SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, JK=10, MN=16, the measure of ⌒KMN is 98, K is the center of circle K, MP⊥KP. Find the measure of arc KNJ.
问题：如图所示，JK=10，MN=16，弧KMN的角度为98，⊙K的圆心为K，MP垂直于KP。求⌒KNJ的角度。
Construction (CDL):
Shape(KJM,MK,KJ)
Shape(KM,MP,PK)
Shape(JK,KN,KNJ)
Shape(KP,PN,NK)
Shape(KML,LP,PM)
Shape(KLN,NP,PL)
Collinear(LPKJ)
Collinear(MPN)
Cocircular(K,JMLN)
Conditions (CDL):
Equal(LengthOfLine(JK),10)
Equal(LengthOfLine(MN),16)
Equal(MeasureOfArc(KMN),98)
IsCentreOfCircle(K,K)
PerpendicularBetweenLine(MP,KP)
Goal: Value(MeasureOfArc(KNJ))
Answer: 131
Theorem sequence:
radius_of_circle_property_length_equal(1,KM,K)
radius_of_circle_property_length_equal(1,KN,K)
arc_property_center_angle(1,KNJ,K)
arc_property_center_angle(1,KMN,K)
perpendicular_bisector_judgment_distance_equal(1,KP,MN)
perpendicular_bisector_property_bisector(1,KP,MN)
angle_addition(1,NKP,PKM)
adjacent_complementary_angle(1,JKN,NKL)
Problem: As shown in the diagram, AB=2, ∠ABC=98°, B is the center of ⊙B. Find the area of the sector BCA.
问题：如图所示，AB=2，∠ABC=98°，圆B的圆心为B。求扇形BCA的面积。
Construction (CDL):
Shape(BAC,CB,BA)
Shape(AB,BC,BCA)
Cocircular(B,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(MeasureOfAngle(ABC),98)
IsCentreOfCircle(B,B)
Goal: Value(AreaOfSector(BCA))
Answer: 49*pi/45
Theorem sequence:
radius_of_circle_property_length_equal(1,BA,B)
arc_property_center_angle(1,BCA,B)
sector_area_formula(1,BCA)
Problem: As shown in the diagram, RS=2*x, ST=4, TU=x+2, VU=3, RV⊥UV, TS⊥US. Find the value of x.
问题：如图所示，RS=2*x，ST=4，TU=x+2，VU=3，RV垂直于UV，TS⊥US。求x的值。
Construction (CDL):
Shape(RV,VU,US,SR)
Shape(SU,UT,TS)
Collinear(VUT)
Collinear(RST)
Conditions (CDL):
Equal(LengthOfLine(RS),2*x)
Equal(LengthOfLine(ST),4)
Equal(LengthOfLine(TU),x+2)
Equal(LengthOfLine(VU),3)
PerpendicularBetweenLine(RV,UV)
PerpendicularBetweenLine(TS,US)
Goal: Value(x)
Answer: 3
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,UTS,RVT)
line_addition(1,VU,UT)
line_addition(1,RS,ST)
mirror_similar_triangle_property_line_ratio(1,UTS,RVT)
mirror_similar_triangle_property_line_ratio(1,SUT,VTR)
Problem: As shown in the diagram, AB=x, AC=12, BC=8, BC⊥AC. Find the value of x.
问题：如图所示，AB=x，AC=12，BC=8，BC垂直于AC。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BC),8)
PerpendicularBetweenLine(BC,AC)
Goal: Value(x)
Answer: 4*sqrt(13)
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, MN=6, MQ=5, ON=3+3/5, QN is parallel to PO. Find the length of line MP.
问题：如图所示，MN=6，MQ=5，ON=3+3/5，QN平行于PO。求直线MP的长度。
Construction (CDL):
Shape(MQ,QN,NM)
Shape(QP,PO,ON,NQ)
Collinear(MQP)
Collinear(MNO)
Conditions (CDL):
Equal(LengthOfLine(MN),6)
Equal(LengthOfLine(MQ),5)
Equal(LengthOfLine(ON),3+3/5)
ParallelBetweenLine(QN,PO)
Goal: Value(LengthOfLine(MP))
Answer: 8
Theorem sequence:
parallel_property_corresponding_angle(1,QN,PO,M)
parallel_property_corresponding_angle(2,OP,NQ,M)
line_addition(1,MN,NO)
similar_triangle_judgment_aa(1,MQN,MPO)
similar_triangle_property_line_ratio(1,QNM,POM)
similar_triangle_property_line_ratio(1,NMQ,OMP)
Problem: As shown in the diagram, SW=14, TW=x, ∠SWT=63°, ∠TSW=31°. Find the value of x.
问题：如图所示，SW=14，TW=x，∠SWT=63°，∠TSW=31°。求x的值。
Construction (CDL):
Shape(TS,SW,WT)
Conditions (CDL):
Equal(LengthOfLine(SW),14)
Equal(LengthOfLine(TW),x)
Equal(MeasureOfAngle(SWT),63)
Equal(MeasureOfAngle(TSW),31)
Goal: Value(x)
Answer: 14*sin(31*pi/180)/cos(pi/45)
Theorem sequence:
triangle_property_angle_sum(1,TSW)
sine_theorem(1,WTS)
Problem: As shown in the diagram, ST=VU, the measure of ⌒MST is 2*x, the measure of arc MVU is 5*x-27, M is the center of circle M. Find the value of x.
问题：如图所示，ST=VU，弧MST的角度为2*x，⌒MVU的角度为5*x-27，M是⊙M的圆心。求x的值。
Construction (CDL):
Shape(MST,TS)
Shape(ST,TM,MS)
Shape(MT,MTV,VM)
Shape(MV,VU,UM)
Shape(MU,MUS,SM)
Shape(UV,MVU)
Collinear(SMV)
Collinear(TMU)
Cocircular(M,STVU)
Conditions (CDL):
Equal(LengthOfLine(ST),LengthOfLine(VU))
Equal(MeasureOfArc(MST),2*x)
Equal(MeasureOfArc(MVU),5*x-27)
IsCentreOfCircle(M,M)
Goal: Value(x)
Answer: 9
Theorem sequence:
congruent_arc_judgment_chord_equal(1,MST,MVU)
congruent_arc_property_measure_equal(1,MST,MVU)
Problem: As shown in the diagram, AD=5, AD=EG, DB=10, Sub(AreaOfQuadrilateral(CBDA)=AreaOfCircle(O)), O is the center of ⊙O, quadrilateral CBDA is a rectangle. Find the value of x.
问题：如图所示，AD=5，AD=EG，DB=10，Sub(AreaOfQuadrilateral(CBDA)=AreaOfCircle(O))，圆O的圆心为O，CBDA是长方形。求x的值。
Construction (CDL):
Shape(GC,CF,OGF)
Shape(FB,BE,OFE)
Shape(AG,OEG,ED,DA)
Shape(OG,OGF,OFE,EO)
Shape(OE,OEG,GO)
Collinear(CGA)
Collinear(CFB)
Collinear(BED)
Collinear(GOE)
Cocircular(O,GFE)
Conditions (CDL):
Equal(LengthOfLine(AD),5)
Equal(LengthOfLine(AD),LengthOfLine(EG))
Equal(LengthOfLine(DB),10)
Equal(Sub(AreaOfQuadrilateral(CBDA),AreaOfCircle(O)),x)
IsCentreOfCircle(O,O)
Rectangle(CBDA)
Goal: Value(x)
Answer: 50-25*pi/4
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,GOE,O)
diameter_of_circle_property_length_equal(1,GE,O)
circle_property_length_of_radius_and_diameter(1,O)
parallelogram_area_formula_sine(1,BDAC)
circle_area_formula(1,O)
Problem: As shown in the diagram, QN=4, QT=8, TM=15, TP=x, PQ is parallel to MN. Find the value of x.
问题：如图所示，QN=4，QT=8，TM=15，TP=x，PQ平行于MN。求x的值。
Construction (CDL):
Shape(MN,NQ,QP,PM)
Shape(PQ,QT,TP)
Collinear(MPT)
Collinear(NQT)
Conditions (CDL):
Equal(LengthOfLine(QN),4)
Equal(LengthOfLine(QT),8)
Equal(LengthOfLine(TM),15)
Equal(LengthOfLine(TP),x)
ParallelBetweenLine(PQ,MN)
Goal: Value(x)
Answer: 10
Theorem sequence:
parallel_property_corresponding_angle(1,PQ,MN,T)
similar_triangle_judgment_aa(1,QTP,NTM)
line_addition(1,NQ,QT)
similar_triangle_property_line_ratio(1,QTP,NTM)
similar_triangle_property_line_ratio(1,PQT,MNT)
Problem: As shown in the diagram, AB=x-2, AD=x, DG=13/2, EB=5, quadrilateral ADCB is similar to quadrilateral AGFE. Find the length of line AB.
问题：如图所示，AB=x-2，AD=x，DG=13/2，EB=5，四边形ADCB相似于四边形AGFE。求直线AB的长度。
Construction (CDL):
Shape(BA,AD,DC,CB)
Shape(EB,BC,CD,DG,GF,FE)
Collinear(ADG)
Collinear(EBA)
Conditions (CDL):
Equal(LengthOfLine(AB),x-2)
Equal(LengthOfLine(AD),x)
Equal(LengthOfLine(DG),13/2)
Equal(LengthOfLine(EB),5)
SimilarBetweenQuadrilateral(ADCB,AGFE)
Goal: Value(LengthOfLine(AB))
Answer: 20/3
Theorem sequence:
line_addition(1,AD,DG)
line_addition(1,EB,BA)
similar_quadrilateral_property_line_ratio(1,ADCB,AGFE)
similar_quadrilateral_property_line_ratio(1,BADC,EAGF)
Problem: As shown in the diagram, the measure of ⌒PNE is 66. Find the measure of ∠EGN.
问题：如图所示，弧PNE的角度为66。求∠EGN的大小。
Construction (CDL):
Shape(PEG,GE)
Shape(EG,GP,PN,NE)
Shape(PNE,EN)
Shape(PG,PGN,NP)
Collinear(GPN)
Cocircular(P,GNE)
Conditions (CDL):
Equal(MeasureOfArc(PNE),66)
Goal: Value(MeasureOfAngle(EGN))
Answer: 33
Theorem sequence:
arc_property_circumference_angle_external(1,PNE,G)
Problem: As shown in the diagram, AB=8, CB=10, CD=9, ED=x, the tangent to circle O is CH, CH is the tangent to ⊙P. Find the value of x.
问题：如图所示，AB=8，CB=10，CD=9，ED=x，CH是圆O的切线，CH是圆O的切线。求x的值。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,OBH,OHA)
Shape(OBH,BC,CH)
Shape(HC,CD,PHD)
Shape(PDE,ED)
Shape(DE,PEH,PHD)
Collinear(CDEF)
Collinear(CBA)
Cocircular(O,ABH)
Cocircular(P,HDE)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(CB),10)
Equal(LengthOfLine(CD),9)
Equal(LengthOfLine(ED),x)
IsTangentOfCircle(CH,O)
IsTangentOfCircle(CH,P)
Goal: Value(x)
Answer: 11
Theorem sequence:
line_addition(1,CD,DE)
line_addition(1,CB,BA)
circle_property_circular_power_tangent_and_segment_line(1,CH,CBA,O)
circle_property_circular_power_tangent_and_segment_line(1,CH,CDE,P)
Problem: As shown in the diagram, DB=70, EC=70, ∠EDC=60°, CB and ED are opposite sides of the ▱ CEDB, CE⊥DE. Find the area of CEDB.
问题：如图所示，DB=70，EC=70，∠EDC=60°，四边形CEDB是▱，CE⊥DE。求四边形CEDB的面积。
Construction (CDL):
Shape(CE,ED,DC)
Shape(BC,CD,DB)
Conditions (CDL):
Equal(LengthOfLine(DB),70)
Equal(LengthOfLine(EC),70)
Equal(MeasureOfAngle(EDC),60)
Parallelogram(CEDB)
PerpendicularBetweenLine(CE,DE)
Goal: Value(AreaOfQuadrilateral(CEDB))
Answer: 4900*sqrt(3)/3
Theorem sequence:
triangle_property_angle_sum(1,CED)
sine_theorem(1,EDC)
parallelogram_area_formula_sine(1,CEDB)
Problem: As shown in the diagram, QV=5, RT=3, ST=12, VS=x, QV is perpendicular to TV, RT⊥ST. Find the length of line VS.
问题：如图所示，QV=5，RT=3，ST=12，VS=x，QV⊥TV，RT⊥ST。求直线VS的长度。
Construction (CDL):
Shape(QV,VT,TR,RQ)
Shape(RT,TS,SR)
Collinear(VTS)
Collinear(QRS)
Conditions (CDL):
Equal(LengthOfLine(QV),5)
Equal(LengthOfLine(RT),3)
Equal(LengthOfLine(ST),12)
Equal(LengthOfLine(VS),x)
PerpendicularBetweenLine(QV,TV)
PerpendicularBetweenLine(RT,ST)
Goal: Value(LengthOfLine(VS))
Answer: 20
Theorem sequence:
similar_triangle_judgment_aa(1,RTS,QVS)
similar_triangle_property_line_ratio(1,RTS,QVS)
similar_triangle_property_line_ratio(1,SRT,SQV)
Problem: As shown in the diagram, ∠ABD=125°, ∠CBA=x°, ∠DBC=155°, B is the center of circle B. Find the value of x.
问题：如图所示，∠ABD=125°，∠CBA=x°，∠DBC=155°，圆B的圆心为B。求x的值。
Construction (CDL):
Shape(BD,BDA,AB)
Shape(BA,BAC,CB)
Shape(BC,BCD,DB)
Cocircular(B,ACD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),125)
Equal(MeasureOfAngle(CBA),x)
Equal(MeasureOfAngle(DBC),155)
IsCentreOfCircle(B,B)
Goal: Value(x)
Answer: 80
Theorem sequence:
round_angle(1,DBC,CBD)
angle_addition(1,CBA,ABD)
Problem: As shown in the diagram, PQ=17, PR=8, QR=15, QR⊥PR. Find the value of tan(PQ).
问题：如图所示，PQ=17，PR=8，QR=15，QR⊥PR。求tan(PQ)的值。
Construction (CDL):
Shape(QR,RP,PQ)
Conditions (CDL):
Equal(LengthOfLine(PQ),17)
Equal(LengthOfLine(PR),8)
Equal(LengthOfLine(QR),15)
PerpendicularBetweenLine(QR,PR)
Goal: Value(Tan(MeasureOfAngle(PQR)))
Answer: 8/15
Theorem sequence:
cosine_theorem(1,QRP)
Problem: As shown in the diagram, AB=y, AC=8, BC=x, ∠ABC=30°, BC is perpendicular to AC. Find the value of x.
问题：如图所示，AB=y，AC=8，BC=x，∠ABC=30°，BC垂直于AC。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(BC),x)
Equal(MeasureOfAngle(ABC),30)
PerpendicularBetweenLine(BC,AC)
Goal: Value(x)
Answer: 8*sqrt(3)
Theorem sequence:
triangle_property_angle_sum(1,BCA)
sine_theorem(1,BCA)
sine_theorem(1,ABC)
Problem: As shown in the diagram, RS=8, TS=12, RS is perpendicular to TS. Find the measure of ∠STR.
问题：如图所示，RS=8，TS=12，RS垂直于TS。求∠STR的大小。
Construction (CDL):
Shape(RS,ST,TR)
Conditions (CDL):
Equal(LengthOfLine(RS),8)
Equal(LengthOfLine(TS),12)
PerpendicularBetweenLine(RS,TS)
Goal: Value(MeasureOfAngle(STR))
Answer: 180*acos(3*sqrt(13)/13)/pi
Theorem sequence:
right_triangle_judgment_angle(1,RST)
right_triangle_property_pythagorean(1,RST)
cosine_theorem(1,TRS)
Problem: As shown in the diagram, BA=5*x, BC=3*y-4, DA=29, DC=25, ∠ADB=34°, ∠DCF=54°, ∠FAD=49°, AB and DC are opposite sides of the ▱ ADCB. Find the value of x.
问题：如图所示，BA=5*x，BC=3*y-4，DA=29，DC=25，∠ADB=34°，∠DCF=54°，∠FAD=49°，DA和CB是平行四边形ADCB的一组对边。求x的值。
Construction (CDL):
Shape(BA,AF,FB)
Shape(FA,AD,DF)
Shape(CF,FD,DC)
Shape(CB,BF,FC)
Collinear(BFD)
Collinear(AFC)
Conditions (CDL):
Equal(LengthOfLine(BA),5*x)
Equal(LengthOfLine(BC),3*y-4)
Equal(LengthOfLine(DA),29)
Equal(LengthOfLine(DC),25)
Equal(MeasureOfAngle(ADB),34)
Equal(MeasureOfAngle(DCF),54)
Equal(MeasureOfAngle(FAD),49)
Parallelogram(ADCB)
Goal: Value(x)
Answer: 5
Theorem sequence:
parallelogram_property_opposite_line_equal(1,DCBA)
Problem: As shown in the diagram, FK=3*x-1, JK=6*y-2, KG=4*y+3, KH=2*x+3, FG and JH are opposite sides of the ▱ FJHG. Find the value of y.
问题：如图所示，FK=3*x-1，JK=6*y-2，KG=4*y+3，KH=2*x+3，FJHG是▱。求y的值。
Construction (CDL):
Shape(FJ,JK,KF)
Shape(KJ,JH,HK)
Shape(KH,HG,GK)
Shape(FK,KG,GF)
Collinear(FKH)
Collinear(JKG)
Conditions (CDL):
Equal(LengthOfLine(FK),3*x-1)
Equal(LengthOfLine(JK),6*y-2)
Equal(LengthOfLine(KG),4*y+3)
Equal(LengthOfLine(KH),2*x+3)
Parallelogram(FJHG)
Goal: Value(y)
Answer: 5/2
Theorem sequence:
parallelogram_property_diagonal_bisection(1,JHGF,K)
Problem: As shown in the diagram, AI=10, BI=13, CI=10, DI=13, AD and AB are one pair of adjacent sides of the kite CDAB, DI is perpendicular to CI. Find the area of CDAB.
问题：如图所示，AI=10，BI=13，CI=10，DI=13，AD和AB是风筝形CDAB的一组临边，DI⊥CI。求四边形CDAB的面积。
Construction (CDL):
Shape(CD,DI,IC)
Shape(ID,DA,AI)
Shape(IA,AB,BI)
Shape(CI,IB,BC)
Collinear(CIA)
Collinear(DIB)
Conditions (CDL):
Equal(LengthOfLine(AI),10)
Equal(LengthOfLine(BI),13)
Equal(LengthOfLine(CI),10)
Equal(LengthOfLine(DI),13)
Kite(CDAB)
PerpendicularBetweenLine(DI,CI)
Goal: Value(AreaOfQuadrilateral(CDAB))
Answer: 260
Theorem sequence:
line_addition(1,CI,IA)
line_addition(1,DI,IB)
kite_area_formula_diagonal(1,CDAB)
Problem: As shown in the diagram, ∠PXR=x°, ∠QPX=4*x°, ∠RQP=2*x°, ∠XRQ=5*x°. Find the measure of ∠QPX.
问题：如图所示，∠PXR=x°，∠QPX=4*x°，∠RQP=2*x°，∠XRQ=5*x°。求∠QPX的大小。
Construction (CDL):
Shape(XR,RQ,QP,PX)
Conditions (CDL):
Equal(MeasureOfAngle(PXR),x)
Equal(MeasureOfAngle(QPX),4*x)
Equal(MeasureOfAngle(RQP),2*x)
Equal(MeasureOfAngle(XRQ),5*x)
Goal: Value(MeasureOfAngle(QPX))
Answer: 120
Theorem sequence:
quadrilateral_property_angle_sum(1,XRQP)
Problem: As shown in the diagram, AB=9, AC=x, BC=12, EF=4, GE=3, GF=4, ∠ABC=44°, ∠BCA=51°, ∠EGF=51°, ∠FEG=85°. Find the value of x.
问题：如图所示，AB=9，AC=x，BC=12，EF=4，GE=3，GF=4，∠ABC=44°，∠BCA=51°，∠EGF=51°，∠FEG=85°。求x的值。
Construction (CDL):
Shape(GF,FE,EG)
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),9)
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BC),12)
Equal(LengthOfLine(EF),4)
Equal(LengthOfLine(GE),3)
Equal(LengthOfLine(GF),4)
Equal(MeasureOfAngle(ABC),44)
Equal(MeasureOfAngle(BCA),51)
Equal(MeasureOfAngle(EGF),51)
Equal(MeasureOfAngle(FEG),85)
Goal: Value(x)
Answer: 9
Theorem sequence:
triangle_property_angle_sum(1,GFE)
mirror_similar_triangle_judgment_aa(1,EGF,ABC)
mirror_similar_triangle_property_line_ratio(1,EGF,ABC)
mirror_similar_triangle_property_line_ratio(1,FEG,BCA)
Problem: As shown in the diagram, ∠DBF=x°, the measure of ⌒ECA is 68, the measure of ⌒EDF is 24, E is the center of ⊙E. Find the value of x.
问题：如图所示，∠DBF=x°，⌒ECA的角度为68，弧EDF的角度为24，圆E的圆心为E。求x的值。
Construction (CDL):
Shape(DB,BF,EDF)
Shape(EFC,CF)
Shape(EAD,DA)
Shape(EDF,FC,ECA,AD)
Collinear(BDA)
Collinear(BFC)
Cocircular(E,DFCA)
Conditions (CDL):
Equal(MeasureOfAngle(DBF),x)
Equal(MeasureOfArc(ECA),68)
Equal(MeasureOfArc(EDF),24)
IsCentreOfCircle(E,E)
Goal: Value(x)
Answer: 22
Theorem sequence:
circle_property_circular_power_segment_and_segment_angle(1,BDA,BFC,E)
Problem: As shown in the diagram, NZ=t+1, XM=2, XN=t-2, XY=10, YZ is parallel to MN. Find the value of t.
问题：如图所示，NZ=t+1，XM=2，XN=t-2，XY=10，YZ平行于MN。求t的值。
Construction (CDL):
Shape(MX,XN,NM)
Shape(MN,NZ,ZY,YM)
Collinear(XMY)
Collinear(XNZ)
Conditions (CDL):
Equal(LengthOfLine(NZ),t+1)
Equal(LengthOfLine(XM),2)
Equal(LengthOfLine(XN),t-2)
Equal(LengthOfLine(XY),10)
ParallelBetweenLine(YZ,MN)
Goal: Value(t)
Answer: 3
Theorem sequence:
line_addition(1,XN,NZ)
parallel_property_corresponding_angle(1,NM,ZY,X)
similar_triangle_judgment_aa(1,MXN,YXZ)
similar_triangle_property_line_ratio(1,MXN,YXZ)
similar_triangle_property_line_ratio(1,NMX,ZYX)
Problem: As shown in the diagram, the radius of circle A is 4. Find the circumference of the ⊙A.
问题：如图所示，⊙A的半径为4。求圆A的周长。
Construction (CDL):
Shape(AN)
Cocircular(A,N)
Conditions (CDL):
Equal(RadiusOfCircle(A),4)
Goal: Value(PerimeterOfCircle(A))
Answer: 8*pi
Theorem sequence:
circle_perimeter_formula(1,A)
Problem: As shown in the diagram, JM=8, KM=16, JL⊥KL, KM⊥LM. Find the length of line LM.
问题：如图所示，JM=8，KM=16，JL⊥KL，KM⊥LM。求直线LM的长度。
Construction (CDL):
Shape(JL,LM,MJ)
Shape(ML,LK,KM)
Collinear(JMK)
Conditions (CDL):
Equal(LengthOfLine(JM),8)
Equal(LengthOfLine(KM),16)
PerpendicularBetweenLine(JL,KL)
PerpendicularBetweenLine(KM,LM)
Goal: Value(LengthOfLine(LM))
Answer: 8*sqrt(2)
Theorem sequence:
adjacent_complementary_angle(1,KML,LMJ)
mirror_similar_triangle_judgment_aa(1,LKM,JLK)
mirror_similar_triangle_property_angle_equal(1,LKM,JLK)
similar_triangle_judgment_aa(1,LMJ,KML)
similar_triangle_property_line_ratio(1,LMJ,KML)
similar_triangle_property_line_ratio(1,JLM,LKM)
Problem: As shown in the diagram, ∠EAB=40°, AB is perpendicular to DB, ABDC is a rectangle. Find the measure of ∠DCB.
问题：如图所示，∠EAB=40°，AB垂直于DB，四边形ABDC是长方形。求∠DCB的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BD,DE)
Shape(ED,DC,CE)
Shape(AE,EC,CA)
Collinear(AED)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),40)
PerpendicularBetweenLine(AB,DB)
Rectangle(ABDC)
Goal: Value(MeasureOfAngle(DCB))
Answer: 40
Theorem sequence:
parallel_property_alternate_interior_angle(2,BA,DC)
parallelogram_property_diagonal_bisection(1,ABDC,E)
parallelogram_property_diagonal_bisection(1,BDCA,E)
rectangle_property_diagonal_equal(1,ABDC)
line_addition(1,AE,ED)
line_addition(1,BE,EC)
isosceles_triangle_judgment_line_equal(1,EDC)
isosceles_triangle_property_angle_equal(1,EDC)
Problem: As shown in the diagram, AD=DE, BC=CE, ∠ADE=x°, ∠CED=72°, ∠ECB=116°. Find the value of x.
问题：如图所示，AD=DE，BC=CE，∠ADE=x°，∠CED=72°，∠ECB=116°。求x的值。
Construction (CDL):
Shape(CB,BE,EC)
Shape(DE,EA,AD)
Shape(CE,ED)
Shape(BE,ED)
Collinear(BEA)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(DE))
Equal(LengthOfLine(BC),LengthOfLine(CE))
Equal(MeasureOfAngle(ADE),x)
Equal(MeasureOfAngle(CED),72)
Equal(MeasureOfAngle(ECB),116)
Goal: Value(x)
Answer: 28
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,CBE)
isosceles_triangle_property_angle_equal(1,CBE)
triangle_property_angle_sum(1,CBE)
flat_angle(1,BEA)
angle_addition(1,BEC,CED)
angle_addition(1,BED,DEA)
isosceles_triangle_judgment_line_equal(1,DEA)
isosceles_triangle_property_angle_equal(1,DEA)
triangle_property_angle_sum(1,DEA)
Problem: As shown in the diagram, AC=x, BD=6, CD=BC, ∠CDA=58°, CA is perpendicular to BA. Find the value of x.
问题：如图所示，AC=x，BD=6，CD=BC，∠CDA=58°，CA⊥BA。求x的值。
Construction (CDL):
Shape(CD,DA,AC)
Shape(CA,AB,BC)
Collinear(DAB)
Conditions (CDL):
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BD),6)
Equal(LengthOfLine(CD),LengthOfLine(BC))
Equal(MeasureOfAngle(CDA),58)
PerpendicularBetweenLine(CA,BA)
Goal: Value(x)
Answer: 3/tan(8*pi/45)
Theorem sequence:
adjacent_complementary_angle(1,DAC,CAB)
perpendicular_bisector_judgment_distance_equal(1,CA,DB)
triangle_property_angle_sum(1,ACD)
line_addition(1,DA,AB)
sine_theorem(1,ACD)
Problem: As shown in the diagram, BE=9, the centroid of △BAC is Q. Find the length of line QE.
问题：如图所示，BE=9，Q是△BAC的重心。求直线QE的长度。
Construction (CDL):
Shape(BF,FQ,QB)
Shape(FA,AQ,QF)
Shape(QA,AE,EQ)
Shape(QE,EC,CQ)
Shape(BQ,QD,DB)
Shape(QC,CD,DQ)
Collinear(BFA)
Collinear(AEC)
Collinear(BDC)
Collinear(BQE)
Collinear(FQC)
Collinear(AQD)
Conditions (CDL):
Equal(LengthOfLine(BE),9)
IsCentroidOfTriangle(Q,BAC)
Goal: Value(LengthOfLine(QE))
Answer: 3
Theorem sequence:
line_addition(1,BQ,QE)
centroid_of_triangle_property_line_ratio(1,Q,BAC,E)
Problem: As shown in the diagram, AD=12, BD=10, DC=15, DE=x, the center of circle F is F. Find the value of x.
问题：如图所示，AD=12，BD=10，DC=15，DE=x，圆F的圆心为F。求x的值。
Construction (CDL):
Shape(DB,FBE,ED)
Shape(DE,FEA,AD)
Shape(DA,FAC,CD)
Shape(DC,FCB,BD)
Collinear(BDA)
Collinear(EDC)
Cocircular(F,BEAC)
Conditions (CDL):
Equal(LengthOfLine(AD),12)
Equal(LengthOfLine(BD),10)
Equal(LengthOfLine(DC),15)
Equal(LengthOfLine(DE),x)
IsCentreOfCircle(F,F)
Goal: Value(x)
Answer: 8
Theorem sequence:
circle_property_circular_power_chord_and_chord(1,BDA,EDC,F)
Problem: As shown in the diagram, ∠AZB=104°, the measure of ⌒ZCB is 94, Z is the center of ⊙Z, AB is parallel to DC. Find the measure of ⌒ZAC.
问题：如图所示，∠AZB=104°，弧ZCB的角度为94，Z是⊙Z的圆心，AB∥DC。求⌒ZAC的角度。
Construction (CDL):
Shape(BA,AZ,ZB)
Shape(BE,EC,CB)
Shape(EA,AD,DE)
Shape(CE,ED,DC)
Shape(BZ,ZA,AE,EB)
Shape(AB,ZBA)
Shape(BC,ZCB)
Shape(CD,ZDC)
Shape(DA,ZAD)
Collinear(BED)
Collinear(AEC)
Cocircular(Z,BADC)
Conditions (CDL):
Equal(MeasureOfAngle(AZB),104)
Equal(MeasureOfArc(ZCB),94)
IsCentreOfCircle(Z,Z)
ParallelBetweenLine(AB,DC)
Goal: Value(MeasureOfArc(ZAC))
Answer: 162
Theorem sequence:
arc_property_center_angle(1,ZBA,Z)
arc_property_circumference_angle_external(1,ZBA,C)
arc_property_circumference_angle_external(1,ZCB,A)
triangle_property_angle_sum(1,ACB)
arc_property_circumference_angle_external(1,ZAC,B)
Problem: As shown in the diagram, the center of ⊙K is K, KL is perpendicular to GL, quadrilateral FJHG is a square. Find the measure of ∠GKH.
问题：如图所示，⊙K的圆心为K，KL垂直于GL，FJHG是正方形。求∠GKH的大小。
Construction (CDL):
Shape(GF,FK,KG)
Shape(FJ,JH,HK,KF)
Shape(GK,KL,LG)
Shape(KH,HL,LK)
Shape(FG,KGF)
Shape(JF,KFJ)
Shape(HJ,KJH)
Shape(GH,KHG)
Collinear(GLH)
Collinear(FKH)
Cocircular(K,FJHG)
Conditions (CDL):
IsCentreOfCircle(K,K)
PerpendicularBetweenLine(KL,GL)
Square(FJHG)
Goal: Value(MeasureOfAngle(GKH))
Answer: 90
Theorem sequence:
radius_of_circle_property_length_equal(1,KF,K)
radius_of_circle_property_length_equal(1,KH,K)
isosceles_triangle_judgment_line_equal(1,GFH)
median_of_triangle_judgment(1,GK,GFH)
isosceles_triangle_property_line_coincidence(2,GFH,K)
Problem: As shown in the diagram, MP=10, ∠MPQ=8*x°, ∠QLM=2*x**2-10°, quadrilateral LMPQ is a rhombus. Find the measure of ∠LPQ.
问题：如图所示，MP=10，∠MPQ=8*x°，∠QLM=2*x**2-10°，四边形LMPQ是菱形。求∠LPQ的大小。
Construction (CDL):
Shape(QL,LR,RQ)
Shape(QR,RP,PQ)
Shape(RL,LM,MR)
Shape(RM,MP,PR)
Collinear(LRP)
Collinear(QRM)
Conditions (CDL):
Equal(LengthOfLine(MP),10)
Equal(MeasureOfAngle(MPQ),8*x)
Equal(MeasureOfAngle(QLM),2*x**2-10)
Rhombus(LMPQ)
Goal: Value(MeasureOfAngle(LPQ))
Answer: 20
Theorem sequence:
angle_addition(1,MPR,RPQ)
parallelogram_property_opposite_angle_equal(1,LMPQ)
kite_property_diagonal_perpendicular_bisection(1,PQLM,R)
perpendicular_bisector_property_bisector(1,PR,QM)
Problem: As shown in the diagram, AB=6*x-10, AB=BD, AE=7*y-11, BD=4*x+2, CE=3*y+5, BE⊥AE, DC⊥EC. Find the value of x.
问题：如图所示，AB=6*x-10，AB=BD，AE=7*y-11，BD=4*x+2，CE=3*y+5，BE⊥AE，DC垂直于EC。求x的值。
Construction (CDL):
Shape(DC,CE,EB,BD)
Shape(BE,EA,AB)
Collinear(CEA)
Collinear(DBA)
Conditions (CDL):
Equal(LengthOfLine(AB),6*x-10)
Equal(LengthOfLine(AB),LengthOfLine(BD))
Equal(LengthOfLine(AE),7*y-11)
Equal(LengthOfLine(BD),4*x+2)
Equal(LengthOfLine(CE),3*y+5)
PerpendicularBetweenLine(BE,AE)
PerpendicularBetweenLine(DC,EC)
Goal: Value(x)
Answer: 6
Theorem sequence:
Problem: As shown in the diagram, DE=7, EX=24, DA⊥XA, XE⊥DE. Find the length of line DX.
问题：如图所示，DE=7，EX=24，DA⊥XA，XE垂直于DE。求直线DX的长度。
Construction (CDL):
Shape(AX,XQ,QD,DA)
Shape(XE,ED,DQ,QX)
Shape(QA,DAQ)
Shape(EQ,DQE)
Shape(TE,DET)
Shape(AT,DTA)
Collinear(XQDT)
Cocircular(D,AETQ)
Conditions (CDL):
Equal(LengthOfLine(DE),7)
Equal(LengthOfLine(EX),24)
PerpendicularBetweenLine(DA,XA)
PerpendicularBetweenLine(XE,DE)
Goal: Value(LengthOfLine(DX))
Answer: 25
Theorem sequence:
right_triangle_judgment_angle(1,XED)
right_triangle_property_pythagorean(1,XED)
Problem: As shown in the diagram, ∠VWX=67°, ∠XWZ=∠WZX, ∠XZY=65°, ∠ZYX=58°. Find the measure of ∠XWZ.
问题：如图所示，∠VWX=67°，∠XWZ=∠WZX，∠XZY=65°，∠ZYX=58°。求∠XWZ的大小。
Construction (CDL):
Shape(VW,WX,XV)
Shape(WZ,ZX,XW)
Shape(YX,XZ,ZY)
Collinear(VXZ)
Collinear(YXW)
Conditions (CDL):
Equal(MeasureOfAngle(VWX),67)
Equal(MeasureOfAngle(XWZ),MeasureOfAngle(WZX))
Equal(MeasureOfAngle(XZY),65)
Equal(MeasureOfAngle(ZYX),58)
Goal: Value(MeasureOfAngle(XWZ))
Answer: 57/2
Theorem sequence:
triangle_property_angle_sum(1,YXZ)
flat_angle(1,YXW)
angle_addition(1,YXZ,ZXW)
triangle_property_angle_sum(1,XWZ)
Problem: As shown in the diagram, ∠ADB=64°, AC⊥DC, BA is perpendicular to CA, BF is perpendicular to DF, CD⊥BD. Find the measure of ∠DAC.
问题：如图所示，∠ADB=64°，AC⊥DC，BA⊥CA，BF⊥DF，CD垂直于BD。求∠DAC的大小。
Construction (CDL):
Shape(BA,AF,FB)
Shape(BF,FD,DB)
Shape(AC,CE,EG,GD,DF,FA)
Collinear(AFD)
Collinear(CEGD)
Conditions (CDL):
Equal(MeasureOfAngle(ADB),64)
PerpendicularBetweenLine(AC,DC)
PerpendicularBetweenLine(BA,CA)
PerpendicularBetweenLine(BF,DF)
PerpendicularBetweenLine(CD,BD)
Goal: Value(MeasureOfAngle(DAC))
Answer: 64
Theorem sequence:
parallel_judgment_per_per(1,CA,DB)
parallel_property_alternate_interior_angle(2,CA,DB)