SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, RT=6, the measure of arc SRP is 98, S is the center of circle S, PT⊥QT. Find the measure of ⌒SQP.
问题：如图所示，RT=6，弧SRP的角度为98，S是圆S的圆心，PT垂直于QT。求弧SQP的角度。
Construction (CDL):
Shape(PT,TQ,SQP)
Shape(QT,TR,SRQ)
Shape(PS,ST,TP)
Shape(RT,TS,SR)
Shape(RS,SP,SPR)
Collinear(STQ)
Collinear(PTR)
Cocircular(S,RQP)
Conditions (CDL):
Equal(LengthOfLine(RT),6)
Equal(MeasureOfArc(SRP),98)
IsCentreOfCircle(S,S)
PerpendicularBetweenLine(PT,QT)
Goal: Value(MeasureOfArc(SQP))
Answer: 49
Theorem sequence:
adjacent_complementary_angle(1,STP,PTQ)
adjacent_complementary_angle(1,RTS,STP)
circle_property_chord_perpendicular_bisect_arc(1,SRP,STQ)
congruent_arc_judgment_length_equal(1,SRQ,SQP)
congruent_arc_property_measure_equal(1,SRQ,SQP)
arc_addition_measure(1,SRQ,SQP)
Problem: As shown in the diagram, AB=z, AD=9, BC=8, BD=y, CD=x, AD⊥BD, CB⊥AB. Find the value of y.
问题：如图所示，AB=z，AD=9，BC=8，BD=y，CD=x，AD⊥BD，CB⊥AB。求y的值。
Construction (CDL):
Shape(DC,CB,BD)
Shape(AD,DB,BA)
Collinear(CDA)
Conditions (CDL):
Equal(LengthOfLine(AB),z)
Equal(LengthOfLine(AD),9)
Equal(LengthOfLine(BC),8)
Equal(LengthOfLine(BD),y)
Equal(LengthOfLine(CD),x)
PerpendicularBetweenLine(AD,BD)
PerpendicularBetweenLine(CB,AB)
Goal: Value(y)
Answer: sqrt(-81/2+9*sqrt(337)/2)
Theorem sequence:
line_addition(1,CD,DA)
adjacent_complementary_angle(1,ADB,BDC)
right_triangle_judgment_angle(1,BDC)
right_triangle_judgment_angle(1,ADB)
right_triangle_judgment_angle(1,CBA)
right_triangle_property_pythagorean(1,BDC)
right_triangle_property_pythagorean(1,ADB)
right_triangle_property_pythagorean(1,CBA)
Problem: As shown in the diagram, AC=11, BC=12, ∠ABC=x°, ∠CAB=60°. Find the value of x.
问题：如图所示，AC=11，BC=12，∠ABC=x°，∠CAB=60°。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AC),11)
Equal(LengthOfLine(BC),12)
Equal(MeasureOfAngle(ABC),x)
Equal(MeasureOfAngle(CAB),60)
Goal: Value(x)
Answer: 180*asin(11*sqrt(3)/24)/pi
Theorem sequence:
sine_theorem(1,CAB)
Problem: As shown in the diagram, ∠ABD=x°, ∠CDA=61°, ∠DAB=69°, ∠DBC=55°, BC⊥DC. Find the measure of ∠ABC.
问题：如图所示，∠ABD=x°，∠CDA=61°，∠DAB=69°，∠DBC=55°，BC⊥DC。求∠ABC的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(BC,CD,DB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),x)
Equal(MeasureOfAngle(CDA),61)
Equal(MeasureOfAngle(DAB),69)
Equal(MeasureOfAngle(DBC),55)
PerpendicularBetweenLine(BC,DC)
Goal: Value(MeasureOfAngle(ABC))
Answer: 140
Theorem sequence:
quadrilateral_property_angle_sum(1,DABC)
Problem: As shown in the diagram, AB=2*y+1, CB=x-w+1, DA=3*x-2, DC=3-4*w, the perimeter of ADCB is 22, AB and DC are opposite sides of the ▱ ADCB. Find the length of line AB.
问题：如图所示，AB=2*y+1，CB=x-w+1，DA=3*x-2，DC=3-4*w，ADCB的周长为22，四边形ADCB是平行四边形。求直线AB的长度。
Construction (CDL):
Shape(AD,DC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),2*y+1)
Equal(LengthOfLine(CB),x-w+1)
Equal(LengthOfLine(DA),3*x-2)
Equal(LengthOfLine(DC),3-4*w)
Equal(PerimeterOfQuadrilateral(ADCB),22)
Parallelogram(ADCB)
Goal: Value(LengthOfLine(AB))
Answer: 7
Theorem sequence:
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, AC=27, BA=2*x+5, BA=BC, BC=3*x-4. Find the length of line AB.
问题：如图所示，AC=27，BA=2*x+5，BA=BC，BC=3*x-4。求直线AB的长度。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AC),27)
Equal(LengthOfLine(BA),2*x+5)
Equal(LengthOfLine(BA),LengthOfLine(BC))
Equal(LengthOfLine(BC),3*x-4)
Goal: Value(LengthOfLine(AB))
Answer: 23
Theorem sequence:
Problem: As shown in the diagram, the area of ADNB is Add(AreaOfTriangle(ADN),AreaOfTriangle(ANB)), AB=3, AB=DN, AD=BN, ∠ANB=62°, AD⊥ND, NB⊥AB. Find the area of quadrilateral ADNB.
问题：如图所示，四边形ADNB的面积为Add(AreaOfTriangle(ADN),AreaOfTriangle(ANB))，AB=3，AB=DN，AD=BN，∠ANB=62°，AD⊥ND，NB垂直于AB。求四边形ADNB的面积。
Construction (CDL):
Shape(AD,DN,NA)
Shape(AN,NB,BA)
Conditions (CDL):
Equal(AreaOfQuadrilateral(ADNB),Add(AreaOfTriangle(ADN),AreaOfTriangle(ANB)))
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AB),LengthOfLine(DN))
Equal(LengthOfLine(AD),LengthOfLine(BN))
Equal(MeasureOfAngle(ANB),62)
PerpendicularBetweenLine(AD,ND)
PerpendicularBetweenLine(NB,AB)
Goal: Value(AreaOfQuadrilateral(ADNB))
Answer: 9*tan(7*pi/45)
Theorem sequence:
triangle_property_angle_sum(1,ANB)
sine_theorem(1,BAN)
triangle_area_formula_sine(1,BAN)
triangle_area_formula_sine(1,DNA)
Problem: As shown in the diagram, PY=2*x+5, ZP=4*x-9, ZW⊥XW, ZWXY is a rectangle. Find the length of line ZX.
问题：如图所示，PY=2*x+5，ZP=4*x-9，ZW垂直于XW，ZWXY是矩形。求直线ZX的长度。
Construction (CDL):
Shape(ZW,WP,PZ)
Shape(ZP,PY,YZ)
Shape(YP,PX,XY)
Shape(PW,WX,XP)
Collinear(ZPX)
Collinear(YPW)
Conditions (CDL):
Equal(LengthOfLine(PY),2*x+5)
Equal(LengthOfLine(ZP),4*x-9)
PerpendicularBetweenLine(ZW,XW)
Rectangle(ZWXY)
Goal: Value(LengthOfLine(ZX))
Answer: 38
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ZWXY,P)
parallelogram_property_diagonal_bisection(1,WXYZ,P)
line_addition(1,ZP,PX)
line_addition(1,YP,PW)
rectangle_property_diagonal_equal(1,ZWXY)
Problem: As shown in the diagram, EH=14, EH=AB, EL=9, EL=CD, Sub(AreaOfQuadrilateral(EGFH)=AreaOfQuadrilateral(CADB)), BCAD is a kite, AB is parallel to EH, EGFH is a parallelogram, AI is perpendicular to CI, EL⊥GL. Find the value of x.
问题：如图所示，EH=14，EH=AB，EL=9，EL=CD，Sub(AreaOfQuadrilateral(EGFH)=AreaOfQuadrilateral(CADB))，BC和BD是风筝形BCAD的一组临边，AB平行于EH，EGFH是平行四边形，AI⊥CI，EL⊥GL。求x的值。
Construction (CDL):
Shape(EA,AC,CE)
Shape(CA,AI,IC)
Shape(CI,IB,BC)
Shape(CB,BH,HC)
Shape(AG,GD,DA)
Shape(AD,DI,IA)
Shape(ID,DB,BI)
Shape(BD,DF,FB)
Shape(EL,LG,GE)
Collinear(ECH)
Collinear(AIB)
Collinear(LGDF)
Collinear(EAG)
Collinear(CID)
Collinear(HBF)
Conditions (CDL):
Equal(LengthOfLine(EH),14)
Equal(LengthOfLine(EH),LengthOfLine(AB))
Equal(LengthOfLine(EL),9)
Equal(LengthOfLine(EL),LengthOfLine(CD))
Equal(Sub(AreaOfQuadrilateral(EGFH),AreaOfQuadrilateral(CADB)),x)
Kite(BCAD)
ParallelBetweenLine(AB,EH)
Parallelogram(EGFH)
PerpendicularBetweenLine(AI,CI)
PerpendicularBetweenLine(EL,GL)
Goal: Value(x)
Answer: 63
Theorem sequence:
altitude_of_quadrilateral_judgment_left_vertex(3,EL,EGFH)
kite_area_formula_diagonal(1,BCAD)
parallelogram_property_opposite_line_equal(1,HEGF)
parallelogram_area_formula_common(1,EGFH)
Problem: As shown in the diagram, ∠WXJ=130°, ∠YZI=20°, Mul(LengthOfLine(JH)=LengthOfLine(JX)). Find the measure of ∠ZIY.
问题：如图所示，∠WXJ=130°，∠YZI=20°，Mul(LengthOfLine(JH)=LengthOfLine(JX))。求∠ZIY的大小。
Construction (CDL):
Shape(JH,HI,IJ)
Shape(WX,XH)
Shape(IY,YZ,ZI)
Shape(HX,XY,YI,IH)
Collinear(HIZ)
Collinear(XHJ)
Collinear(JIY)
Collinear(WXYZ)
Conditions (CDL):
Equal(MeasureOfAngle(WXJ),130)
Equal(MeasureOfAngle(YZI),20)
Equal(Mul(LengthOfLine(JH),LengthOfLine(JX)),Mul(LengthOfLine(JI),LengthOfLine(JY)))
Goal: Value(MeasureOfAngle(ZIY))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,WXJ,JXY)
mirror_similar_triangle_judgment_sas(1,JHI,JXY)
mirror_similar_triangle_property_angle_equal(1,IJH,XYJ)
vertical_angle(1,HIJ,ZIY)
Problem: As shown in the diagram, BG=17, GC=x, GD=8, IG=10, E is the center of ⊙E. Find the value of x.
问题：如图所示，BG=17，GC=x，GD=8，IG=10，E是圆E的圆心。求x的值。
Construction (CDL):
Shape(CG,GB,EBC)
Shape(IG,GC,ECI)
Shape(GI,EID,DG)
Shape(GD,EDB,BG)
Collinear(IGB)
Collinear(DGC)
Cocircular(E,CIDB)
Conditions (CDL):
Equal(LengthOfLine(BG),17)
Equal(LengthOfLine(GC),x)
Equal(LengthOfLine(GD),8)
Equal(LengthOfLine(IG),10)
IsCentreOfCircle(E,E)
Goal: Value(x)
Answer: 85/4
Theorem sequence:
circle_property_circular_power_chord_and_chord(1,DGC,IGB,E)
Problem: As shown in the diagram, ∠BCE=120°, ∠DBC=5*y°, ∠DFE=108°, ∠FDB=2*x°, BD is parallel to CE. Find the value of x.
问题：如图所示，∠BCE=120°，∠DBC=5*y°，∠DFE=108°，∠FDB=2*x°，BD∥CE。求x的值。
Construction (CDL):
Shape(BC,CF,FD,DB)
Shape(DF,FE,ED)
Collinear(CFE)
Conditions (CDL):
Equal(MeasureOfAngle(BCE),120)
Equal(MeasureOfAngle(DBC),5*y)
Equal(MeasureOfAngle(DFE),108)
Equal(MeasureOfAngle(FDB),2*x)
ParallelBetweenLine(BD,CE)
Goal: Value(x)
Answer: 54
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,BD,CE)
flat_angle(1,CFE)
angle_addition(1,CFD,DFE)
quadrilateral_property_angle_sum(1,BCFD)
Problem: As shown in the diagram, ST=6, WX=5, the perimeter of triangle SRT is 15, the perimeter of triangle WZX is Mul(PerimeterOfTriangle(SRT),RatioOfSimilarTriangle(WZX,SRT)), triangle WZX is similar to triangle SRT.. Find the perimeter of triangle WZX.
问题：如图所示，ST=6，WX=5，△SRT的周长为15，三角形WZX的周长为Mul(PerimeterOfTriangle(SRT),RatioOfSimilarTriangle(WZX,SRT))，三角形WZX相似于三角形SRT。求△WZX的周长。
Construction (CDL):
Shape(WZ,ZX,XW)
Shape(SR,RT,TS)
Conditions (CDL):
Equal(LengthOfLine(ST),6)
Equal(LengthOfLine(WX),5)
Equal(PerimeterOfTriangle(SRT),15)
Equal(PerimeterOfTriangle(WZX),Mul(PerimeterOfTriangle(SRT),RatioOfSimilarTriangle(WZX,SRT)))
SimilarBetweenTriangle(WZX,SRT)
Goal: Value(PerimeterOfTriangle(WZX))
Answer: 25/2
Theorem sequence:
similar_triangle_property_line_ratio(1,ZXW,RTS)
Problem: As shown in the diagram, ∠ABH=3*p-10°, ∠CDB=6*r+5°, ∠JAE=4*p+15°, AJ∥BH, AL is parallel to CI. Find the value of p.
问题：如图所示，∠ABH=3*p-10°，∠CDB=6*r+5°，∠JAE=4*p+15°，AJ∥BH，AL∥CI。求p的值。
Construction (CDL):
Shape(JA,AB,BH,HJ)
Shape(CD,DB,BA,AC)
Shape(HB,BE,EH)
Shape(LA,AJ,JL)
Shape(IC,CA,AL,LI)
Shape(IG,GC,CI)
Shape(CG,GF,FD,DC)
Shape(DK,KE,EB,BD)
Shape(DF,FK,KD)
Collinear(GCAJ)
Collinear(FDBH)
Collinear(ICDK)
Collinear(LABE)
Conditions (CDL):
Equal(MeasureOfAngle(ABH),3*p-10)
Equal(MeasureOfAngle(CDB),6*r+5)
Equal(MeasureOfAngle(JAE),4*p+15)
ParallelBetweenLine(AJ,BH)
ParallelBetweenLine(AL,CI)
Goal: Value(p)
Answer: 25
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AJ,BH)
Problem: As shown in the diagram, the area of GKJH is 75, GH=17, KJ=13, GH and KJ are the parallel sides of trapezoid GKJH. Find the height of GKJH.
问题：如图所示，四边形GKJH的面积为75，GH=17，KJ=13，GH和KJ是梯形GKJH的平行边。求GKJH的高。
Construction (CDL):
Shape(GK,KJ,JH,HG)
Conditions (CDL):
Equal(AreaOfQuadrilateral(GKJH),75)
Equal(LengthOfLine(GH),17)
Equal(LengthOfLine(KJ),13)
Trapezoid(GKJH)
Goal: Value(HeightOfQuadrilateral(GKJH))
Answer: 5
Theorem sequence:
trapezoid_area_formula(1,GKJH)
Problem: As shown in the diagram, EJ=6, LK=7.2, ML=4, JK and ML are opposite sides of the parallelogram JMLK, JE⊥LE. Find the area of JMLK.
问题：如图所示，EJ=6，LK=7.2，ML=4，JK和ML是平行四边形JMLK的一组对边，JE⊥LE。求四边形JMLK的面积。
Construction (CDL):
Shape(JM,ME,EJ)
Shape(JE,EL,LK,KJ)
Collinear(MEL)
Conditions (CDL):
Equal(LengthOfLine(EJ),6)
Equal(LengthOfLine(LK),7.2)
Equal(LengthOfLine(ML),4)
Parallelogram(JMLK)
PerpendicularBetweenLine(JE,LE)
Goal: Value(AreaOfQuadrilateral(JMLK))
Answer: 24
Theorem sequence:
flat_angle(1,MEL)
angle_addition(1,MEJ,JEL)
altitude_of_quadrilateral_judgment_left_vertex(1,JE,JMLK)
parallelogram_area_formula_common(1,JMLK)
Problem: As shown in the diagram, the diameter of circle A is 10, the center of circle A is A. Find the length of line DA.
问题：如图所示，⊙A的直径为10，A是⊙A的圆心。求直线DA的长度。
Construction (CDL):
Shape(AP,APD,DA)
Shape(AD,ADL,LA)
Shape(AL,ALG,GA)
Shape(AG,AGF,FA)
Shape(AF,AFP,PA)
Collinear(DAF)
Collinear(GAP)
Cocircular(A,PDLGF)
Conditions (CDL):
Equal(DiameterOfCircle(A),10)
IsCentreOfCircle(A,A)
Goal: Value(LengthOfLine(DA))
Answer: 5
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,A)
radius_of_circle_property_length_equal(1,AD,A)
Problem: As shown in the diagram, AX=CX, BC=3*x, BX=8, FG=6, FH=x, GH=HE, ∠ACB=∠EGF, ∠BAC=∠FEG. Find the value of x.
问题：如图所示，AX=CX，BC=3*x，BX=8，FG=6，FH=x，GH=HE，∠ACB=∠EGF，∠BAC=∠FEG。求x的值。
Construction (CDL):
Shape(CB,BX,XC)
Shape(BA,AX,XB)
Shape(FH,HG,GF)
Shape(FE,EH,HF)
Collinear(CXA)
Collinear(GHE)
Conditions (CDL):
Equal(LengthOfLine(AX),LengthOfLine(CX))
Equal(LengthOfLine(BC),3*x)
Equal(LengthOfLine(BX),8)
Equal(LengthOfLine(FG),6)
Equal(LengthOfLine(FH),x)
Equal(LengthOfLine(GH),LengthOfLine(HE))
Equal(MeasureOfAngle(ACB),MeasureOfAngle(EGF))
Equal(MeasureOfAngle(BAC),MeasureOfAngle(FEG))
Goal: Value(x)
Answer: 4
Theorem sequence:
similar_triangle_judgment_aa(1,BAC,FEG)
similar_triangle_property_line_ratio(1,BAC,FEG)
similar_triangle_property_line_ratio(1,ACB,EGF)
line_addition(1,AX,XC)
line_addition(1,EH,HG)
similar_triangle_judgment_sas(1,CBX,GFH)
similar_triangle_property_line_ratio(1,CBX,GFH)
similar_triangle_property_line_ratio(1,XCB,HGF)
Problem: As shown in the diagram, AB=15, AP=x, DF=12, QF=8, ∠GFQ=∠QFD, ∠PAC=∠BAP, Triangle ACB is mirror similar to triangle FDG.. Find the value of x.
问题：如图所示，AB=15，AP=x，DF=12，QF=8，∠GFQ=∠QFD，∠PAC=∠BAP，△ACB镜像相似于△FDG。求x的值。
Construction (CDL):
Shape(AC,CP,PA)
Shape(AP,PB,BA)
Shape(DQ,QF,FD)
Shape(QG,GF,FQ)
Collinear(CPB)
Collinear(DQG)
Conditions (CDL):
Equal(LengthOfLine(AB),15)
Equal(LengthOfLine(AP),x)
Equal(LengthOfLine(DF),12)
Equal(LengthOfLine(QF),8)
Equal(MeasureOfAngle(GFQ),MeasureOfAngle(QFD))
Equal(MeasureOfAngle(PAC),MeasureOfAngle(BAP))
MirrorSimilarBetweenTriangle(ACB,FDG)
Goal: Value(x)
Answer: 10
Theorem sequence:
mirror_similar_triangle_property_angle_equal(1,BAC,DGF)
mirror_similar_triangle_property_angle_equal(1,ACB,FDG)
angle_addition(1,BAP,PAC)
angle_addition(1,GFQ,QFD)
mirror_similar_triangle_judgment_aa(1,PBA,QFD)
mirror_similar_triangle_property_line_ratio(1,PBA,QFD)
mirror_similar_triangle_property_line_ratio(1,BAP,DQF)
Problem: As shown in the diagram, ∠CDE=116°, BADC is a rhombus. Find the measure of ∠DCA.
问题：如图所示，∠CDE=116°，BADC是菱形。求∠DCA的大小。
Construction (CDL):
Shape(BO,OC,CB)
Shape(CO,OD,DC)
Shape(BA,AO,OB)
Shape(OA,AD,DO)
Shape(CD,DE,EC)
Collinear(ADE)
Collinear(COA)
Collinear(BOD)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),116)
Rhombus(BADC)
Goal: Value(MeasureOfAngle(DCA))
Answer: 58
Theorem sequence:
flat_angle(1,ADE)
angle_addition(1,ADC,CDE)
parallelogram_property_opposite_angle_equal(1,BADC)
parallelogram_property_opposite_angle_equal(1,CBAD)
quadrilateral_property_angle_sum(1,BADC)
kite_property_diagonal_perpendicular_bisection(1,CBAD,O)
perpendicular_bisector_property_bisector(1,CO,BD)
angle_addition(1,DCO,OCB)
Problem: As shown in the diagram, ∠AYG=y+25°, ∠BYD=3*x-15°, ∠GBE=105°, DY∥EB. Find the value of y.
问题：如图所示，∠AYG=y+25°，∠BYD=3*x-15°，∠GBE=105°，DY平行于EB。求y的值。
Construction (CDL):
Shape(HD,DY,YH)
Shape(HY,YA,AH)
Shape(YB,BI,IA,AY)
Shape(DE,EB,BY,YD)
Shape(EG,GB,BE)
Shape(BG,GI,IB)
Collinear(HYBG)
Collinear(DYA)
Collinear(EBI)
Conditions (CDL):
Equal(MeasureOfAngle(AYG),y+25)
Equal(MeasureOfAngle(BYD),3*x-15)
Equal(MeasureOfAngle(GBE),105)
ParallelBetweenLine(DY,EB)
Goal: Value(y)
Answer: 50
Theorem sequence:
parallel_property_corresponding_angle(1,BE,YD,G)
flat_angle(1,AYD)
angle_addition(1,AYB,BYD)
Problem: As shown in the diagram, BQ=6, QE=8, ∠BDQ=∠EAQ, ∠DQB=∠AQE, ∠QEA=∠QBD, the ratio of similar triangles AQE and DQB is 4/3. Find the ratio of similar △AQE and △DQB.
问题：如图所示，BQ=6，QE=8，∠BDQ=∠EAQ，∠DQB=∠AQE，∠QEA=∠QBD，三角形AQE和三角形DQB的相似比为4/3。求△AQE和△DQB的相似比。
Construction (CDL):
Shape(DQ,QB,BD)
Shape(AD,DB,BE,EA)
Collinear(ADQ)
Collinear(QBE)
Conditions (CDL):
Equal(LengthOfLine(BQ),6)
Equal(LengthOfLine(QE),8)
Equal(MeasureOfAngle(BDQ),MeasureOfAngle(EAQ))
Equal(MeasureOfAngle(DQB),MeasureOfAngle(AQE))
Equal(MeasureOfAngle(QEA),MeasureOfAngle(QBD))
Equal(RatioOfSimilarTriangle(AQE,DQB),4/3)
Goal: Value(RatioOfSimilarTriangle(AQE,DQB))
Answer: 4/3
Theorem sequence:
Problem: As shown in the diagram, AB=15, AC=x, BC=8, CD=16, AB⊥CB. Find the length of line AC.
问题：如图所示，AB=15，AC=x，BC=8，CD=16，AB⊥CB。求直线AC的长度。
Construction (CDL):
Shape(DB,BA,AD)
Shape(BC,CA,AB)
Collinear(DBC)
Conditions (CDL):
Equal(LengthOfLine(AB),15)
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BC),8)
Equal(LengthOfLine(CD),16)
PerpendicularBetweenLine(AB,CB)
Goal: Value(LengthOfLine(AC))
Answer: 17
Theorem sequence:
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
Problem: As shown in the diagram, ∠ABD=x°, ∠CBA=65°, ∠DBC=132°, the center of ⊙B is B. Find the value of x.
问题：如图所示，∠ABD=x°，∠CBA=65°，∠DBC=132°，圆B的圆心为B。求x的值。
Construction (CDL):
Shape(CB,BA,BAC)
Shape(BC,BCD,DB)
Shape(AB,BD,BDA)
Cocircular(B,ACD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),x)
Equal(MeasureOfAngle(CBA),65)
Equal(MeasureOfAngle(DBC),132)
IsCentreOfCircle(B,B)
Goal: Value(x)
Answer: 163
Theorem sequence:
angle_addition(1,DBC,CBA)
round_angle(1,ABD,DBA)
Problem: As shown in the diagram, ∠ACB=100°, ∠BCD=10*y°, ∠CBA=40°, ∠DBC=25*x°, BACD is a parallelogram. Find the value of x.
问题：如图所示，∠ACB=100°，∠BCD=10*y°，∠CBA=40°，∠DBC=25*x°，四边形BACD是平行四边形。求x的值。
Construction (CDL):
Shape(BA,AC,CB)
Shape(BC,CD,DB)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),100)
Equal(MeasureOfAngle(BCD),10*y)
Equal(MeasureOfAngle(CBA),40)
Equal(MeasureOfAngle(DBC),25*x)
Parallelogram(BACD)
Goal: Value(x)
Answer: 4
Theorem sequence:
parallel_property_alternate_interior_angle(1,BD,AC)
Problem: As shown in the diagram, ∠ADC=x°, ∠BAD=x°, ∠CBA=2*x°, ∠DCB=2*x°. Find the measure of ∠BAD.
问题：如图所示，∠ADC=x°，∠BAD=x°，∠CBA=2*x°，∠DCB=2*x°。求∠BAD的大小。
Construction (CDL):
Shape(BA,AD,DC,CB)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),x)
Equal(MeasureOfAngle(BAD),x)
Equal(MeasureOfAngle(CBA),2*x)
Equal(MeasureOfAngle(DCB),2*x)
Goal: Value(MeasureOfAngle(BAD))
Answer: 60
Theorem sequence:
quadrilateral_property_angle_sum(1,BADC)
Problem: As shown in the diagram, ∠ZYX=68°, the measure of ⌒CXE is 224, the center of ⊙C is C. Find the measure of arc CZX.
问题：如图所示，∠ZYX=68°，弧CXE的角度为224，圆C的圆心为C。求弧CZX的角度。
Construction (CDL):
Shape(YX,CZX,ZY)
Shape(ZE,CEZ)
Shape(EZ,CZD)
Collinear(BXY)
Collinear(YZED)
Cocircular(C,EZXW)
Conditions (CDL):
Equal(MeasureOfAngle(ZYX),68)
Equal(MeasureOfArc(CXE),224)
IsCentreOfCircle(C,C)
Goal: Value(MeasureOfArc(CZX))
Answer: 88
Theorem sequence:
circle_property_circular_power_tangent_and_segment_angle(2,YX,YZE,C)
Problem: As shown in the diagram, AB=11, EP=15, PA=13, ∠DCP=28°, ∠PAF=33°, P is the center of the inscribed circle of △AEC, ED⊥PD, PB is perpendicular to AB, PF is perpendicular to EF. Find the measure of ∠PED.
问题：如图所示，AB=11，EP=15，PA=13，∠DCP=28°，∠PAF=33°，三角形AEC的内切圆圆心为P，ED垂直于PD，PB⊥AB，PF⊥EF。求∠PED的大小。
Construction (CDL):
Shape(AF,FP,PA)
Shape(AP,PB,BA)
Shape(FE,EP,PF)
Shape(ED,DP,PE)
Shape(PD,DC,CP)
Shape(PC,CB,BP)
Collinear(AFE)
Collinear(EDC)
Collinear(CBA)
Collinear(DPA)
Collinear(CPF)
Collinear(EPB)
Conditions (CDL):
Equal(LengthOfLine(AB),11)
Equal(LengthOfLine(EP),15)
Equal(LengthOfLine(PA),13)
Equal(MeasureOfAngle(DCP),28)
Equal(MeasureOfAngle(PAF),33)
IsIncenterOfTriangle(P,AEC)
PerpendicularBetweenLine(ED,PD)
PerpendicularBetweenLine(PB,AB)
PerpendicularBetweenLine(PF,EF)
Goal: Value(MeasureOfAngle(PED))
Answer: 29
Theorem sequence:
angle_addition(1,AEP,PED)
angle_addition(1,BAP,PAF)
angle_addition(1,DCP,PCB)
triangle_property_angle_sum(1,AEC)
Problem: As shown in the diagram, NA=x, PN=20*sqrt(2), ∠NPA=38°, PA⊥NA. Find the value of x.
问题：如图所示，NA=x，PN=20*sqrt(2)，∠NPA=38°，PA垂直于NA。求x的值。
Construction (CDL):
Shape(NP,PA,AN)
Conditions (CDL):
Equal(LengthOfLine(NA),x)
Equal(LengthOfLine(PN),20*sqrt(2))
Equal(MeasureOfAngle(NPA),38)
PerpendicularBetweenLine(PA,NA)
Goal: Value(x)
Answer: 20*sqrt(2)*sin(19*pi/90)
Theorem sequence:
sine_theorem(1,NPA)
Problem: As shown in the diagram, AC=2*x-5, AD=5*y, CB=2*y+12, DB=3*x-18, quadrilateral ACBD is a parallelogram. Find the value of x.
问题：如图所示，AC=2*x-5，AD=5*y，CB=2*y+12，DB=3*x-18，ACBD是▱。求x的值。
Construction (CDL):
Shape(AC,CB,BD,DA)
Conditions (CDL):
Equal(LengthOfLine(AC),2*x-5)
Equal(LengthOfLine(AD),5*y)
Equal(LengthOfLine(CB),2*y+12)
Equal(LengthOfLine(DB),3*x-18)
Parallelogram(ACBD)
Goal: Value(x)
Answer: 13
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ACBD)
Problem: As shown in the diagram, ∠EAB=40°, AB⊥DB, ABDC is a rectangle. Find the measure of ∠CED.
问题：如图所示，∠EAB=40°，AB垂直于DB，ABDC是长方形。求∠CED的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BD,DE)
Shape(CE,ED,DC)
Shape(AE,EC,CA)
Collinear(AED)
Collinear(CEB)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),40)
PerpendicularBetweenLine(AB,DB)
Rectangle(ABDC)
Goal: Value(MeasureOfAngle(CED))
Answer: 100
Theorem sequence:
rectangle_property_diagonal_equal(1,ABDC)
parallelogram_property_diagonal_bisection(1,CABD,E)
parallelogram_property_diagonal_bisection(1,ABDC,E)
line_addition(1,AE,ED)
line_addition(1,BE,EC)
isosceles_triangle_judgment_line_equal(1,EAB)
isosceles_triangle_property_angle_equal(1,EAB)
triangle_property_angle_sum(1,EAB)
vertical_angle(1,BEA,CED)
Problem: As shown in the diagram, AB=y, AF=z, AL=sqrt(3), FB=x, FL=2*sqrt(3), FB is perpendicular to AB, LA⊥FA. Find the value of z.
问题：如图所示，AB=y，AF=z，AL=sqrt(3)，FB=x，FL=2*sqrt(3)，FB⊥AB，LA⊥FA。求z的值。
Construction (CDL):
Shape(FB,BA,AF)
Shape(BL,LA,AB)
Collinear(LBF)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AF),z)
Equal(LengthOfLine(AL),sqrt(3))
Equal(LengthOfLine(FB),x)
Equal(LengthOfLine(FL),2*sqrt(3))
PerpendicularBetweenLine(FB,AB)
PerpendicularBetweenLine(LA,FA)
Goal: Value(z)
Answer: 3
Theorem sequence:
right_triangle_judgment_angle(1,LAF)
right_triangle_property_pythagorean(1,LAF)
Problem: As shown in the diagram, TW=7, TY=6, XW=12, ZY=x, A is the center of circle A. Find the value of x.
问题：如图所示，TW=7，TY=6，XW=12，ZY=x，A是圆A的圆心。求x的值。
Construction (CDL):
Shape(XW,AWX)
Shape(AYW,YT,TW)
Shape(ZY,AYW,WX,AXZ)
Shape(YZ,AZY)
Collinear(XWT)
Collinear(TYZ)
Cocircular(A,YWXZ)
Conditions (CDL):
Equal(LengthOfLine(TW),7)
Equal(LengthOfLine(TY),6)
Equal(LengthOfLine(XW),12)
Equal(LengthOfLine(ZY),x)
IsCentreOfCircle(A,A)
Goal: Value(x)
Answer: 97/6
Theorem sequence:
line_addition(1,TW,WX)
line_addition(1,TY,YZ)
circle_property_circular_power_segment_and_segment_line(1,TYZ,TWX,A)
Problem: As shown in the diagram, WT=x, XY=16, YW=20, ZT=10, ZT is parallel to XY. Find the value of x.
问题：如图所示，WT=x，XY=16，YW=20，ZT=10，ZT平行于XY。求x的值。
Construction (CDL):
Shape(WZ,ZT,TW)
Shape(ZX,XY,YT,TZ)
Collinear(WZX)
Collinear(WTY)
Conditions (CDL):
Equal(LengthOfLine(WT),x)
Equal(LengthOfLine(XY),16)
Equal(LengthOfLine(YW),20)
Equal(LengthOfLine(ZT),10)
ParallelBetweenLine(ZT,XY)
Goal: Value(x)
Answer: 25/2
Theorem sequence:
parallel_property_corresponding_angle(1,ZT,XY,W)
similar_triangle_judgment_aa(1,TWZ,YWX)
similar_triangle_property_line_ratio(1,WZT,WXY)
similar_triangle_property_line_ratio(1,ZTW,XYW)
Problem: As shown in the diagram, ∠CBA=70°, CBAD is a rhombus. Find the measure of ∠BAE.
问题：如图所示，∠CBA=70°，四边形CBAD是菱形。求∠BAE的大小。
Construction (CDL):
Shape(CE,ED,DC)
Shape(CB,BE,EC)
Shape(EB,BA,AE)
Shape(EA,AD,DE)
Collinear(CEA)
Collinear(DEB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),70)
Rhombus(CBAD)
Goal: Value(MeasureOfAngle(BAE))
Answer: 55
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,CBAD)
parallelogram_property_opposite_angle_equal(1,DCBA)
quadrilateral_property_angle_sum(1,CBAD)
kite_property_diagonal_perpendicular_bisection(1,ADCB,E)
perpendicular_bisector_property_bisector(1,AE,DB)
angle_addition(1,BAE,EAD)
Problem: As shown in the diagram, AB=y, AC=x, BC=14*sqrt(3), ∠ABC=30°, BC⊥AC. Find the value of x.
问题：如图所示，AB=y，AC=x，BC=14*sqrt(3)，∠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),x)
Equal(LengthOfLine(BC),14*sqrt(3))
Equal(MeasureOfAngle(ABC),30)
PerpendicularBetweenLine(BC,AC)
Goal: Value(x)
Answer: 14
Theorem sequence:
sine_theorem(1,ABC)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, the diameter of circle C is equal to the length of line AJ, AJ=15, the measure of ⌒CKJ is 105, C is the center of ⊙C. Find the length of arc CKJ.
问题：如图所示，圆C的直径与直线AJ的长度相等，AJ=15，弧CKJ的角度为105，圆C的圆心为C。求弧CKJ的长度。
Construction (CDL):
Shape(AJ,CJA)
Shape(CA,CAK,KC)
Shape(CK,CKJ,JC)
Collinear(ACJ)
Cocircular(C,AKJ)
Conditions (CDL):
Equal(DiameterOfCircle(C),LengthOfLine(AJ))
Equal(LengthOfLine(AJ),15)
Equal(MeasureOfArc(CKJ),105)
IsCentreOfCircle(C,C)
Goal: Value(LengthOfArc(CKJ))
Answer: 35*pi/8
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,C)
arc_length_formula(1,CKJ)