ratio(1,CAB,EGH)
Problem: As shown in the diagram, RT=3*x-9, SR=x+9, TS=2*x, triangle SRT is an equilateral triangle. Find the value of x.
问题：如图所示，RT=3*x-9，SR=x+9，TS=2*x，三角形SRT为等边三角形。求x的值。
Construction (CDL):
Shape(SR,RT,TS)
Conditions (CDL):
Equal(LengthOfLine(RT),3*x-9)
Equal(LengthOfLine(SR),x+9)
Equal(LengthOfLine(TS),2*x)
EquilateralTriangle(SRT)
Goal: Value(x)
Answer: 9
Theorem sequence:
Problem: As shown in the diagram, AE=3, BA=2, CB=6, ED=2*x-3, EB∥DC. Find the value of x.
问题：如图所示，AE=3，BA=2，CB=6，ED=2*x-3，EB∥DC。求x的值。
Construction (CDL):
Shape(DC,CB,BE,ED)
Shape(EB,BA,AE)
Collinear(CBA)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AE),3)
Equal(LengthOfLine(BA),2)
Equal(LengthOfLine(CB),6)
Equal(LengthOfLine(ED),2*x-3)
ParallelBetweenLine(EB,DC)
Goal: Value(x)
Answer: 6
Theorem sequence:
parallel_property_corresponding_angle(2,CD,BE,A)
similar_triangle_judgment_aa(1,EBA,DCA)
line_addition(1,CB,BA)
line_addition(1,AE,ED)
similar_triangle_property_line_ratio(1,EBA,DCA)
similar_triangle_property_line_ratio(1,BAE,CAD)
Problem: As shown in the diagram, ∠UZY=2*x+24°, ∠VZU=4*x°, ∠XZW=∠YZX, the center of circle Z is Z. Find the measure of arc ZXW.
问题：如图所示，∠UZY=2*x+24°，∠VZU=4*x°，∠XZW=∠YZX，圆Z的圆心为Z。求弧ZXW的角度。
Construction (CDL):
Shape(ZVW,WZ,ZV)
Shape(ZWX,XZ,ZW)
Shape(ZXY,YZ,ZX)
Shape(ZYU,UZ,ZY)
Shape(ZUV,VZ,ZU)
Collinear(WZU)
Collinear(VZY)
Cocircular(Z,VWXYU)
Conditions (CDL):
Equal(MeasureOfAngle(UZY),2*x+24)
Equal(MeasureOfAngle(VZU),4*x)
Equal(MeasureOfAngle(XZW),MeasureOfAngle(YZX))
IsCentreOfCircle(Z,Z)
Goal: Value(MeasureOfArc(ZXW))
Answer: 308
Theorem sequence:
adjacent_complementary_angle(1,VZU,UZY)
angle_addition(1,YZX,XZW)
adjacent_complementary_angle(1,UZY,YZW)
round_angle(1,XZW,WZX)
arc_property_center_angle(1,ZXW,Z)
Problem: As shown in the diagram, AB=3*x, BA=BC, CA=7, the perimeter of △ACB is 25, the center of circle E is E, the tangent to ⊙E is BA, the tangent to circle E is BC. Find the value of x.
问题：如图所示，AB=3*x，BA=BC，CA=7，三角形ACB的周长为25，⊙E的圆心为E，BA是⊙O的切线，BC是⊙O的切线。求x的值。
Construction (CDL):
Shape(CA,EAC)
Shape(AC,ECA)
Shape(CB,BA,ECA)
Cocircular(E,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),3*x)
Equal(LengthOfLine(BA),LengthOfLine(BC))
Equal(LengthOfLine(CA),7)
Equal(PerimeterOfTriangle(ACB),25)
IsCentreOfCircle(E,E)
IsTangentOfCircle(BA,E)
IsTangentOfCircle(BC,E)
Goal: Value(x)
Answer: 3
Theorem sequence:
triangle_perimeter_formula(1,ACB)