SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BEF=78°, ∠DGF=120°, ∠FBE=50°, ∠GDA=56°. Find the measure of ∠GFE.
问题：如图所示，∠BEF=78°，∠DGF=120°，∠FBE=50°，∠GDA=56°。求∠GFE的大小。
Construction (CDL):
Shape(FB,BE,EF)
Shape(BF,FC)
Shape(CF,FG,GC)
Shape(GF,FE)
Shape(CG,GA)
Shape(AG,GD,DA)
Shape(HA,AD)
Collinear(CFB)
Collinear(CGD)
Collinear(BFGAH)
Conditions (CDL):
Equal(MeasureOfAngle(BEF),78)
Equal(MeasureOfAngle(DGF),120)
Equal(MeasureOfAngle(FBE),50)
Equal(MeasureOfAngle(GDA),56)
Goal: Value(MeasureOfAngle(GFE))
Answer: 128
Theorem sequence:
triangle_property_angle_sum(1,FBE)
adjacent_complementary_angle(1,GFE,EFB)
Problem: As shown in the diagram, ∠FOE=118°, ∠LDA=104°, EA∥IB. Find the measure of ∠DIB.
问题：如图所示，∠FOE=118°，∠LDA=104°，EA∥IB。求∠DIB的大小。
Construction (CDL):
Shape(LD,DA)
Shape(AD,DI)
Shape(ID,DO)
Shape(OD,DL)
Shape(HO,OD)
Shape(DO,OF)
Shape(FO,OE)
Shape(EO,OH)
Shape(DI,IB)
Shape(BI,IC)
Shape(CI,IF)
Shape(FI,ID)
Collinear(ADOE)
Collinear(LDIC)
Collinear(HOF)
Collinear(BIF)
Conditions (CDL):
Equal(MeasureOfAngle(FOE),118)
Equal(MeasureOfAngle(LDA),104)
ParallelBetweenLine(EA,IB)
Goal: Value(MeasureOfAngle(DIB))
Answer: 104
Theorem sequence:
parallel_property_collinear_extend(3,EA,IB,D)
parallel_property_corresponding_angle(1,DA,IB,L)
Problem: As shown in the diagram, CB=18, CD=12, ∠BAD=115°, ADCB is a ▱. Find the measure of ∠DCB.
问题：如图所示，CB=18，CD=12，∠BAD=115°，ADCB是▱。求∠DCB的大小。
Construction (CDL):
Shape(AD,DC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(CB),18)
Equal(LengthOfLine(CD),12)
Equal(MeasureOfAngle(BAD),115)
Parallelogram(ADCB)
Goal: Value(MeasureOfAngle(DCB))
Answer: 115
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,ADCB)
Problem: As shown in the diagram, ∠ADE=57°, ∠BCE=55°, ∠EAD=47°. Find the measure of ∠DEA.
问题：如图所示，∠ADE=57°，∠BCE=55°，∠EAD=47°。求∠DEA的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(CE,EB,BC)
Collinear(DEC)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),57)
Equal(MeasureOfAngle(BCE),55)
Equal(MeasureOfAngle(EAD),47)
Goal: Value(MeasureOfAngle(DEA))
Answer: 76
Theorem sequence:
triangle_property_angle_sum(1,ADE)
Problem: As shown in the diagram, HP=5*x-16, PJ=3*x+8, GP is the median of triangle GJH. Find the length of line HJ.
问题：如图所示，HP=5*x-16，PJ=3*x+8，GP是△GJH的中线。求直线HJ的长度。
Construction (CDL):
Shape(HG,GP,PH)
Shape(PG,GJ,JP)
Collinear(HPJ)
Conditions (CDL):
Equal(LengthOfLine(HP),5*x-16)
Equal(LengthOfLine(PJ),3*x+8)
IsMedianOfTriangle(GP,GJH)
Goal: Value(LengthOfLine(HJ))
Answer: 88
Theorem sequence:
line_addition(1,HP,PJ)
Problem: As shown in the diagram, CF=14, the center of circle F is F. Find the diameter of ⊙F.
问题：如图所示，CF=14，F是⊙F的圆心。求⊙F的直径。
Construction (CDL):
Shape(BF,FC,FCB)
Shape(CF,FD,FDC)
Shape(DF,FE,ED)
Shape(DE,FED)
Shape(FA,AE,EF)
Shape(EA,FAE)
Shape(AF,FB,FBA)
Cocircular(F,AEDCB)
Conditions (CDL):
Equal(LengthOfLine(CF),14)
IsCentreOfCircle(F,F)
Goal: Value(DiameterOfCircle(F))
Answer: 28
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,F)
radius_of_circle_property_length_equal(1,FC,F)
Problem: As shown in the diagram, AD=16, CD=12, ∠ADC=90°. Find the area of △DCA.
问题：如图所示，AD=16，CD=12，∠ADC=90°。求△DCA的面积。
Construction (CDL):
Shape(DC,CA,AD)
Conditions (CDL):
Equal(LengthOfLine(AD),16)
Equal(LengthOfLine(CD),12)
Equal(MeasureOfAngle(ADC),90)
Goal: Value(AreaOfTriangle(DCA))
Answer: 96
Theorem sequence:
triangle_area_formula_sine(1,DCA)
Problem: As shown in the diagram, AB=y, AC=x, AD=4, CD=10, BA is perpendicular to CA, DC⊥BC. Find the value of y.
问题：如图所示，AB=y，AC=x，AD=4，CD=10，BA垂直于CA，DC⊥BC。求y的值。
Construction (CDL):
Shape(CB,BA,AC)
Shape(CA,AD,DC)
Collinear(BAD)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(AD),4)
Equal(LengthOfLine(CD),10)
PerpendicularBetweenLine(BA,CA)
PerpendicularBetweenLine(DC,BC)
Goal: Value(y)
Answer: 21
Theorem sequence:
adjacent_complementary_angle(1,BAC,CAD)
line_addition(1,BA,AD)
right_triangle_judgment_angle(1,BAC)
right_triangle_judgment_angle(1,CAD)
right_triangle_judgment_angle(1,DCB)
right_triangle_property_pythagorean(1,BAC)
right_triangle_property_pythagorean(1,CAD)
right_triangle_property_pythagorean(1,DCB)
Problem: As shown in the diagram, ∠CDF=y°, ∠EFG=114°, ∠FDI=x+12°, DC is parallel to BE. Find the value of y.
问题：如图所示，∠CDF=y°，∠EFG=114°，∠FDI=x+12°，DC∥BE。求y的值。
Construction (CDL):
Shape(ID,DJ)
Shape(JD,DC)
Shape(CD,DF)
Shape(FD,DI)
Shape(BF,FD)
Shape(DF,FE)
Shape(EF,FG)
Shape(GF,FB)
Collinear(IDC)
Collinear(BFE)
Collinear(JDFG)
Conditions (CDL):
Equal(MeasureOfAngle(CDF),y)
Equal(MeasureOfAngle(EFG),114)
Equal(MeasureOfAngle(FDI),x+12)
ParallelBetweenLine(DC,BE)
Goal: Value(y)
Answer: 114
Theorem sequence:
parallel_property_collinear_extend(3,EB,CD,F)
parallel_property_corresponding_angle(2,DC,FE,G)
Problem: As shown in the diagram, FJ=y+12, FK=3*x+7, GK=4*x-1, HJ=2*y-5, HJ=FJ, FG⊥HG, FK is perpendicular to JK. Find the value of x.
问题：如图所示，FJ=y+12，FK=3*x+7，GK=4*x-1，HJ=2*y-5，HJ=FJ，FG垂直于HG，FK垂直于JK。求x的值。
Construction (CDL):
Shape(JF,FK,KJ)
Shape(HJ,JK,KG,GH)
Collinear(FKG)
Collinear(FJH)
Conditions (CDL):
Equal(LengthOfLine(FJ),y+12)
Equal(LengthOfLine(FK),3*x+7)
Equal(LengthOfLine(GK),4*x-1)
Equal(LengthOfLine(HJ),2*y-5)
Equal(LengthOfLine(HJ),LengthOfLine(FJ))
PerpendicularBetweenLine(FG,HG)
PerpendicularBetweenLine(FK,JK)
Goal: Value(x)
Answer: 8
Theorem sequence:
similar_triangle_judgment_aa(1,JFK,HFG)
line_addition(1,FK,KG)
line_addition(1,FJ,JH)
similar_triangle_property_line_ratio(1,JFK,HFG)
similar_triangle_property_line_ratio(1,KJF,GHF)
Problem: As shown in the diagram, JL=4, KJ=x, LP=6, PM=12, ∠KJL=∠LPM. Find the length of line JK.
问题：如图所示，JL=4，KJ=x，LP=6，PM=12，∠KJL=∠LPM。求直线JK的长度。
Construction (CDL):
Shape(KJ,JL,LK)
Shape(LP,PM,ML)
Collinear(JLM)
Collinear(KLP)
Conditions (CDL):
Equal(LengthOfLine(JL),4)
Equal(LengthOfLine(KJ),x)
Equal(LengthOfLine(LP),6)
Equal(LengthOfLine(PM),12)
Equal(MeasureOfAngle(KJL),MeasureOfAngle(LPM))
Goal: Value(LengthOfLine(JK))
Answer: 8
Theorem sequence:
vertical_angle(1,JLK,MLP)
mirror_similar_triangle_judgment_aa(1,KJL,MLP)
mirror_similar_triangle_property_line_ratio(1,KJL,MLP)
mirror_similar_triangle_property_line_ratio(1,LKJ,LPM)
Problem: As shown in the diagram, YX=9, the measure of arc XWY is 228, the center of circle X is X. Find the area of the sector XWY.
问题：如图所示，YX=9，⌒XWY的角度为228，X是圆X的圆心。求扇形XWY的面积。
Construction (CDL):
Shape(WX,XY,XYW)
Shape(YX,XW,XWY)
Cocircular(X,WY)
Conditions (CDL):
Equal(LengthOfLine(YX),9)
Equal(MeasureOfArc(XWY),228)
IsCentreOfCircle(X,X)
Goal: Value(AreaOfSector(XWY))
Answer: 513*pi/10
Theorem sequence:
radius_of_circle_property_length_equal(1,XY,X)
sector_area_formula(1,XWY)
Problem: As shown in the diagram, BA=15, DA=x, the center of ⊙C is C, the tangent to ⊙C is AB, AD is the tangent to ⊙C, AB is perpendicular to CB, CD is perpendicular to AD. Find the value of x.
问题：如图所示，BA=15，DA=x，圆C的圆心为C，⊙O的切线为AB，圆O的切线为AD，AB⊥CB，CD垂直于AD。求x的值。
Construction (CDL):
Shape(EA,AB,CEB)
Shape(DA,AE,CDE)
Shape(DC,CB,CBD)
Shape(BC,CE,CEB)
Shape(EC,CD,CDE)
Collinear(CEA)
Cocircular(C,BED)
Conditions (CDL):
Equal(LengthOfLine(BA),15)
Equal(LengthOfLine(DA),x)
IsCentreOfCircle(C,C)
IsTangentOfCircle(AB,C)
IsTangentOfCircle(AD,C)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(CD,AD)
Goal: Value(x)
Answer: 15
Theorem sequence:
radius_of_circle_property_length_equal(1,CB,C)
radius_of_circle_property_length_equal(1,CD,C)
mirror_congruent_triangle_judgment_hl(1,ABC,ACD)
mirror_congruent_triangle_property_line_equal(1,CAB,CDA)
Problem: As shown in the diagram, the diameter of ⊙A is 10, the diameter of circle B is 20, the diameter of circle C is 14, A is the center of circle A, B is the center of ⊙B, the center of circle C is C. Find the length of line XB.
问题：如图所示，圆A的直径为10，⊙B的直径为20，圆C的直径为14，圆A的圆心为A，圆B的圆心为B，C是⊙C的圆心。求直线XB的长度。
Construction (CDL):
Shape(BDA,AX,AXD)
Shape(XA,BAE,AEX)
Shape(AEX,BAE,ADE)
Shape(XB,BY,CFY,BFD,AXD)
Shape(YB,BX,AEX,BEG,CYG)
Shape(YC,BCF,CFY)
Shape(CY,CYG,BGC)
Shape(BCF,BGC,CGF)
Collinear(AXBYC)
Cocircular(A,DXE)
Cocircular(B,ADFCGE)
Cocircular(C,FYG)
Conditions (CDL):
Equal(DiameterOfCircle(A),10)
Equal(DiameterOfCircle(B),20)
Equal(DiameterOfCircle(C),14)
IsCentreOfCircle(A,A)
IsCentreOfCircle(B,B)
IsCentreOfCircle(C,C)
Goal: Value(LengthOfLine(XB))
Answer: 5
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,A)
circle_property_length_of_radius_and_diameter(1,B)
radius_of_circle_property_length_equal(1,AX,A)
radius_of_circle_property_length_equal(1,BA,B)
line_addition(1,AX,XB)
Problem: As shown in the diagram, ∠CBD=70°, ∠DBF=x°, ∠FBC=65°, the center of ⊙B is B. Find the value of x.
问题：如图所示，∠CBD=70°，∠DBF=x°，∠FBC=65°，⊙B的圆心为B。求x的值。
Construction (CDL):
Shape(FB,BC,BCF)
Shape(CB,BD,BDC)
Shape(DB,BF,BFD)
Cocircular(B,FCD)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),70)
Equal(MeasureOfAngle(DBF),x)
Equal(MeasureOfAngle(FBC),65)
IsCentreOfCircle(B,B)
Goal: Value(x)
Answer: 225
Theorem sequence:
round_angle(1,FBC,CBF)
angle_addition(1,CBD,DBF)
Problem: As shown in the diagram, AC=5*y-2*x, AY=4, CB=y+2*x, YB=3*y+2*x, quadrilateral CBYA is a parallelogram. Find the value of y.
问题：如图所示，AC=5*y-2*x，AY=4，CB=y+2*x，YB=3*y+2*x，四边形CBYA是▱。求y的值。
Construction (CDL):
Shape(CB,BY,YA,AC)
Conditions (CDL):
Equal(LengthOfLine(AC),5*y-2*x)
Equal(LengthOfLine(AY),4)
Equal(LengthOfLine(CB),y+2*x)
Equal(LengthOfLine(YB),3*y+2*x)
Parallelogram(CBYA)
Goal: Value(y)
Answer: 2
Theorem sequence:
parallelogram_property_opposite_line_equal(1,CBYA)
parallelogram_property_opposite_line_equal(1,BYAC)
Problem: As shown in the diagram, PS=3, WY=8, XW=10, YX=6, YP∥XW, AQ⊥SQ, SY⊥AY, YS⊥PS. Find the length of line PY.
问题：如图所示，PS=3，WY=8，XW=10，YX=6，YP∥XW，AQ⊥SQ，SY垂直于AY，YS垂直于PS。求直线PY的长度。
Construction (CDL):
Shape(PY,YS,SP)
Shape(SY,YA,AQ,QS)
Shape(YR,RA,AY)
Shape(QA,AX,XQ)
Shape(WS,SQ,QW)
Collinear(YSW)
Collinear(RAQ)
Collinear(PSQ)
Collinear(YAX)
Collinear(XQW)
Collinear(RYP)
Conditions (CDL):
Equal(LengthOfLine(PS),3)
Equal(LengthOfLine(WY),8)
Equal(LengthOfLine(XW),10)
Equal(LengthOfLine(YX),6)
ParallelBetweenLine(YP,XW)
PerpendicularBetweenLine(AQ,SQ)
PerpendicularBetweenLine(SY,AY)
PerpendicularBetweenLine(YS,PS)
Goal: Value(LengthOfLine(PY))
Answer: 5
Theorem sequence:
parallel_property_alternate_interior_angle(1,YP,XW)
similar_triangle_judgment_aa(1,PYS,XWY)
similar_triangle_property_line_ratio(1,YSP,WYX)
similar_triangle_property_line_ratio(1,SPY,YXW)
Problem: As shown in the diagram, AC=y, AD=9, BC=z, CD=x, DB=16, AD is perpendicular to CD, BC is perpendicular to AC. Find the value of x.
问题：如图所示，AC=y，AD=9，BC=z，CD=x，DB=16，AD⊥CD，BC⊥AC。求x的值。
Construction (CDL):
Shape(CA,AD,DC)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),y)
Equal(LengthOfLine(AD),9)
Equal(LengthOfLine(BC),z)
Equal(LengthOfLine(CD),x)
Equal(LengthOfLine(DB),16)
PerpendicularBetweenLine(AD,CD)
PerpendicularBetweenLine(BC,AC)
Goal: Value(x)
Answer: 12
Theorem sequence:
adjacent_complementary_angle(1,ADC,CDB)
line_addition(1,AD,DB)
right_triangle_judgment_angle(1,ADC)
right_triangle_judgment_angle(1,CDB)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,ADC)
right_triangle_property_pythagorean(1,CDB)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, ∠UZY=2*x+24°, ∠VZU=4*x°, ∠XZW=∠YZX, Z is the center of circle Z. Find the measure of ⌒ZYW.
问题：如图所示，∠UZY=2*x+24°，∠VZU=4*x°，∠XZW=∠YZX，Z是⊙Z的圆心。求弧ZYW的角度。
Construction (CDL):
Shape(WZ,ZV,ZVW)
Shape(XZ,ZW,ZWX)
Shape(YZ,ZX,ZXY)
Shape(UZ,ZY,ZYU)
Shape(VZ,ZU,ZUV)
Collinear(VZY)
Collinear(WZU)
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(ZYW))
Answer: 256
Theorem sequence:
adjacent_complementary_angle(1,VZU,UZY)
vertical_angle(1,WZV,UZY)
angle_addition(1,WZU,UZY)
angle_addition(1,WZV,VZU)
arc_property_center_angle(1,ZYW,Z)
Problem: As shown in the diagram, CD=2, ∠BCA=112°, ∠CBD=32°. Find the measure of ∠BDC.
问题：如图所示，CD=2，∠BCA=112°，∠CBD=32°。求∠BDC的大小。
Construction (CDL):
Shape(CB,BD,DC)
Shape(BC,CA)
Collinear(ACD)
Conditions (CDL):
Equal(LengthOfLine(CD),2)
Equal(MeasureOfAngle(BCA),112)
Equal(MeasureOfAngle(CBD),32)
Goal: Value(MeasureOfAngle(BDC))
Answer: 80
Theorem sequence:
flat_angle(1,DCA)
angle_addition(1,DCB,BCA)
triangle_property_angle_sum(1,CBD)
Problem: As shown in the diagram, AB=35, AC=x, BC=21, BC is perpendicular to AC. Find the value of x.
问题：如图所示，AB=35，AC=x，BC=21，BC⊥AC。求x的值。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),35)
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BC),21)
PerpendicularBetweenLine(BC,AC)
Goal: Value(x)
Answer: 28
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, ∠CDF=y°, ∠EFG=114°, ∠FDI=x+12°, DC∥BE. Find the value of x.
问题：如图所示，∠CDF=y°，∠EFG=114°，∠FDI=x+12°，DC平行于BE。求x的值。
Construction (CDL):
Shape(ID,DJ)
Shape(JD,DC)
Shape(CD,DF)
Shape(FD,DI)
Shape(DF,FE)
Shape(EF,FG)
Shape(GF,FB)
Shape(BF,FD)
Collinear(IDC)
Collinear(BFE)
Collinear(JDFG)
Conditions (CDL):
Equal(MeasureOfAngle(CDF),y)
Equal(MeasureOfAngle(EFG),114)
Equal(MeasureOfAngle(FDI),x+12)
ParallelBetweenLine(DC,BE)
Goal: Value(x)
Answer: 54
Theorem sequence:
parallel_property_collinear_extend(3,EB,CD,F)
parallel_property_corresponding_angle(2,DC,FE,G)
adjacent_complementary_angle(1,CDF,FDI)
Problem: As shown in the diagram, AB=13, AC=9, ∠CAB=65°, ABDC is a ▱, BE⊥CE. Find the area of ABDC.
问题：如图所示，AB=13，AC=9，∠CAB=65°，AC和BD是平行四边形ABDC的一组对边，BE垂直于CE。求四边形ABDC的面积。
Construction (CDL):
Shape(AB,BE,EC,CA)
Shape(BD,DE,EB)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(AB),13)
Equal(LengthOfLine(AC),9)
Equal(MeasureOfAngle(CAB),65)
Parallelogram(ABDC)
PerpendicularBetweenLine(BE,CE)
Goal: Value(AreaOfQuadrilateral(ABDC))
Answer: 117*sin(13*pi/36)
Theorem sequence:
parallelogram_area_formula_sine(1,CABD)
Problem: As shown in the diagram, JK=2*b+3, JM=3*a, LK=21, LM=45, ∠MLR=30°, ∠RLK=70°, quadrilateral JMLK is a ▱. Find the measure of ∠KJM.
问题：如图所示，JK=2*b+3，JM=3*a，LK=21，LM=45，∠MLR=30°，∠RLK=70°，JMLK是▱。求∠KJM的大小。
Construction (CDL):
Shape(JR,RK,KJ)
Shape(JM,MR,RJ)
Shape(RM,ML,LR)
Shape(KR,RL,LK)
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(KJM))
Answer: 100
Theorem sequence:
angle_addition(1,MLR,RLK)
parallelogram_property_opposite_angle_equal(1,JMLK)
Problem: As shown in the diagram, FM=6, JL=8.5, JM=8.5, JF⊥LF. Find the length of line LM.
问题：如图所示，FM=6，JL=8.5，JM=8.5，JF⊥LF。求直线LM的长度。
Construction (CDL):
Shape(JM,MF,FJ)
Shape(JF,FL,LJ)
Collinear(MFL)
Conditions (CDL):
Equal(LengthOfLine(FM),6)
Equal(LengthOfLine(JL),8.5)
Equal(LengthOfLine(JM),8.5)
PerpendicularBetweenLine(JF,LF)
Goal: Value(LengthOfLine(LM))
Answer: 12
Theorem sequence:
adjacent_complementary_angle(1,MFJ,JFL)
mirror_congruent_triangle_judgment_hl(1,MFJ,LJF)
mirror_congruent_triangle_property_line_equal(1,JMF,JFL)
line_addition(1,MF,FL)
Problem: As shown in the diagram, AC=10, BD=18, quadrilateral ABCD is a rhombus. Find the area of ABCD.
问题：如图所示，AC=10，BD=18，四边形ABCD是菱形。求ABCD的面积。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Shape(AE,ED,DA)
Shape(EC,CD,DE)
Collinear(BED)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AC),10)
Equal(LengthOfLine(BD),18)
Rhombus(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 90
Theorem sequence:
kite_area_formula_diagonal(1,ABCD)
Problem: As shown in the diagram, AB=8, XY=20, ∠XYA=70°, BX and YA are the non-parallel sides of isosceles trapezoid BXYA. Find the measure of ∠ABX.
问题：如图所示，AB=8，XY=20，∠XYA=70°，BX和YA是等腰梯形BXYA的腰。求∠ABX的大小。
Construction (CDL):
Shape(AB,BX,XY,YA)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(XY),20)
Equal(MeasureOfAngle(XYA),70)
IsoscelesTrapezoid(BXYA)
Goal: Value(MeasureOfAngle(ABX))
Answer: 110
Theorem sequence:
isosceles_trapezoid_property_angle_equal(1,BXYA)
parallel_property_ipsilateral_internal_angle(1,BA,XY)
Problem: As shown in the diagram, ∠LBJ=47°, the measure of arc CLJ is 111, C is the center of circle C, BJ is the tangent to circle C. Find the measure of ⌒CMJ.
问题：如图所示，∠LBJ=47°，⌒CLJ的角度为111，C是圆C的圆心，BJ是⊙O的切线。求⌒CMJ的角度。
Construction (CDL):
Shape(JC,CL,CLJ)
Shape(LC,CM,CML)
Shape(MC,CJ,CJM)
Shape(LM,CML)
Shape(LB,BJ,CLJ)
Collinear(BJ)
Collinear(BLM)
Cocircular(C,JML)
Conditions (CDL):
Equal(MeasureOfAngle(LBJ),47)
Equal(MeasureOfArc(CLJ),111)
IsCentreOfCircle(C,C)
IsTangentOfCircle(BJ,C)
Goal: Value(MeasureOfArc(CMJ))
Answer: 155
Theorem sequence:
circle_property_circular_power_tangent_and_segment_angle(2,BJ,BLM,C)
arc_property_center_angle(1,CJM,C)
arc_property_center_angle(1,CMJ,C)
round_angle(1,MCJ,JCM)
Problem: As shown in the diagram, AB=8, BC=10, BD=y, CA=6, CD=x, BD⊥AD, CA⊥BA. Find the value of y.
问题：如图所示，AB=8，BC=10，BD=y，CA=6，CD=x，BD⊥AD，CA⊥BA。求y的值。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BC),10)
Equal(LengthOfLine(BD),y)
Equal(LengthOfLine(CA),6)
Equal(LengthOfLine(CD),x)
PerpendicularBetweenLine(BD,AD)
PerpendicularBetweenLine(CA,BA)
Goal: Value(y)
Answer: 32/5
Theorem sequence:
adjacent_complementary_angle(1,BDA,ADC)
line_addition(1,BD,DC)
right_triangle_judgment_angle(1,BDA)
right_triangle_judgment_angle(1,ADC)
right_triangle_property_pythagorean(1,BDA)
right_triangle_property_pythagorean(1,ADC)
Problem: As shown in the diagram, EI=JI, HI=FI, ∠HFI=x+5°, ∠IHF=2*x-25°, EJ∥HF. Find the value of x.
问题：如图所示，EI=JI，HI=FI，∠HFI=x+5°，∠IHF=2*x-25°，EJ∥HF。求x的值。
Construction (CDL):
Shape(EI,IJ,JE)
Shape(IH,HF,FI)
Collinear(EIF)
Collinear(JIH)
Conditions (CDL):
Equal(LengthOfLine(EI),LengthOfLine(JI))
Equal(LengthOfLine(HI),LengthOfLine(FI))
Equal(MeasureOfAngle(HFI),x+5)
Equal(MeasureOfAngle(IHF),2*x-25)
ParallelBetweenLine(EJ,HF)
Goal: Value(x)
Answer: 30
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,IHF)
isosceles_triangle_property_angle_equal(1,IHF)
Problem: As shown in the diagram, DB=2*x-4, PB=2*x-9, DP⊥AP, ADCB is a rhombus. Find the length of line PD.
问题：如图所示，DB=2*x-4，PB=2*x-9，DP垂直于AP，四边形ADCB是菱形。求直线PD的长度。
Construction (CDL):
Shape(AD,DP,PA)
Shape(AP,PB,BA)
Shape(PD,DC,CP)
Shape(PC,CB,BP)
Collinear(APC)
Collinear(DPB)
Conditions (CDL):
Equal(LengthOfLine(DB),2*x-4)
Equal(LengthOfLine(PB),2*x-9)
PerpendicularBetweenLine(DP,AP)
Rhombus(ADCB)
Goal: Value(LengthOfLine(PD))
Answer: 5
Theorem sequence:
line_addition(1,DP,PB)
Problem: As shown in the diagram, AC=17, AD=15, AD=EH, EH=15/2, AD is the altitude of triangle ABC, EH is the altitude of triangle EGF, △ABC is mirror similar to △EGF., CD⊥AD, EH⊥GH. Find the length of line EG.
问题：如图所示，AC=17，AD=15，AD=EH，EH=15/2，AD是△ABC的高，EH是△EGF的高，△ABC与△EGF是镜像相似三角形，CD垂直于AD，EH⊥GH。求直线EG的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(EH,HG,GE)
Shape(EG,GF,FE)
Collinear(BCD)
Collinear(HGF)
Conditions (CDL):
Equal(LengthOfLine(AC),17)
Equal(LengthOfLine(AD),15)
Equal(LengthOfLine(AD),LengthOfLine(EH))
Equal(LengthOfLine(EH),15/2)
IsAltitudeOfTriangle(AD,ABC)
IsAltitudeOfTriangle(EH,EGF)
MirrorSimilarBetweenTriangle(ABC,EGF)
PerpendicularBetweenLine(CD,AD)
PerpendicularBetweenLine(EH,GH)
Goal: Value(LengthOfLine(EG))
Answer: 17/2
Theorem sequence:
mirror_similar_triangle_property_angle_equal(1,CAB,GFE)
flat_angle(1,BCD)
flat_angle(1,HGF)
angle_addition(1,BCA,ACD)
angle_addition(1,HGE,EGF)
mirror_similar_triangle_judgment_aa(1,ACD,EHG)
mirror_similar_triangle_property_line_ratio(1,CDA,GEH)
mirror_similar_triangle_property_line_ratio(1,DAC,HGE)
Problem: As shown in the diagram, TR=23, TS=11, TS is perpendicular to RS. Find the measure of ∠RTS.
问题：如图所示，TR=23，TS=11，TS垂直于RS。求∠RTS的大小。
Construction (CDL):
Shape(TS,SR,RT)
Conditions (CDL):
Equal(LengthOfLine(TR),23)
Equal(LengthOfLine(TS),11)
PerpendicularBetweenLine(TS,RS)
Goal: Value(MeasureOfAngle(RTS))
Answer: 180*asin(2*sqrt(102)/23)/pi
Theorem sequence:
right_triangle_judgment_angle(1,TSR)
right_triangle_property_pythagorean(1,TSR)
sine_theorem(1,TSR)
sine_theorem(1,SRT)
Problem: As shown in the diagram, ∠BFA=x+84°, ∠TAF=21+2*x°, AT∥BN. Find the value of x.
问题：如图所示，∠BFA=x+84°，∠TAF=21+2*x°，AT平行于BN。求x的值。
Construction (CDL):
Shape(EA,AT)
Shape(TA,AF)
Shape(FA,AC)
Shape(CA,AE)
Shape(NF,FD)
Shape(DF,FB)
Shape(BF,FA)
Shape(AF,FN)
Collinear(EAFD)
Collinear(TAC)
Collinear(NFB)
Conditions (CDL):
Equal(MeasureOfAngle(BFA),x+84)
Equal(MeasureOfAngle(TAF),21+2*x)
ParallelBetweenLine(AT,BN)
Goal: Value(x)
Answer: 63
Theorem sequence:
parallel_property_collinear_extend(3,NB,TA,F)
parallel_property_alternate_interior_angle(1,FB,TA)
Problem: As shown in the diagram, ∠TZY=30*x°, ∠TZY=∠ZYW, ∠WTZ=20*x°, ∠WTZ=∠YWT, TZ and YW are the non-parallel sides of isosceles trapezoid TZYW. Find the measure of ∠ZYW.
问题：如图所示，∠TZY=30*x°，∠TZY=∠ZYW，∠WTZ=20*x°，∠WTZ=∠YWT，TW和ZY是等腰梯形TZYW的平行边。求∠ZYW的大小。
Construction (CDL):
Shape(TZ,ZY,YW,WT)
Conditions (CDL):
Equal(MeasureOfAngle(TZY),30*x)
Equal(MeasureOfAngle(TZY),MeasureOfAngle(ZYW))
Equal(MeasureOfAngle(WTZ),20*x)
Equal(MeasureOfAngle(WTZ),MeasureOfAngle(YWT))
IsoscelesTrapezoid(TZYW)
Goal: Value(MeasureOfAngle(ZYW))
Answer: 108
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,TW,ZY)
Problem: As shown in the diagram, the length of arc AEC is equal to the length of arc ADB, BD=5*x+13, EC=12*x-8, the center of circle A is A. Find the value of x.
问题：如图所示，⌒AEC的长度与⌒ADB的长度相等，BD=5*x+13，EC=12*x-8，圆A的圆心为A。求x的值。
Construction (CDL):
Shape(CE,AEC)
Shape(BD,ADB)
Shape(ABE,EC,ACD,DB)
Cocircular(A,DBEC)
Conditions (CDL):
Equal(LengthOfArc(AEC),LengthOfArc(ADB))
Equal(LengthOfLine(BD),5*x+13)
Equal(LengthOfLine(EC),12*x-8)
IsCentreOfCircle(A,A)
Goal: Value(x)
Answer: 3
Theorem sequence:
congruent_arc_judgment_length_equal(1,ADB,AEC)
congruent_arc_property_chord_equal(1,ADB,AEC)
Problem: As shown in the diagram, ∠DFE=36°, the center of ⊙F is F. Find the measure of ⌒FAE.
问题：如图所示，∠DFE=36°，圆F的圆心为F。求弧FAE的角度。
Construction (CDL):
Shape(EF,FA,FAE)
Shape(CF,FE,FEC)
Shape(BD,FDB)
Shape(AF,FC,FDC,DB,FBA)
Collinear(AFD)
Cocircular(F,AECDB)
Conditions (CDL):
Equal(MeasureOfAngle(DFE),36)
IsCentreOfCircle(F,F)
Goal: Value(MeasureOfArc(FAE))
Answer: 144
Theorem sequence:
arc_property_center_angle(1,FAE,F)
adjacent_complementary_angle(1,DFE,EFA)
Problem: As shown in the diagram, DG=GF, FG=FH, ∠FDH=28°, ∠FGH and ∠GHF are the base angles of the isosceles △ FGH, triangleGFD is an isosceles triangle. Find the measure of ∠DGF.
问题：如图所示，DG=GF，FG=FH，∠FDH=28°，FG和FH是等腰三角形FGH的腰，GF和GD是等腰三角形GFD的腰。求∠DGF的大小。
Construction (CDL):
Shape(FD,DG,GF)
Shape(FG,GH,HF)
Collinear(DGH)
Conditions (CDL):
Equal(LengthOfLine(DG),LengthOfLine(GF))
Equal(LengthOfLine(FG),LengthOfLine(FH))
Equal(MeasureOfAngle(FDH),28)
IsoscelesTriangle(FGH)
IsoscelesTriangle(GFD)
Goal: Value(MeasureOfAngle(DGF))
Answer: 124
Theorem sequence:
isosceles_triangle_property_angle_equal(1,GFD)
triangle_property_angle_sum(1,FDG)
Problem: As shown in the diagram, ∠NJK=101°, KJ∥HF. Find the measure of ∠FNI.
问题：如图所示，∠NJK=101°，KJ平行于HF。求∠FNI的大小。
Construction (CDL):
Shape(KJ,JE)
Shape(EJ,JA)
Shape(AJ,JN)
Shape(NJ,JK)
Shape(HN,NJ)
Shape(JN,NF)
Shape(FN,NI)
Shape(IN,NH)
Collinear(KJA)
Collinear(HNF)
Collinear(EJNI)
Conditions (CDL):
Equal(MeasureOfAngle(NJK),101)
ParallelBetweenLine(KJ,HF)
Goal: Value(MeasureOfAngle(FNI))
Answer: 79
Theorem sequence:
parallel_property_collinear_extend(3,FH,JK,N)
parallel_property_corresponding_angle(1,NH,JK,I)
adjacent_complementary_angle(1,FNI,INH)