SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AF=y, DA=8, DC=z, DF=x, FE=6, ∠DAE=30°, ∠EBC=45°, DC is parallel to FE, DF is perpendicular to EF, FE is perpendicular to CE. Find the value of z.
问题：如图所示，AF=y，DA=8，DC=z，DF=x，FE=6，∠DAE=30°，∠EBC=45°，DC∥FE，DF⊥EF，FE垂直于CE。求z的值。
Construction (CDL):
Shape(AF,FD,DA)
Shape(DF,FE,EC,CD)
Shape(CE,EB,BC)
Collinear(AFEB)
Conditions (CDL):
Equal(LengthOfLine(AF),y)
Equal(LengthOfLine(DA),8)
Equal(LengthOfLine(DC),z)
Equal(LengthOfLine(DF),x)
Equal(LengthOfLine(FE),6)
Equal(MeasureOfAngle(DAE),30)
Equal(MeasureOfAngle(EBC),45)
ParallelBetweenLine(DC,FE)
PerpendicularBetweenLine(DF,EF)
PerpendicularBetweenLine(FE,CE)
Goal: Value(z)
Answer: 6
Theorem sequence:
parallel_judgment_per_per(1,FD,EC)
parallelogram_judgment_parallel_and_parallel(1,DFEC)
parallelogram_property_opposite_line_equal(1,FECD)
Problem: As shown in the diagram, WY=6*x+3, WY=XW, XW=9*x, YX=4*x+5, YX=WY. Find the value of x.
问题：如图所示，WY=6*x+3，WY=XW，XW=9*x，YX=4*x+5，YX=WY。求x的值。
Construction (CDL):
Shape(XW,WY,YX)
Conditions (CDL):
Equal(LengthOfLine(WY),6*x+3)
Equal(LengthOfLine(WY),LengthOfLine(XW))
Equal(LengthOfLine(XW),9*x)
Equal(LengthOfLine(YX),4*x+5)
Equal(LengthOfLine(YX),LengthOfLine(WY))
Goal: Value(x)
Answer: 1
Theorem sequence:
Problem: As shown in the diagram, DF=z+5, FC=3*z-4, CB and AD are opposite sides of the ▱ CADB. Find the value of z.
问题：如图所示，DF=z+5，FC=3*z-4，CB和AD是平行四边形CADB的一组对边。求z的值。
Construction (CDL):
Shape(CA,AF,FC)
Shape(AD,DF,FA)
Shape(FD,DB,BF)
Shape(CF,FB,BC)
Collinear(CFD)
Collinear(AFB)
Conditions (CDL):
Equal(LengthOfLine(DF),z+5)
Equal(LengthOfLine(FC),3*z-4)
Parallelogram(CADB)
Goal: Value(z)
Answer: 9/2
Theorem sequence:
parallelogram_property_diagonal_bisection(1,CADB,F)
Problem: As shown in the diagram, CA=10, CD=2, CE=t-2, EB=t+1, AB is parallel to DE. Find the length of line CE.
问题：如图所示，CA=10，CD=2，CE=t-2，EB=t+1，AB∥DE。求直线CE的长度。
Construction (CDL):
Shape(DC,CE,ED)
Shape(EB,BA,AD,DE)
Collinear(ADC)
Collinear(CEB)
Conditions (CDL):
Equal(LengthOfLine(CA),10)
Equal(LengthOfLine(CD),2)
Equal(LengthOfLine(CE),t-2)
Equal(LengthOfLine(EB),t+1)
ParallelBetweenLine(AB,DE)
Goal: Value(LengthOfLine(CE))
Answer: 1
Theorem sequence:
parallel_property_corresponding_angle(1,ED,BA,C)
similar_triangle_judgment_aa(1,DCE,ACB)
line_addition(1,CE,EB)
similar_triangle_property_line_ratio(1,DCE,ACB)
similar_triangle_property_line_ratio(1,EDC,BAC)
Problem: As shown in the diagram, AB=c, BC=a, CA=b, ∠ABC=60°, ∠CAB=30°, c=5, BC is perpendicular to AC. Find the value of a.
问题：如图所示，AB=c，BC=a，CA=b，∠ABC=60°，∠CAB=30°，c=5，BC垂直于AC。求a的值。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),c)
Equal(LengthOfLine(BC),a)
Equal(LengthOfLine(CA),b)
Equal(MeasureOfAngle(ABC),60)
Equal(MeasureOfAngle(CAB),30)
Equal(c,5)
PerpendicularBetweenLine(BC,AC)
Goal: Value(a)
Answer: 5/2
Theorem sequence:
sine_theorem(1,BCA)
Problem: As shown in the diagram, ∠ABC=70°, the measure of ⌒EAC is x, E is the center of circle E, the tangent to ⊙E is BA, BC is the tangent to ⊙E. Find the value of x.
问题：如图所示，∠ABC=70°，弧EAC的角度为x，圆E的圆心为E，⊙O的切线为BA，BC是圆O的切线。求x的值。
Construction (CDL):
Shape(AB,BC,EAC)
Shape(EAC,CE,EA)
Shape(ECA,AE,EC)
Cocircular(E,AC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),70)
Equal(MeasureOfArc(EAC),x)
IsCentreOfCircle(E,E)
IsTangentOfCircle(BA,E)
IsTangentOfCircle(BC,E)
Goal: Value(x)
Answer: 110
Theorem sequence:
tangent_of_circle_property_perpendicular(1,BA,E,E)
tangent_of_circle_property_perpendicular(2,BC,E,E)
quadrilateral_property_angle_sum(1,ABCE)
arc_property_center_angle(1,EAC,E)
Problem: As shown in the diagram, WX=4, WZ=4, ∠CYB=23°, WZ⊥YZ, YX is perpendicular to WX. Find the measure of ∠ZYW.
问题：如图所示，WX=4，WZ=4，∠CYB=23°，WZ⊥YZ，YX垂直于WX。求∠ZYW的大小。
Construction (CDL):
Shape(XW,WY,YX)
Shape(WZ,ZY,YW)
Collinear(BXY)
Collinear(CWY)
Collinear(AZY)
Conditions (CDL):
Equal(LengthOfLine(WX),4)
Equal(LengthOfLine(WZ),4)
Equal(MeasureOfAngle(CYB),23)
PerpendicularBetweenLine(WZ,YZ)
PerpendicularBetweenLine(YX,WX)
Goal: Value(MeasureOfAngle(ZYW))
Answer: 23
Theorem sequence:
mirror_congruent_triangle_judgment_hl(1,YXW,YWZ)
mirror_congruent_triangle_property_angle_equal(1,YXW,YWZ)
Problem: As shown in the diagram, WZ=4*a, XW=b+11, XY=3*a+7, YZ=2*b, WX and ZY are opposite sides of the ▱ XWZY. Find the value of b.
问题：如图所示，WZ=4*a，XW=b+11，XY=3*a+7，YZ=2*b，XY和WZ是平行四边形XWZY的一组对边。求b的值。
Construction (CDL):
Shape(XW,WZ,ZY,YX)
Conditions (CDL):
Equal(LengthOfLine(WZ),4*a)
Equal(LengthOfLine(XW),b+11)
Equal(LengthOfLine(XY),3*a+7)
Equal(LengthOfLine(YZ),2*b)
Parallelogram(XWZY)
Goal: Value(b)
Answer: 11
Theorem sequence:
parallelogram_property_opposite_line_equal(1,XWZY)
Problem: As shown in the diagram, JV=3, SY=YZ, TS=UT, TZ=18, UJ=9, ZV=VU. Find the length of line SJ.
问题：如图所示，JV=3，SY=YZ，TS=UT，TZ=18，UJ=9，ZV=VU。求直线SJ的长度。
Construction (CDL):
Shape(SY,YJ,JS)
Shape(JY,YZ,ZJ)
Shape(JZ,ZV,VJ)
Shape(JV,VU,UJ)
Shape(JU,UT,TJ)
Shape(SJ,JT,TS)
Collinear(SJV)
Collinear(YJU)
Collinear(ZJT)
Collinear(SYZ)
Collinear(ZVU)
Collinear(SJV)
Collinear(UTS)
Conditions (CDL):
Equal(LengthOfLine(JV),3)
Equal(LengthOfLine(SY),LengthOfLine(YZ))
Equal(LengthOfLine(TS),LengthOfLine(UT))
Equal(LengthOfLine(TZ),18)
Equal(LengthOfLine(UJ),9)
Equal(LengthOfLine(ZV),LengthOfLine(VU))
Goal: Value(LengthOfLine(SJ))
Answer: 6
Theorem sequence:
median_of_triangle_judgment(1,SV,SZU)
median_of_triangle_judgment(1,UY,USZ)
centroid_of_triangle_judgment_intersection(1,J,ZUS,V,Y)
centroid_of_triangle_property_line_ratio(1,J,SZU,V)
Problem: As shown in the diagram, AB=x, BC=5, ED=x-4, FE=3, ∠ABC=∠FED, ∠CAB=∠EDF. Find the length of line AB.
问题：如图所示，AB=x，BC=5，ED=x-4，FE=3，∠ABC=∠FED，∠CAB=∠EDF。求直线AB的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(ED,DF,FE)
Conditions (CDL):
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(ED),x-4)
Equal(LengthOfLine(FE),3)
Equal(MeasureOfAngle(ABC),MeasureOfAngle(FED))
Equal(MeasureOfAngle(CAB),MeasureOfAngle(EDF))
Goal: Value(LengthOfLine(AB))
Answer: 10
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,CAB,FED)
mirror_similar_triangle_property_line_ratio(1,CAB,FED)
mirror_similar_triangle_property_line_ratio(1,ABC,DFE)
Problem: As shown in the diagram, C is the center of circle C, C is the center of the circumscribed circle of quadrilateral PSRQ, PSRQ is a rhombus. Find the measure of ⌒CPS.
问题：如图所示，C是⊙C的圆心，四边形PSRQ外切圆的圆心为C，PSRQ是菱形。求弧CPS的角度。
Construction (CDL):
Shape(CPS,SP)
Shape(CSR,RS)
Shape(CRQ,QR)
Shape(CQP,PQ)
Shape(CP,PS,SC)
Shape(CS,SR,RC)
Shape(CR,RQ,QC)
Shape(CQ,QP,PC)
Collinear(PCR)
Collinear(SCQ)
Cocircular(C,PSRQ)
Conditions (CDL):
IsCentreOfCircle(C,C)
IsCircumcenterOfQuadrilateral(C,PSRQ)
Rhombus(PSRQ)
Goal: Value(MeasureOfArc(CPS))
Answer: 90
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,PSRQ,C)
arc_property_center_angle(1,CPS,C)
Problem: As shown in the diagram, MO=20, MP=16, OC=x, OF=x, OP=x, PB=5, O is the center of circle O, MP is the tangent to ⊙O. Find the value of x.
问题：如图所示，MO=20，MP=16，OC=x，OF=x，OP=x，PB=5，O是⊙O的圆心，圆O的切线为MP。求x的值。
Construction (CDL):
Shape(MC,OCP,PM)
Shape(OPC,CO,OP)
Shape(BP,OPE,EB)
Shape(OEP,PO,OE)
Shape(EO,OF,OFE)
Shape(FO,OC,OCF)
Collinear(MPB)
Collinear(MCOF)
Cocircular(O,EPCF)
Conditions (CDL):
Equal(LengthOfLine(MO),20)
Equal(LengthOfLine(MP),16)
Equal(LengthOfLine(OC),x)
Equal(LengthOfLine(OF),x)
Equal(LengthOfLine(OP),x)
Equal(LengthOfLine(PB),5)
IsCentreOfCircle(O,O)
IsTangentOfCircle(MP,O)
Goal: Value(x)
Answer: 12
Theorem sequence:
line_addition(1,MC,CF)
line_addition(1,MO,OF)
line_addition(1,CO,OF)
circle_property_circular_power_tangent_and_segment_line(1,MP,MCF,O)
Problem: As shown in the diagram, ∠ADC=32°, ∠DAB=22°, ∠DCB=23°. Find the measure of ∠ABF.
问题：如图所示，∠ADC=32°，∠DAB=22°，∠DCB=23°。求∠ABF的大小。
Construction (CDL):
Shape(CF,FD,DC)
Shape(AB,BF,FA)
Collinear(CFB)
Collinear(DFA)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),32)
Equal(MeasureOfAngle(DAB),22)
Equal(MeasureOfAngle(DCB),23)
Goal: Value(MeasureOfAngle(ABF))
Answer: 33
Theorem sequence:
triangle_property_angle_sum(1,CFD)
triangle_property_angle_sum(1,ABF)
vertical_angle(1,CFD,BFA)
Problem: As shown in the diagram, GD=2*x-2, HD=2*x+4, HG=7, JH=10, ∠GDH=∠JHG. Find the length of line GD.
问题：如图所示，GD=2*x-2，HD=2*x+4，HG=7，JH=10，∠GDH=∠JHG。求直线GD的长度。
Construction (CDL):
Shape(HG,GD,DH)
Shape(HD,DJ,JH)
Collinear(GDJ)
Conditions (CDL):
Equal(LengthOfLine(GD),2*x-2)
Equal(LengthOfLine(HD),2*x+4)
Equal(LengthOfLine(HG),7)
Equal(LengthOfLine(JH),10)
Equal(MeasureOfAngle(GDH),MeasureOfAngle(JHG))
Goal: Value(LengthOfLine(GD))
Answer: 14
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,HGD,JHG)
mirror_similar_triangle_property_line_ratio(1,HGD,JHG)
mirror_similar_triangle_property_line_ratio(1,GDH,GJH)
Problem: As shown in the diagram, ∠LPM=65°, LP is perpendicular to NP, NP is perpendicular to JP. Find the measure of ∠MPN.
问题：如图所示，∠LPM=65°，LP垂直于NP，NP⊥JP。求∠MPN的大小。
Construction (CDL):
Shape(PK,PKJ,JP)
Shape(NP,PJ,PJN)
Shape(MP,PN,PNM)
Shape(LP,PM,PML)
Shape(KP,PL,PLK)
Collinear(JPL)
Collinear(KPM)
Cocircular(P,KJNML)
Conditions (CDL):
Equal(MeasureOfAngle(LPM),65)
PerpendicularBetweenLine(LP,NP)
PerpendicularBetweenLine(NP,JP)
Goal: Value(MeasureOfAngle(MPN))
Answer: 25
Theorem sequence:
flat_angle(1,JPL)
flat_angle(1,KPM)
flat_angle(1,MPK)
angle_addition(1,JPL,LPM)
angle_addition(1,JPK,KPM)
angle_addition(1,NPJ,JPK)
angle_addition(1,MPN,NPK)
Problem: As shown in the diagram, YX=6, ∠ZXY=75°, the radius of ⊙X is 6, the center of circle X is X. Find the length of ⌒XYZ.
问题：如图所示，YX=6，∠ZXY=75°，⊙X的半径为6，⊙X的圆心为X。求⌒XYZ的长度。
Construction (CDL):
Shape(XY,XYZ,ZX)
Shape(XZ,XZY,YX)
Cocircular(X,YZ)
Conditions (CDL):
Equal(LengthOfLine(YX),6)
Equal(MeasureOfAngle(ZXY),75)
Equal(RadiusOfCircle(X),6)
IsCentreOfCircle(X,X)
Goal: Value(LengthOfArc(XYZ))
Answer: 5*pi/2
Theorem sequence:
arc_property_center_angle(1,XYZ,X)
arc_length_formula(1,XYZ)
Problem: As shown in the diagram, AC=22, DB=12, quadrilateral ABCD is a kite. Find the area of ABCD.
问题：如图所示，AC=22，DB=12，四边形ABCD是风筝形。求四边形ABCD的面积。
Construction (CDL):
Shape(CO,OB,BC)
Shape(CD,DO,OC)
Shape(AO,OD,DA)
Shape(OA,AB,BO)
Collinear(BOD)
Collinear(COA)
Conditions (CDL):
Equal(LengthOfLine(AC),22)
Equal(LengthOfLine(DB),12)
Kite(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 132
Theorem sequence:
kite_area_formula_diagonal(1,ABCD)
Problem: As shown in the diagram, ∠EDC=42°, CD is parallel to EF. Find the measure of ∠FEG.
问题：如图所示，∠EDC=42°，CD平行于EF。求∠FEG的大小。
Construction (CDL):
Shape(ED,DC)
Shape(DE,EF)
Shape(FE,EG)
Collinear(DEG)
Conditions (CDL):
Equal(MeasureOfAngle(EDC),42)
ParallelBetweenLine(CD,EF)
Goal: Value(MeasureOfAngle(FEG))
Answer: 138
Theorem sequence:
parallel_property_alternate_interior_angle(2,CD,EF)
flat_angle(1,DEG)
angle_addition(1,DEF,FEG)
Problem: As shown in the diagram, ∠WZY=a+2°, ∠XWZ=a/2+8°, ∠YXW=a°, ∠ZYX=a-28°. Find the measure of ∠ZYX.
问题：如图所示，∠WZY=a+2°，∠XWZ=a/2+8°，∠YXW=a°，∠ZYX=a-28°。求∠ZYX的大小。
Construction (CDL):
Shape(XW,WZ,ZY,YX)
Conditions (CDL):
Equal(MeasureOfAngle(WZY),a+2)
Equal(MeasureOfAngle(XWZ),a/2+8)
Equal(MeasureOfAngle(YXW),a)
Equal(MeasureOfAngle(ZYX),a-28)
Goal: Value(MeasureOfAngle(ZYX))
Answer: 80
Theorem sequence:
quadrilateral_property_angle_sum(1,XWZY)
Problem: As shown in the diagram, BC=4, CA=x+7, DC=6, EC=12-x, AB⊥CB, ED⊥CD, △ABC is similar to △EDC.. Find the length of line AC.
问题：如图所示，BC=4，CA=x+7，DC=6，EC=12-x，AB垂直于CB，ED⊥CD，三角形ABC相似于三角形EDC。求直线AC的长度。
Construction (CDL):
Shape(CA,AB,BC)
Shape(DC,CE,ED)
Collinear(ACE)
Collinear(DCB)
Conditions (CDL):
Equal(LengthOfLine(BC),4)
Equal(LengthOfLine(CA),x+7)
Equal(LengthOfLine(DC),6)
Equal(LengthOfLine(EC),12-x)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(ED,CD)
SimilarBetweenTriangle(ABC,EDC)
Goal: Value(LengthOfLine(AC))
Answer: 38/5
Theorem sequence:
similar_triangle_property_line_ratio(1,ABC,EDC)
similar_triangle_property_line_ratio(1,BCA,DCE)
Problem: As shown in the diagram, SR=9, TS=5, UT=x, the tangent to circle A is SR. Find the value of x.
问题：如图所示，SR=9，TS=5，UT=x，SR是⊙O的切线。求x的值。
Construction (CDL):
Shape(TS,SR,ART)
Shape(ATR,ARU,UT)
Shape(AUT,TU)
Collinear(UTS)
Cocircular(A,UTR)
Conditions (CDL):
Equal(LengthOfLine(SR),9)
Equal(LengthOfLine(TS),5)
Equal(LengthOfLine(UT),x)
IsTangentOfCircle(SR,A)
Goal: Value(x)
Answer: 56/5
Theorem sequence:
line_addition(1,UT,TS)
circle_property_circular_power_tangent_and_segment_line(1,SR,STU,A)
Problem: As shown in the diagram, AB=2, CB=x, KA=5, KC=5, CB⊥KB. Find the length of line BC.
问题：如图所示，AB=2，CB=x，KA=5，KC=5，CB垂直于KB。求直线BC的长度。
Construction (CDL):
Shape(KC,CB,BK)
Shape(BC,KCA,AB)
Shape(KB,BA,KAC,CK)
Collinear(KBA)
Cocircular(K,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(CB),x)
Equal(LengthOfLine(KA),5)
Equal(LengthOfLine(KC),5)
PerpendicularBetweenLine(CB,KB)
Goal: Value(LengthOfLine(BC))
Answer: 4
Theorem sequence:
line_addition(1,KB,BA)
right_triangle_judgment_angle(1,CBK)
right_triangle_property_pythagorean(1,CBK)
Problem: As shown in the diagram, AC=x, BA=11, ∠ACB=21°, CB⊥AB. Find the value of x.
问题：如图所示，AC=x，BA=11，∠ACB=21°，CB⊥AB。求x的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BA),11)
Equal(MeasureOfAngle(ACB),21)
PerpendicularBetweenLine(CB,AB)
Goal: Value(x)
Answer: 176/(-sqrt(30)-sqrt(3)*sqrt(sqrt(5)+5)-sqrt(sqrt(5)+5)-sqrt(6)+sqrt(2)+sqrt(10)+sqrt(5)*sqrt(sqrt(5)+5)+sqrt(15)*sqrt(sqrt(5)+5))
Theorem sequence:
sine_theorem(1,ACB)
Problem: As shown in the diagram, the area of quadrilateral ADBC is 400, the area of quadrilateral EHFG is 64, CB=40, GF=x, quadrilateral ADBC is similar to quadrilateral EHFG. Find the value of x.
问题：如图所示，四边形ADBC的面积为400，四边形EHFG的面积为64，CB=40，GF=x，四边形ADBC相似于四边形EHFG。求x的值。
Construction (CDL):
Shape(AD,DB,BC,CA)
Shape(EH,HF,FG,GE)
Conditions (CDL):
Equal(AreaOfQuadrilateral(ADBC),400)
Equal(AreaOfQuadrilateral(EHFG),64)
Equal(LengthOfLine(CB),40)
Equal(LengthOfLine(GF),x)
SimilarBetweenQuadrilateral(ADBC,EHFG)
Goal: Value(x)
Answer: 16
Theorem sequence:
similar_quadrilateral_property_area_square_ratio(1,ADBC,EHFG)
similar_quadrilateral_property_line_ratio(1,BCAD,FGEH)
Problem: As shown in the diagram, BC=5, CD=5, ∠BAC=38°, AD⊥CD, CB is perpendicular to AB. Find the measure of ∠CAD.
问题：如图所示，BC=5，CD=5，∠BAC=38°，AD⊥CD，CB垂直于AB。求∠CAD的大小。
Construction (CDL):
Shape(AD,DC,CA)
Shape(AC,CB,BA)
Collinear(ADE)
Collinear(ACG)
Collinear(ABF)
Conditions (CDL):
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(CD),5)
Equal(MeasureOfAngle(BAC),38)
PerpendicularBetweenLine(AD,CD)
PerpendicularBetweenLine(CB,AB)
Goal: Value(MeasureOfAngle(CAD))
Answer: 38
Theorem sequence:
mirror_congruent_triangle_judgment_hl(1,ADC,ACB)
mirror_congruent_triangle_property_angle_equal(1,ADC,ACB)
Problem: As shown in the diagram, the center of ⊙O is O, CO is perpendicular to AO. Find the measure of ⌒OAC.
问题：如图所示，O是⊙O的圆心，CO⊥AO。求弧OAC的角度。
Construction (CDL):
Shape(AO,OF,OFA)
Shape(FO,FE,OEF)
Shape(EO,OD,ODE)
Shape(DO,DB,OBD)
Shape(BO,OC,OCB)
Shape(CO,OA,OAC)
Collinear(EOC)
Collinear(BOA)
Cocircular(O,BDEFAC)
Conditions (CDL):
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CO,AO)
Goal: Value(MeasureOfArc(OAC))
Answer: 90
Theorem sequence:
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, AB=41, AD=y, CH=5*x-8, DB=39, DC=72-3*x, the tangent to circle A is CD, the tangent to circle A is CH. Find the value of x.
问题：如图所示，AB=41，AD=y，CH=5*x-8，DB=39，DC=72-3*x，CD是⊙O的切线，CH是⊙O的切线。求x的值。
Construction (CDL):
Shape(HC,CD,ADH)
Shape(AHD,DA,AF,FH)
Shape(FA,AD,ADF)
Shape(AFD,DB,BF)
Collinear(CDB)
Collinear(AFB)
Cocircular(A,HDF)
Conditions (CDL):
Equal(LengthOfLine(AB),41)
Equal(LengthOfLine(AD),y)
Equal(LengthOfLine(CH),5*x-8)
Equal(LengthOfLine(DB),39)
Equal(LengthOfLine(DC),72-3*x)
IsTangentOfCircle(CD,A)
IsTangentOfCircle(CH,A)
Goal: Value(x)
Answer: 10
Theorem sequence:
tangent_of_circle_property_length_equal(1,CD,CH,A)
Problem: As shown in the diagram, CB=11, WX=2*x+5, ∠CBD=24*x+5°, ∠DCB=20*x-4°, ∠YWX=77°, △CBD is mirror congruent to △XYW. Find the value of x.
问题：如图所示，CB=11，WX=2*x+5，∠CBD=24*x+5°，∠DCB=20*x-4°，∠YWX=77°，三角形CBD与三角形XYW是镜像全等三角形。求x的值。
Construction (CDL):
Shape(CB,BD,DC)
Shape(XY,YW,WX)
Conditions (CDL):
Equal(LengthOfLine(CB),11)
Equal(LengthOfLine(WX),2*x+5)
Equal(MeasureOfAngle(CBD),24*x+5)
Equal(MeasureOfAngle(DCB),20*x-4)
Equal(MeasureOfAngle(YWX),77)
MirrorCongruentBetweenTriangle(CBD,XYW)
Goal: Value(x)
Answer: 3
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,BDC,WXY)
Problem: As shown in the diagram, RS=5, SP=3, ∠PQR=128°, quadrilateral QRSP is a ▱. Find the length of line PQ.
问题：如图所示，RS=5，SP=3，∠PQR=128°，四边形QRSP是平行四边形。求直线PQ的长度。
Construction (CDL):
Shape(QR,RS,SP,PQ)
Conditions (CDL):
Equal(LengthOfLine(RS),5)
Equal(LengthOfLine(SP),3)
Equal(MeasureOfAngle(PQR),128)
Parallelogram(QRSP)
Goal: Value(LengthOfLine(PQ))
Answer: 5
Theorem sequence:
parallelogram_property_opposite_line_equal(1,RSPQ)
Problem: As shown in the diagram, JH=FG, ∠GBF=70°, ∠HBJ=x°, B is the center of ⊙B. Find the value of x.
问题：如图所示，JH=FG，∠GBF=70°，∠HBJ=x°，圆B的圆心为B。求x的值。
Construction (CDL):
Shape(BFG,GF)
Shape(FG,GB,BF)
Shape(FB,BH,BHF)
Shape(HB,BJ,JH)
Shape(XJH,HJ)
Shape(BG,BGJ,JB)
Cocircular(B,HFGJ)
Conditions (CDL):
Equal(LengthOfLine(JH),LengthOfLine(FG))
Equal(MeasureOfAngle(GBF),70)
Equal(MeasureOfAngle(HBJ),x)
IsCentreOfCircle(B,B)
Goal: Value(x)
Answer: 70
Theorem sequence:
arc_property_center_angle(1,BJH,B)
arc_property_center_angle(1,BFG,B)
congruent_arc_judgment_chord_equal(1,BJH,BFG)
congruent_arc_property_measure_equal(1,BJH,BFG)
Problem: As shown in the diagram, AB=14, BC=y, CA=x, ∠BCA=60°, CA is perpendicular to BA. Find the value of y.
问题：如图所示，AB=14，BC=y，CA=x，∠BCA=60°，CA垂直于BA。求y的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),14)
Equal(LengthOfLine(BC),y)
Equal(LengthOfLine(CA),x)
Equal(MeasureOfAngle(BCA),60)
PerpendicularBetweenLine(CA,BA)
Goal: Value(y)
Answer: 28*sqrt(3)/3
Theorem sequence:
sine_theorem(1,BCA)
Problem: As shown in the diagram, the diameter of ⊙H is 14, DF=14, the center of circle H is H, AD⊥FD, CA is perpendicular to DA, DF⊥CF, FC is perpendicular to CC. Find the circumference of the circle H.
问题：如图所示，⊙H的直径为14，DF=14，⊙H的圆心为H，AD垂直于FD，CA⊥DA，DF垂直于CF，FC⊥CC。求圆H的周长。
Construction (CDL):
Shape(AG,HEG,EA)
Shape(CE,HIE,IC)
Shape(HMI,MF,FI)
Shape(HGM,GD,DM)
Shape(H)
Collinear(AEC)
Collinear(CIF)
Collinear(DMF)
Collinear(AGD)
Cocircular(H,EGMI)
Conditions (CDL):
Equal(DiameterOfCircle(H),14)
Equal(LengthOfLine(DF),14)
IsCentreOfCircle(H,H)
PerpendicularBetweenLine(AD,FD)
PerpendicularBetweenLine(CA,DA)
PerpendicularBetweenLine(DF,CF)
PerpendicularBetweenLine(FC,CA)
Goal: Value(PerimeterOfCircle(H))
Answer: 14*pi
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,H)
circle_perimeter_formula(1,H)
Problem: As shown in the diagram, BB=15, PA=12, ∠DBA=24°, ADCB is a rhombus. Find the measure of ∠ADB.
问题：如图所示，BB=15，PA=12，∠DBA=24°，四边形ADCB是菱形。求∠ADB的大小。
Construction (CDL):
Shape(AD,DP,PA)
Shape(PD,DC,CP)
Shape(PC,CB,BP)
Shape(PB,BA,AP)
Collinear(APC)
Collinear(DPB)
Conditions (CDL):
Equal(LengthOfLine(BB),15)
Equal(LengthOfLine(PA),12)
Equal(MeasureOfAngle(DBA),24)
Rhombus(ADCB)
Goal: Value(MeasureOfAngle(ADB))
Answer: 24
Theorem sequence:
parallelogram_property_diagonal_bisection(1,DCBA,P)
mirror_similar_triangle_judgment_sss(1,ADP,APB)
mirror_similar_triangle_property_angle_equal(1,DPA,BAP)
Problem: As shown in the diagram, CK=2, ∠JCK=30°, the radius of circle C is 2, C is the center of circle C. Find the length of ⌒CKJ.
问题：如图所示，CK=2，∠JCK=30°，圆C的半径为2，圆C的圆心为C。求⌒CKJ的长度。
Construction (CDL):
Shape(CKJ,JC,CK)
Shape(CJK,KC,CJ)
Cocircular(C,KJ)
Conditions (CDL):
Equal(LengthOfLine(CK),2)
Equal(MeasureOfAngle(JCK),30)
Equal(RadiusOfCircle(C),2)
IsCentreOfCircle(C,C)
Goal: Value(LengthOfArc(CKJ))
Answer: pi/3
Theorem sequence:
arc_property_center_angle(1,CKJ,C)
arc_length_formula(1,CKJ)
Problem: As shown in the diagram, ∠DBC=60°, ∠EAD=80°, DE∥BC. Find the measure of ∠CED.
问题：如图所示，∠DBC=60°，∠EAD=80°，DE平行于BC。求∠CED的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(DBC),60)
Equal(MeasureOfAngle(EAD),80)
ParallelBetweenLine(DE,BC)
Goal: Value(MeasureOfAngle(CED))
Answer: 140
Theorem sequence:
triangle_property_angle_sum(1,ABC)
parallel_property_ipsilateral_internal_angle(1,CB,ED)
Problem: As shown in the diagram, ∠EFG=50°, EG is the angle bisector of ∠BEF, AB is parallel to CD. Find the measure of ∠FGE.
问题：如图所示，∠EFG=50°，EG平分∠BEF，AB∥CD。求∠FGE的大小。
Construction (CDL):
Shape(FE,EA)
Shape(CF,FE)
Shape(EF,FG,GE)
Shape(EG,GD)
Shape(BE,EG)
Collinear(AEB)
Collinear(CFGD)
Conditions (CDL):
Equal(MeasureOfAngle(EFG),50)
IsBisectorOfAngle(EG,BEF)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(FGE))
Answer: 65
Theorem sequence:
angle_addition(1,BEG,GEF)
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_collinear_extend(3,DC,BE,F)
parallel_property_ipsilateral_internal_angle(1,EB,FD)
triangle_property_angle_sum(1,EFG)
Problem: As shown in the diagram, OA=OB, OB=OC, ∠BCD=30°. Find the measure of ∠DOA.
问题：如图所示，OA=OB，OB=OC，∠BCD=30°。求∠DOA的大小。
Construction (CDL):
Shape(AD,DO,OA)
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Shape(DC,CO,OD)
Collinear(ODB)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(OA),LengthOfLine(OB))
Equal(LengthOfLine(OB),LengthOfLine(OC))
Equal(MeasureOfAngle(BCD),30)
Goal: Value(MeasureOfAngle(DOA))
Answer: 60
Theorem sequence:
angle_addition(1,BCD,DCO)
angle_addition(1,OAD,DAB)
angle_addition(1,ABD,DBC)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OAB)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠GDA=70°, CM is perpendicular to DM, MN⊥EN. Find the measure of ∠FEN.
问题：如图所示，∠GDA=70°，CM⊥DM，MN垂直于EN。求∠FEN的大小。
Construction (CDL):
Shape(CM,MD)
Shape(MD,DG)
Shape(GD,DA)
Shape(AD,DE)
Shape(DE,EB)
Shape(BE,EF)
Shape(MN,NE,ED,DM)
Shape(NF,FE,EN)
Collinear(CMNF)
Collinear(GDEF)
Collinear(MDA)
Collinear(NEB)
Conditions (CDL):
Equal(MeasureOfAngle(GDA),70)
PerpendicularBetweenLine(CM,DM)
PerpendicularBetweenLine(MN,EN)
Goal: Value(MeasureOfAngle(FEN))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,CMD,DMN)
adjacent_complementary_angle(1,GDA,ADE)
adjacent_complementary_angle(1,ADE,EDM)
perpendicular_judgment_angle(1,DM,NM)
parallel_judgment_per_per(1,MD,NE)
parallel_property_corresponding_angle(1,EN,DM,F)
Problem: As shown in the diagram, ∠GBD=120°, FE∥HG. Find the measure of ∠FAC.
问题：如图所示，∠GBD=120°，FE平行于HG。求∠FAC的大小。
Construction (CDL):
Shape(FA,AC)
Shape(CA,AE)
Shape(HB,BA)
Shape(BA,AF)
Shape(EA,AB)
Shape(AB,BG)
Shape(DB,BH)
Shape(GB,BD)
Collinear(CABD)
Collinear(FAE)
Collinear(HBG)
Conditions (CDL):
Equal(MeasureOfAngle(GBD),120)
ParallelBetweenLine(FE,HG)
Goal: Value(MeasureOfAngle(FAC))
Answer: 120
Theorem sequence:
parallel_property_collinear_extend(3,FE,HG,A)
parallel_property_collinear_extend(3,GH,EA,B)
vertical_angle(1,FAC,EAB)
parallel_property_corresponding_angle(2,AE,BG,D)
Problem: As shown in the diagram, ∠CGF=135°, AB∥CD. Find the measure of ∠EFB.
问题：如图所示，∠CGF=135°，AB∥CD。求∠EFB的大小。
Construction (CDL):
Shape(AF,FE)
Shape(EF,FB)
Shape(GF,FA)
Shape(BF,FG)
Shape(CG,GF)
Shape(FG,GD)
Shape(HG,GC)
Shape(DG,GH)
Collinear(AFB)
Collinear(CGD)
Collinear(EFGH)
Conditions (CDL):
Equal(MeasureOfAngle(CGF),135)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EFB))
Answer: 45
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_collinear_extend(3,DC,BF,G)
adjacent_complementary_angle(1,CGF,FGD)
parallel_property_corresponding_angle(1,FB,GD,E)