SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, FA=4*w-7, GA=2*b+5, HA=2*w+3, JA=3*b+1, JF and HG are opposite sides of the ▱ FJHG. Find the value of b.
问题：如图所示，FA=4*w-7，GA=2*b+5，HA=2*w+3，JA=3*b+1，四边形FJHG是▱。求b的值。
Construction (CDL):
Shape(FJ,JA,AF)
Shape(AJ,JH,HA)
Shape(AH,HG,GA)
Shape(FA,AG,GF)
Collinear(FAH)
Collinear(JAG)
Conditions (CDL):
Equal(LengthOfLine(FA),4*w-7)
Equal(LengthOfLine(GA),2*b+5)
Equal(LengthOfLine(HA),2*w+3)
Equal(LengthOfLine(JA),3*b+1)
Parallelogram(FJHG)
Goal: Value(b)
Answer: 4
Theorem sequence:
parallelogram_property_diagonal_bisection(1,GFJH,A)
Problem: As shown in the diagram, NP=10, NQ=x, QP=24, N is the center of circle N, QP is the tangent to circle N. Find the value of x.
问题：如图所示，NP=10，NQ=x，QP=24，圆N的圆心为N，⊙O的切线为QP。求x的值。
Construction (CDL):
Shape(QP,PN,NQ)
Shape(NG,GP,PN)
Shape(NP,NPG,GN)
Collinear(NGQ)
Cocircular(N,GP)
Conditions (CDL):
Equal(LengthOfLine(NP),10)
Equal(LengthOfLine(NQ),x)
Equal(LengthOfLine(QP),24)
IsCentreOfCircle(N,N)
IsTangentOfCircle(QP,N)
Goal: Value(x)
Answer: 26
Theorem sequence:
tangent_of_circle_property_perpendicular(2,QP,N,N)
right_triangle_judgment_angle(1,QPN)
right_triangle_property_pythagorean(1,QPN)
Problem: As shown in the diagram, PS=3, RY=5, WX=10, WY=8, XY=6, RP is parallel to XW, RQ is perpendicular to PQ, WY⊥XY, YS is perpendicular to PS. Find the length of line PY.
问题：如图所示，PS=3，RY=5，WX=10，WY=8，XY=6，RP∥XW，RQ垂直于PQ，WY⊥XY，YS垂直于PS。求直线PY的长度。
Construction (CDL):
Shape(PY,YS,SP)
Shape(YR,RB,BY)
Shape(SY,YB,BQ,QS)
Shape(BX,XQ,QB)
Shape(SQ,QW,WS)
Collinear(RYP)
Collinear(YBX)
Collinear(RBQ)
Collinear(PSQ)
Collinear(XQW)
Collinear(YSW)
Conditions (CDL):
Equal(LengthOfLine(PS),3)
Equal(LengthOfLine(RY),5)
Equal(LengthOfLine(WX),10)
Equal(LengthOfLine(WY),8)
Equal(LengthOfLine(XY),6)
ParallelBetweenLine(RP,XW)
PerpendicularBetweenLine(RQ,PQ)
PerpendicularBetweenLine(WY,XY)
PerpendicularBetweenLine(YS,PS)
Goal: Value(LengthOfLine(PY))
Answer: 5
Theorem sequence:
parallel_property_collinear_extend(3,RP,XW,Y)
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, the diameter of ⊙B is equal to the length of line AD, the diameter of circle B is equal to the length of line CG, ∠CBD=55°, ∠GBF=35°, the center of ⊙B is B. Find the measure of arc BCA.
问题：如图所示，⊙B的直径与直线AD的长度相等，圆B的直径与直线CG的长度相等，∠CBD=55°，∠GBF=35°，B是⊙B的圆心。求弧BCA的角度。
Construction (CDL):
Shape(BC,BCA,AB)
Shape(BA,BAG,GB)
Shape(BG,BGF,FB)
Shape(BF,BFD,DB)
Shape(BD,BDC,CB)
Collinear(CBG)
Collinear(ABD)
Cocircular(B,CAGFD)
Conditions (CDL):
Equal(DiameterOfCircle(B),LengthOfLine(AD))
Equal(DiameterOfCircle(B),LengthOfLine(CG))
Equal(MeasureOfAngle(CBD),55)
Equal(MeasureOfAngle(GBF),35)
IsCentreOfCircle(B,B)
Goal: Value(MeasureOfArc(BCA))
Answer: 125
Theorem sequence:
flat_angle(1,GBC)
vertical_angle(1,CBD,GBA)
angle_addition(1,GBA,ABC)
arc_property_center_angle(1,BCA,B)
Problem: As shown in the diagram, ∠EAC=x+15°, ∠FCB=106°, ∠GDA=x-10°, ∠HBD=2*x-35°. Find the value of x.
问题：如图所示，∠EAC=x+15°，∠FCB=106°，∠GDA=x-10°，∠HBD=2*x-35°。求x的值。
Construction (CDL):
Shape(EA,AC,CE)
Shape(AD,DB,BC,CA)
Shape(AG,GD,DA)
Shape(FC,CB,BF)
Shape(DH,HB,BD)
Collinear(EAD)
Collinear(ACF)
Collinear(CBH)
Collinear(GDB)
Conditions (CDL):
Equal(MeasureOfAngle(EAC),x+15)
Equal(MeasureOfAngle(FCB),106)
Equal(MeasureOfAngle(GDA),x-10)
Equal(MeasureOfAngle(HBD),2*x-35)
Goal: Value(x)
Answer: 71
Theorem sequence:
flat_angle(1,EAD)
flat_angle(1,GDB)
flat_angle(1,HBC)
flat_angle(1,FCA)
angle_addition(1,EAC,CAD)
angle_addition(1,GDA,ADB)
angle_addition(1,HBD,DBC)
angle_addition(1,FCB,BCA)
quadrilateral_property_angle_sum(1,ADBC)
Problem: As shown in the diagram, AB=32, AC=x, BC=y, ∠ABC=60°, ∠BCA=90°. Find the value of x.
问题：如图所示，AB=32，AC=x，BC=y，∠ABC=60°，∠BCA=90°。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),32)
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BC),y)
Equal(MeasureOfAngle(ABC),60)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(x)
Answer: 16*sqrt(3)
Theorem sequence:
sine_theorem(1,ABC)
Problem: As shown in the diagram, AB=c, BC=a, CA=b, ∠ABC=60°, ∠CAB=30°, b=3, CB⊥CB. Find the value of a.
问题：如图所示，AB=c，BC=a，CA=b，∠ABC=60°，∠CAB=30°，b=3，CB垂直于CB。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(b,3)
PerpendicularBetweenLine(CB,CA)
Goal: Value(a)
Answer: sqrt(3)
Theorem sequence:
sine_theorem(1,CAB)
Problem: As shown in the diagram, ∠TYZ=52°, ∠ZYX=38°, YZ is perpendicular to TZ, TYXW is a rectangle. Find the measure of ∠ZTY.
问题：如图所示，∠TYZ=52°，∠ZYX=38°，YZ垂直于TZ，四边形TYXW是长方形。求∠ZTY的大小。
Construction (CDL):
Shape(TY,YZ,ZT)
Shape(YX,XW,WY)
Shape(TZ,ZW,WT)
Collinear(YZW)
Conditions (CDL):
Equal(MeasureOfAngle(TYZ),52)
Equal(MeasureOfAngle(ZYX),38)
PerpendicularBetweenLine(YZ,TZ)
Rectangle(TYXW)
Goal: Value(MeasureOfAngle(ZTY))
Answer: 38
Theorem sequence:
triangle_property_angle_sum(1,TYZ)
Problem: As shown in the diagram, QS=TS, RQ=2*x+3, RT=4*x-7, RS is perpendicular to TS. Find the length of line RT.
问题：如图所示，QS=TS，RQ=2*x+3，RT=4*x-7，RS⊥TS。求直线RT的长度。
Construction (CDL):
Shape(QS,SR,RQ)
Shape(ST,TR,RS)
Collinear(ACSRB)
Collinear(QST)
Conditions (CDL):
Equal(LengthOfLine(QS),LengthOfLine(TS))
Equal(LengthOfLine(RQ),2*x+3)
Equal(LengthOfLine(RT),4*x-7)
PerpendicularBetweenLine(RS,TS)
Goal: Value(LengthOfLine(RT))
Answer: 13
Theorem sequence:
adjacent_complementary_angle(1,QSR,RST)
mirror_congruent_triangle_judgment_sas(1,SRQ,STR)
mirror_congruent_triangle_property_line_equal(1,SRQ,STR)
Problem: As shown in the diagram, Add(AreaOfSector(EDB)=AreaOfSector(ECA)), the circumference of the ⊙E is 16*pi, E is the center of ⊙E, AE is perpendicular to CE, BE is perpendicular to DE. Find the value of x.
问题：如图所示，Add(AreaOfSector(EDB)=AreaOfSector(ECA))，⊙E的周长为16*pi，⊙E的圆心为E，AE垂直于CE，BE⊥DE。求x的值。
Construction (CDL):
Shape(ED,EDB,BE)
Shape(EB,EBC,CE)
Shape(EC,ECA,AE)
Shape(EA,EAD,DE)
Collinear(BEA)
Collinear(DEC)
Cocircular(E,DBCA)
Conditions (CDL):
Equal(Add(AreaOfSector(EDB),AreaOfSector(ECA)),x)
Equal(PerimeterOfCircle(E),16*pi)
IsCentreOfCircle(E,E)
PerpendicularBetweenLine(AE,CE)
PerpendicularBetweenLine(BE,DE)
Goal: Value(x)
Answer: 32*pi
Theorem sequence:
arc_property_center_angle(1,EDB,E)
arc_property_center_angle(1,ECA,E)
circle_perimeter_formula(1,E)
sector_area_formula(1,EDB)
sector_area_formula(1,ECA)
Problem: As shown in the diagram, the area of BADC is 2000, BC=x+10, BE=x, BADC is a ▱, AE⊥BE. Find the length of line BC.
问题：如图所示，四边形BADC的面积为2000，BC=x+10，BE=x，AB和DC是▱BADC的一组对边，AE垂直于BE。求直线BC的长度。
Construction (CDL):
Shape(BA,AE,EB)
Shape(BE,ED,DC,CB)
Collinear(AED)
Conditions (CDL):
Equal(AreaOfQuadrilateral(BADC),2000)
Equal(LengthOfLine(BC),x+10)
Equal(LengthOfLine(BE),x)
Parallelogram(BADC)
PerpendicularBetweenLine(AE,BE)
Goal: Value(LengthOfLine(BC))
Answer: 50
Theorem sequence:
altitude_of_quadrilateral_judgment_left_vertex(1,BE,BADC)
parallelogram_property_opposite_line_equal(1,CBAD)
parallelogram_area_formula_common(1,BADC)
Problem: As shown in the diagram, AB=13, AD=x, AF=3, BE=15, CE=15/2, DC=12, BF⊥AF, DE⊥BE. Find the value of x.
问题：如图所示，AB=13，AD=x，AF=3，BE=15，CE=15/2，DC=12，BF⊥AF，DE⊥BE。求x的值。
Construction (CDL):
Shape(DC,CE,ED)
Shape(DA,AF,FD)
Shape(FA,AB,BF)
Shape(DE,EB,BA,AD)
Collinear(DFB)
Collinear(CEB)
Conditions (CDL):
Equal(LengthOfLine(AB),13)
Equal(LengthOfLine(AD),x)
Equal(LengthOfLine(AF),3)
Equal(LengthOfLine(BE),15)
Equal(LengthOfLine(CE),15/2)
Equal(LengthOfLine(DC),12)
PerpendicularBetweenLine(BF,AF)
PerpendicularBetweenLine(DE,BE)
Goal: Value(x)
Answer: sqrt(1927-48*sqrt(1390))/2
Theorem sequence:
line_addition(1,DF,FB)
adjacent_complementary_angle(1,BFA,AFD)
adjacent_complementary_angle(1,CED,DEB)
right_triangle_judgment_angle(1,CED)
right_triangle_judgment_angle(1,DEB)
right_triangle_judgment_angle(1,AFD)
right_triangle_judgment_angle(1,BFA)
right_triangle_property_pythagorean(1,CED)
right_triangle_property_pythagorean(1,DEB)
right_triangle_property_pythagorean(1,BFA)
right_triangle_property_pythagorean(1,AFD)
Problem: As shown in the diagram, AC=9, DAEB is a square. Find the area of DAEB.
问题：如图所示，AC=9，四边形DAEB是正方形。求DAEB的面积。
Construction (CDL):
Shape(DA,AC,CD)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(DC,CB,BD)
Collinear(DCE)
Collinear(ACB)
Conditions (CDL):
Equal(LengthOfLine(AC),9)
Square(DAEB)
Goal: Value(AreaOfQuadrilateral(DAEB))
Answer: 162
Theorem sequence:
parallelogram_property_diagonal_bisection(1,AEBD,C)
line_addition(1,AC,CB)
isosceles_triangle_judgment_line_equal(1,DAB)
isosceles_triangle_property_angle_equal(1,DAB)
triangle_property_angle_sum(1,DAB)
sine_theorem(1,ABD)
parallelogram_area_formula_sine(1,DAEB)
Problem: As shown in the diagram, AD=6, ∠ACB=88°, ∠DCA=32°, DA and CB are opposite sides of the parallelogram ADCB. Find the perimeter of ADCB.
问题：如图所示，AD=6，∠ACB=88°，∠DCA=32°，DA和CB是▱ADCB的一组对边。求ADCB的周长。
Construction (CDL):
Shape(AD,DC,CA)
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AD),6)
Equal(MeasureOfAngle(ACB),88)
Equal(MeasureOfAngle(DCA),32)
Parallelogram(ADCB)
Goal: Value(PerimeterOfQuadrilateral(ADCB))
Answer: 12+12*cos(pi/90)/sin(8*pi/45)
Theorem sequence:
parallel_property_alternate_interior_angle(2,BC,AD)
sine_theorem(1,DCA)
parallelogram_property_opposite_line_equal(1,ADCB)
parallelogram_property_opposite_line_equal(1,BADC)
quadrilateral_perimeter_formula(1,ADCB)
Problem: As shown in the diagram, CB=6, ∠DAB=Mul(MeasureOfAngle(CDA),2)°, DABC is a rhombus. Find the measure of ∠DAB.
问题：如图所示，CB=6，∠DAB=Mul(MeasureOfAngle(CDA),2)°，DABC是菱形。求∠DAB的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EA,AB,BE)
Shape(CE,EB,BC)
Shape(DE,EC,CD)
Collinear(DEB)
Collinear(CEA)
Conditions (CDL):
Equal(LengthOfLine(CB),6)
Equal(MeasureOfAngle(DAB),Mul(MeasureOfAngle(CDA),2))
Rhombus(DABC)
Goal: Value(MeasureOfAngle(DAB))
Answer: 120
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,DABC,E)
kite_property_diagonal_perpendicular_bisection(1,ABCD,E)
perpendicular_bisector_property_bisector(1,DE,AC)
perpendicular_bisector_property_bisector(1,AE,BD)
angle_addition(1,CDE,EDA)
angle_addition(1,DAE,EAB)
triangle_property_angle_sum(1,DAE)
Problem: As shown in the diagram, PY=3*x-5, WP=2*x+11, quadrilateral ZWXY is a rectangle. Find the length of line ZP.
问题：如图所示，PY=3*x-5，WP=2*x+11，四边形ZWXY是长方形。求直线ZP的长度。
Construction (CDL):
Shape(ZW,WP,PZ)
Shape(PW,WX,XP)
Shape(PX,XY,YP)
Shape(ZP,PY,YZ)
Collinear(ZPX)
Collinear(WPY)
Conditions (CDL):
Equal(LengthOfLine(PY),3*x-5)
Equal(LengthOfLine(WP),2*x+11)
Rectangle(ZWXY)
Goal: Value(LengthOfLine(ZP))
Answer: 43
Theorem sequence:
parallelogram_property_diagonal_bisection(1,YZWX,P)
line_addition(1,WP,PY)
rectangle_property_diagonal_equal(1,ZWXY)
parallelogram_property_diagonal_bisection(1,ZWXY,P)
line_addition(1,ZP,PX)
Problem: As shown in the diagram, AB=c, BC=a, CA=b, ∠ABC=60°, ∠CAB=30°, a=4, BC⊥AC. Find the value of c.
问题：如图所示，AB=c，BC=a，CA=b，∠ABC=60°，∠CAB=30°，a=4，BC⊥AC。求c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(a,4)
PerpendicularBetweenLine(BC,AC)
Goal: Value(c)
Answer: 8
Theorem sequence:
sine_theorem(1,BCA)
Problem: As shown in the diagram, BE=7, BE=AE, IE=6, IE=EO, IAOB is a kite. Find the area of IAOB.
问题：如图所示，BE=7，BE=AE，IE=6，IE=EO，四边形IAOB是风筝形。求IAOB的面积。
Construction (CDL):
Shape(IA,AE,EI)
Shape(EA,AO,OE)
Shape(IE,EB,BI)
Shape(EO,OB,BE)
Collinear(IEO)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(BE),7)
Equal(LengthOfLine(BE),LengthOfLine(AE))
Equal(LengthOfLine(IE),6)
Equal(LengthOfLine(IE),LengthOfLine(EO))
Kite(IAOB)
Goal: Value(AreaOfQuadrilateral(IAOB))
Answer: 84
Theorem sequence:
line_addition(1,IE,EO)
line_addition(1,AE,EB)
kite_area_formula_diagonal(1,IAOB)
Problem: As shown in the diagram, WT=3, WX⊥YX, XY is perpendicular to ZY, YZ is perpendicular to WZ, ZW⊥XW, quadrilateral XYZW is a square. Find the measure of ∠XYW.
问题：如图所示，WT=3，WX⊥YX，XY垂直于ZY，YZ⊥WZ，ZW⊥XW，XYZW是正方形。求∠XYW的大小。
Construction (CDL):
Shape(XY,YT,TX)
Shape(YZ,ZT,TY)
Shape(TZ,ZW,WT)
Shape(XT,TW,WX)
Collinear(XTZ)
Collinear(WTY)
Conditions (CDL):
Equal(LengthOfLine(WT),3)
PerpendicularBetweenLine(WX,YX)
PerpendicularBetweenLine(XY,ZY)
PerpendicularBetweenLine(YZ,WZ)
PerpendicularBetweenLine(ZW,XW)
Square(XYZW)
Goal: Value(MeasureOfAngle(XYW))
Answer: 45
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,XYW)
isosceles_triangle_property_angle_equal(1,XYW)
triangle_property_angle_sum(1,XYW)
Problem: As shown in the diagram, ∠EJB=133°, JC is parallel to EK. Find the measure of ∠JEK.
问题：如图所示，∠EJB=133°，JC平行于EK。求∠JEK的大小。
Construction (CDL):
Shape(DB,BJ,JD)
Shape(DJ.JC,CD)
Shape(BH,HE,EJ,JB)
Shape(JE,EK,KC,CJ)
Shape(HI,IE,EH)
Shape(EI,IK,KE)
Collinear(BJC)
Collinear(HEK)
Collinear(DJEI)
Conditions (CDL):
Equal(MeasureOfAngle(EJB),133)
ParallelBetweenLine(JC,EK)
Goal: Value(MeasureOfAngle(JEK))
Answer: 133
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,JC,EK)
flat_angle(1,CJB)
angle_addition(1,CJE,EJB)
Problem: As shown in the diagram, AD=22, BD=10, BX=8, CX=x. Find the value of x.
问题：如图所示，AD=22，BD=10，BX=8，CX=x。求x的值。
Construction (CDL):
Collinear(BDA)
Collinear(BXC)
Cocircular(O,ADXC)
Conditions (CDL):
Equal(LengthOfLine(AD),22)
Equal(LengthOfLine(BD),10)
Equal(LengthOfLine(BX),8)
Equal(LengthOfLine(CX),x)
Goal: Value(x)
Answer: 32
Theorem sequence:
line_addition(1,BD,DA)
circle_property_circular_power_segment_and_segment_line(1,BDA,BXC,O)
line_addition(1,BX,XC)
Problem: As shown in the diagram, NM=11, ∠MPN=∠NMP, ∠PNM=120°. Find the length of line PN.
问题：如图所示，NM=11，∠MPN=∠NMP，∠PNM=120°。求直线PN的长度。
Construction (CDL):
Shape(PN,NM,MP)
Conditions (CDL):
Equal(LengthOfLine(NM),11)
Equal(MeasureOfAngle(MPN),MeasureOfAngle(NMP))
Equal(MeasureOfAngle(PNM),120)
Goal: Value(LengthOfLine(PN))
Answer: 11
Theorem sequence:
sine_theorem(1,MPN)
sine_theorem(1,PNM)
Problem: As shown in the diagram, AD=11, ∠AEF=32°, ∠DEA=32°, AD⊥ED, IF is perpendicular to GF. Find the length of line AF.
问题：如图所示，AD=11，∠AEF=32°，∠DEA=32°，AD垂直于ED，IF垂直于GF。求直线AF的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(AE,EF,FA)
Collinear(EDJB)
Collinear(EAH)
Collinear(EIFLK)
Collinear(AGF)
Conditions (CDL):
Equal(LengthOfLine(AD),11)
Equal(MeasureOfAngle(AEF),32)
Equal(MeasureOfAngle(DEA),32)
PerpendicularBetweenLine(AD,ED)
PerpendicularBetweenLine(IF,GF)
Goal: Value(LengthOfLine(AF))
Answer: 11
Theorem sequence:
sine_theorem(1,ADE)
sine_theorem(1,AEF)
Problem: As shown in the diagram, FM=BM, ∠MHN=40°, the center of ⊙M is M, EF is perpendicular to MF, HB is perpendicular to MB. Find the measure of ⌒MES.
问题：如图所示，FM=BM，∠MHN=40°，M是圆M的圆心，EF⊥MF，HB⊥MB。求⌒MES的角度。
Construction (CDL):
Shape(ME,EF,FM)
Shape(MF,FS,SM)
Shape(MH,HB,BM)
Shape(MB,BN,NM)
Shape(FE,MEA,AF)
Shape(SF,FA,MAS)
Collinear(SFE)
Collinear(AFM)
Collinear(HBN)
Cocircular(M,EASHN)
Conditions (CDL):
Equal(LengthOfLine(FM),LengthOfLine(BM))
Equal(MeasureOfAngle(MHN),40)
IsCentreOfCircle(M,M)
PerpendicularBetweenLine(EF,MF)
PerpendicularBetweenLine(HB,MB)
Goal: Value(MeasureOfArc(MES))
Answer: 100
Theorem sequence:
radius_of_circle_property_length_equal(1,ME,M)
radius_of_circle_property_length_equal(1,MS,M)
radius_of_circle_property_length_equal(1,MH,M)
congruent_triangle_judgment_hl(2,HBM,EFM)
congruent_triangle_property_angle_equal(1,HBM,EFM)
isosceles_triangle_judgment_line_equal(1,MES)
isosceles_triangle_property_angle_equal(1,MES)
triangle_property_angle_sum(1,MES)
arc_property_center_angle(1,MES,M)
Problem: As shown in the diagram, AB=y, AC=x, AD=z, BC=9, CD=4, AC is perpendicular to DC, BC is perpendicular to AC, DA⊥BA. Find the value of z.
问题：如图所示，AB=y，AC=x，AD=z，BC=9，CD=4，AC⊥DC，BC垂直于AC，DA⊥BA。求z的值。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Collinear(BCD)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(AD),z)
Equal(LengthOfLine(BC),9)
Equal(LengthOfLine(CD),4)
PerpendicularBetweenLine(AC,DC)
PerpendicularBetweenLine(BC,AC)
PerpendicularBetweenLine(DA,BA)
Goal: Value(z)
Answer: 2*sqrt(13)
Theorem sequence:
line_addition(1,BC,CD)
right_triangle_judgment_angle(1,BCA)
right_triangle_judgment_angle(1,ACD)
right_triangle_judgment_angle(1,DAB)
right_triangle_property_pythagorean(1,BCA)
right_triangle_property_pythagorean(1,ACD)
right_triangle_property_pythagorean(1,DAB)
Problem: As shown in the diagram, RS=5, SP=3, ∠PQR=128°, QP and RS are opposite sides of the parallelogram QRSP. Find the measure of ∠RSP.
问题：如图所示，RS=5，SP=3，∠PQR=128°，四边形QRSP是平行四边形。求∠RSP的大小。
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(MeasureOfAngle(RSP))
Answer: 128
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,QRSP)
Problem: As shown in the diagram, XY=2, the radius of ⊙W is 4, the radius of circle Z is 7, the center of ⊙W is W, the center of ⊙Z is Z, the diameter of circle W is IY. Find the length of line IX.
问题：如图所示，XY=2，⊙W的半径为4，⊙Z的半径为7，⊙W的圆心为W，⊙Z的圆心为Z，IY是圆W的直径。求直线IX的长度。
Construction (CDL):
Collinear(IWXYZC)
Cocircular(W,YI)
Cocircular(Z,CX)
Conditions (CDL):
Equal(LengthOfLine(XY),2)
Equal(RadiusOfCircle(W),4)
Equal(RadiusOfCircle(Z),7)
IsCentreOfCircle(W,W)
IsCentreOfCircle(Z,Z)
IsDiameterOfCircle(IY,W)
Goal: Value(LengthOfLine(IX))
Answer: 6
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,W)
diameter_of_circle_property_length_equal(1,IY,W)
line_addition(1,IX,XY)
Problem: As shown in the diagram, ∠ADC=120°, ∠CIA=3*y+40°, ∠DCI=x°, ∠EIC=3*x-70°, IC∥AD. Find the value of x.
问题：如图所示，∠ADC=120°，∠CIA=3*y+40°，∠DCI=x°，∠EIC=3*x-70°，IC平行于AD。求x的值。
Construction (CDL):
Shape(AD,DC,CI,IA)
Shape(IC,CE,EI)
Collinear(EIA)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),120)
Equal(MeasureOfAngle(CIA),3*y+40)
Equal(MeasureOfAngle(DCI),x)
Equal(MeasureOfAngle(EIC),3*x-70)
ParallelBetweenLine(IC,AD)
Goal: Value(x)
Answer: 60
Theorem sequence:
parallel_property_corresponding_angle(1,IC,AD,E)
flat_angle(1,EIA)
angle_addition(1,EIC,CIA)
quadrilateral_property_angle_sum(1,ADCI)
Problem: As shown in the diagram, GH=10, G is the center of circle G, J is the center of ⊙J, the center of ⊙K is K. Find the length of line FG.
问题：如图所示，GH=10，⊙G的圆心为G，圆J的圆心为J，K是⊙K的圆心。求直线FG的长度。
Construction (CDL):
Shape(GH,GHB,BG)
Shape(GB,GBF,FG)
Shape(GF,GFL,LG)
Shape(GK,GKH,HG)
Collinear(BGJKL)
Collinear(FGH)
Cocircular(G,HBFL)
Cocircular(J,GL)
Cocircular(K,JICL)
Conditions (CDL):
Equal(LengthOfLine(GH),10)
IsCentreOfCircle(G,G)
IsCentreOfCircle(J,J)
IsCentreOfCircle(K,K)
Goal: Value(LengthOfLine(FG))
Answer: 10
Theorem sequence:
radius_of_circle_property_length_equal(1,GH,G)
radius_of_circle_property_length_equal(1,GF,G)
Problem: As shown in the diagram, AB=17, AC=BD, AD=x, BD=AD, BD=BC, ∠BAC=y°, AC⊥BC. Find the value of y.
问题：如图所示，AB=17，AC=BD，AD=x，BD=AD，BD=BC，∠BAC=y°，AC⊥BC。求y的值。
Construction (CDL):
Shape(AC,CB,BA)
Shape(AB,BD,DA)
Conditions (CDL):
Equal(LengthOfLine(AB),17)
Equal(LengthOfLine(AC),LengthOfLine(BD))
Equal(LengthOfLine(AD),x)
Equal(LengthOfLine(BD),LengthOfLine(AD))
Equal(LengthOfLine(BD),LengthOfLine(BC))
Equal(MeasureOfAngle(BAC),y)
PerpendicularBetweenLine(AC,BC)
Goal: Value(y)
Answer: 45
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,CBA)
isosceles_triangle_property_angle_equal(1,CBA)
triangle_property_angle_sum(1,CBA)
Problem: As shown in the diagram, AB=y, AC=3, BC=x, ∠CBA=45°, AC⊥BC. Find the value of y.
问题：如图所示，AB=y，AC=3，BC=x，∠CBA=45°，AC垂直于BC。求y的值。
Construction (CDL):
Shape(CB,BA,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AC),3)
Equal(LengthOfLine(BC),x)
Equal(MeasureOfAngle(CBA),45)
PerpendicularBetweenLine(AC,BC)
Goal: Value(y)
Answer: 3*sqrt(2)
Theorem sequence:
sine_theorem(1,ACB)
Problem: As shown in the diagram, JL=5*x-6, KJ=3*x+6, KJ=KL, KL=4*x, KL=JL. Find the length of line JL.
问题：如图所示，JL=5*x-6，KJ=3*x+6，KJ=KL，KL=4*x，KL=JL。求直线JL的长度。
Construction (CDL):
Shape(KJ,JL,LK)
Conditions (CDL):
Equal(LengthOfLine(JL),5*x-6)
Equal(LengthOfLine(KJ),3*x+6)
Equal(LengthOfLine(KJ),LengthOfLine(KL))
Equal(LengthOfLine(KL),4*x)
Equal(LengthOfLine(KL),LengthOfLine(JL))
Goal: Value(LengthOfLine(JL))
Answer: 24
Theorem sequence:
Problem: As shown in the diagram, the measure of arc NAE is 75. Find the measure of ∠EDA.
问题：如图所示，⌒NAE的角度为75。求∠EDA的大小。
Construction (CDL):
Shape(AE,ED,DA)
Collinear(AND)
Cocircular(N,AED)
Conditions (CDL):
Equal(MeasureOfArc(NAE),75)
Goal: Value(MeasureOfAngle(EDA))
Answer: 75/2
Theorem sequence:
arc_property_circumference_angle_external(1,NAE,D)
Problem: As shown in the diagram, AE=3*y+5, BC=6*x-10, BE=7*y-11, DC=4*x+2, DC=CB, CE is perpendicular to BE, DA is perpendicular to EA. Find the value of y.
问题：如图所示，AE=3*y+5，BC=6*x-10，BE=7*y-11，DC=4*x+2，DC=CB，CE⊥BE，DA⊥EA。求y的值。
Construction (CDL):
Shape(DA,AE,EC,CD)
Shape(CE,EB,BC)
Collinear(AEB)
Collinear(DCB)
Conditions (CDL):
Equal(LengthOfLine(AE),3*y+5)
Equal(LengthOfLine(BC),6*x-10)
Equal(LengthOfLine(BE),7*y-11)
Equal(LengthOfLine(DC),4*x+2)
Equal(LengthOfLine(DC),LengthOfLine(CB))
PerpendicularBetweenLine(CE,BE)
PerpendicularBetweenLine(DA,EA)
Goal: Value(y)
Answer: 4
Theorem sequence:
line_addition(1,DC,CB)
line_addition(1,AE,EB)
similar_triangle_judgment_aa(1,CEB,DAB)
similar_triangle_property_line_ratio(1,EBC,ABD)
similar_triangle_property_line_ratio(1,CEB,DAB)
Problem: As shown in the diagram, the area of △CBA is 150, the area of △FDE is 96, AB=x, DE=14, △CBA is similar to △FDE.. Find the value of x.
问题：如图所示，△CBA的面积为150，△FDE的面积为96，AB=x，DE=14，△CBA相似于△FDE。求x的值。
Construction (CDL):
Shape(BA,AC,CB)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(AreaOfTriangle(CBA),150)
Equal(AreaOfTriangle(FDE),96)
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(DE),14)
SimilarBetweenTriangle(CBA,FDE)
Goal: Value(x)
Answer: 35/2
Theorem sequence:
similar_triangle_property_area_square_ratio(1,CBA,FDE)
similar_triangle_property_line_ratio(1,CBA,FDE)
Problem: As shown in the diagram, ∠CBA=28°, ∠DAE=68°, ∠ECB=82°. Find the measure of ∠AED.
问题：如图所示，∠CBA=28°，∠DAE=68°，∠ECB=82°。求∠AED的大小。
Construction (CDL):
Shape(CB,BE,EC)
Shape(ED,DA,AE)
Collinear(CED)
Collinear(BEA)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),28)
Equal(MeasureOfAngle(DAE),68)
Equal(MeasureOfAngle(ECB),82)
Goal: Value(MeasureOfAngle(AED))
Answer: 70
Theorem sequence:
triangle_property_angle_sum(1,CBE)
vertical_angle(1,BEC,AED)
Problem: As shown in the diagram, AC=11, AD=y, CD=x, ∠ABC=60°, ∠BCA=60°, ∠CAD=30°, BD is perpendicular to AD. Find the value of y.
问题：如图所示，AC=11，AD=y，CD=x，∠ABC=60°，∠BCA=60°，∠CAD=30°，BD⊥AD。求y的值。
Construction (CDL):
Shape(CA,AD,DC)
Shape(DA,AB,BD)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(AC),11)
Equal(LengthOfLine(AD),y)
Equal(LengthOfLine(CD),x)
Equal(MeasureOfAngle(ABC),60)
Equal(MeasureOfAngle(BCA),60)
Equal(MeasureOfAngle(CAD),30)
PerpendicularBetweenLine(BD,AD)
Goal: Value(y)
Answer: 11*sqrt(3)/2
Theorem sequence:
adjacent_complementary_angle(1,BDA,ADC)
sine_theorem(1,ADC)
Problem: As shown in the diagram, AN=10, BC=30, CN=5, AC⊥NC. Find the perimeter of triangle ABC.
问题：如图所示，AN=10，BC=30，CN=5，AC⊥NC。求三角形ABC的周长。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CN,NA)
Collinear(BCN)
Conditions (CDL):
Equal(LengthOfLine(AN),10)
Equal(LengthOfLine(BC),30)
Equal(LengthOfLine(CN),5)
PerpendicularBetweenLine(AC,NC)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 5*sqrt(3)+30+5*sqrt(39)
Theorem sequence:
adjacent_complementary_angle(1,BCA,ACN)
right_triangle_judgment_angle(1,ACN)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,ACN)
right_triangle_property_pythagorean(1,BCA)
triangle_perimeter_formula(1,ABC)
Problem: As shown in the diagram, LN=3*x+5, QS=2*x+10, ∠MNL=y+6°, ∠QSR=2*y-12°, triangle LMN is mirror congruent to triangle QSR. Find the value of y.
问题：如图所示，LN=3*x+5，QS=2*x+10，∠MNL=y+6°，∠QSR=2*y-12°，三角形LMN与三角形QSR是镜像全等三角形。求y的值。
Construction (CDL):
Shape(NL,LM,MN)
Shape(SR,RQ,QS)
Conditions (CDL):
Equal(LengthOfLine(LN),3*x+5)
Equal(LengthOfLine(QS),2*x+10)
Equal(MeasureOfAngle(MNL),y+6)
Equal(MeasureOfAngle(QSR),2*y-12)
MirrorCongruentBetweenTriangle(LMN,QSR)
Goal: Value(y)
Answer: 18
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,NLM,SRQ)