SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AC=5, BC=12, the center of circle T is T, AB is the diameter of ⊙T, AC⊥BC. Find the circumference of the circle T.
问题：如图所示，AC=5，BC=12，圆T的圆心为T，圆T的直径为AB，AC垂直于BC。求圆T的周长。
Construction (CDL):
Shape(CA,TAC)
Shape(BC,TCB)
Shape(AB,TBA)
Shape(AC,CB,BA)
Collinear(ATB)
Cocircular(T,ACB)
Conditions (CDL):
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(BC),12)
IsCentreOfCircle(T,T)
IsDiameterOfCircle(AB,T)
PerpendicularBetweenLine(AC,BC)
Goal: Value(PerimeterOfCircle(T))
Answer: 13*pi
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
diameter_of_circle_property_length_equal(1,AB,T)
circle_property_length_of_radius_and_diameter(1,T)
circle_perimeter_formula(1,T)
Problem: As shown in the diagram, XY=4, the measure of ⌒XYZ is 75, the center of circle X is X. Find the length of arc XYZ.
问题：如图所示，XY=4，弧XYZ的角度为75，X是⊙X的圆心。求弧XYZ的长度。
Construction (CDL):
Shape(ZX,XY,XYZ)
Shape(XZ,XZY,YX)
Cocircular(X,ZY)
Conditions (CDL):
Equal(LengthOfLine(XY),4)
Equal(MeasureOfArc(XYZ),75)
IsCentreOfCircle(X,X)
Goal: Value(LengthOfArc(XYZ))
Answer: 5*pi/3
Theorem sequence:
radius_of_circle_property_length_equal(1,XY,X)
arc_length_formula(1,XYZ)
Problem: As shown in the diagram, JK=JL, JK=KL, JK=x+7, JL=4*x-8, LK=JL. Find the value of x.
问题：如图所示，JK=JL，JK=KL，JK=x+7，JL=4*x-8，LK=JL。求x的值。
Construction (CDL):
Shape(KJ,JL,LK)
Conditions (CDL):
Equal(LengthOfLine(JK),LengthOfLine(JL))
Equal(LengthOfLine(JK),LengthOfLine(KL))
Equal(LengthOfLine(JK),x+7)
Equal(LengthOfLine(JL),4*x-8)
Equal(LengthOfLine(LK),LengthOfLine(JL))
Goal: Value(x)
Answer: 5
Theorem sequence:
Problem: As shown in the diagram, CA=11, DA=13, CDOB is a rhombus. Find the area of quadrilateral CDOB.
问题：如图所示，CA=11，DA=13，四边形CDOB是菱形。求四边形CDOB的面积。
Construction (CDL):
Shape(CD,DA,AC)
Shape(CA,AB,BC)
Shape(DO,OA,AD)
Shape(AO,OB,BA)
Collinear(CAO)
Collinear(DAB)
Conditions (CDL):
Equal(LengthOfLine(CA),11)
Equal(LengthOfLine(DA),13)
Rhombus(CDOB)
Goal: Value(AreaOfQuadrilateral(CDOB))
Answer: 286
Theorem sequence:
parallelogram_property_diagonal_bisection(1,CDOB,A)
parallelogram_property_diagonal_bisection(1,DOBC,A)
line_addition(1,CA,AO)
line_addition(1,DA,AB)
kite_area_formula_diagonal(1,CDOB)
Problem: As shown in the diagram, AE=10, NB=7, NC=12.2, AE is the height of ABNC, ABNC is a ▱. Find the area of quadrilateral ABNC.
问题：如图所示，AE=10，NB=7，NC=12.2，ABNC的高为AE，四边形ABNC是▱。求ABNC的面积。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EN,NC,CA)
Collinear(BEN)
Conditions (CDL):
Equal(LengthOfLine(AE),10)
Equal(LengthOfLine(NB),7)
Equal(LengthOfLine(NC),12.2)
IsAltitudeOfQuadrilateral(AE,ABNC)
Parallelogram(ABNC)
Goal: Value(AreaOfQuadrilateral(ABNC))
Answer: 70
Theorem sequence:
parallelogram_area_formula_common(1,ABNC)
Problem: As shown in the diagram, AC=x+11, DB=2*x+9, ∠CAD=3*y+19°, ∠DBC=106°, quadrilateral BCAD is a parallelogram. Find the value of x.
问题：如图所示，AC=x+11，DB=2*x+9，∠CAD=3*y+19°，∠DBC=106°，BCAD是▱。求x的值。
Construction (CDL):
Shape(BC,CA,AD,DB)
Conditions (CDL):
Equal(LengthOfLine(AC),x+11)
Equal(LengthOfLine(DB),2*x+9)
Equal(MeasureOfAngle(CAD),3*y+19)
Equal(MeasureOfAngle(DBC),106)
Parallelogram(BCAD)
Goal: Value(x)
Answer: 2
Theorem sequence:
parallelogram_property_opposite_line_equal(1,CADB)
Problem: As shown in the diagram, ∠ABC=24°. Find the measure of ⌒DCA.
问题：如图所示，∠ABC=24°。求弧DCA的角度。
Construction (CDL):
Shape(AB,BC,DCA)
Shape(BA,DAB)
Shape(CB,DBC)
Cocircular(D,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),24)
Goal: Value(MeasureOfArc(DCA))
Answer: 48
Theorem sequence:
arc_property_circumference_angle_external(1,DCA,B)
Problem: As shown in the diagram, DC=GC, FD=FG, ∠DCG=67°, ∠GFD=49°, CG and CD are one pair of adjacent sides of the kite CGFD. Find the measure of ∠FDC.
问题：如图所示，DC=GC，FD=FG，∠DCG=67°，∠GFD=49°，四边形CGFD是风筝形。求∠FDC的大小。
Construction (CDL):
Shape(DC,CG,GF,FD)
Conditions (CDL):
Equal(LengthOfLine(DC),LengthOfLine(GC))
Equal(LengthOfLine(FD),LengthOfLine(FG))
Equal(MeasureOfAngle(DCG),67)
Equal(MeasureOfAngle(GFD),49)
Kite(CGFD)
Goal: Value(MeasureOfAngle(FDC))
Answer: 122
Theorem sequence:
kite_property_opposite_angle_equal(1,CGFD)
quadrilateral_property_angle_sum(1,DCGF)
Problem: As shown in the diagram, RS=3*x+2, SR=RT, TR=2*x+4. Find the length of line RT.
问题：如图所示，RS=3*x+2，SR=RT，TR=2*x+4。求直线RT的长度。
Construction (CDL):
Shape(RT,TS,SR)
Conditions (CDL):
Equal(LengthOfLine(RS),3*x+2)
Equal(LengthOfLine(SR),LengthOfLine(RT))
Equal(LengthOfLine(TR),2*x+4)
Goal: Value(LengthOfLine(RT))
Answer: 8
Theorem sequence:
Problem: As shown in the diagram, ML=w, NM=2*y+5, NQ=3*x+2, QL=12, RM=4*x-2, RQ=3*y, quadrilateral NMRQ is a ▱. Find the value of x.
问题：如图所示，ML=w，NM=2*y+5，NQ=3*x+2，QL=12，RM=4*x-2，RQ=3*y，NQ和MR是平行四边形NMRQ的一组对边。求x的值。
Construction (CDL):
Shape(NL,LQ,QN)
Shape(LN,NM,ML)
Shape(LM,MR,RL)
Shape(LR,RQ,QL)
Collinear(NLR)
Collinear(MLQ)
Conditions (CDL):
Equal(LengthOfLine(ML),w)
Equal(LengthOfLine(NM),2*y+5)
Equal(LengthOfLine(NQ),3*x+2)
Equal(LengthOfLine(QL),12)
Equal(LengthOfLine(RM),4*x-2)
Equal(LengthOfLine(RQ),3*y)
Parallelogram(NMRQ)
Goal: Value(x)
Answer: 4
Theorem sequence:
parallelogram_property_opposite_line_equal(1,MRQN)
Problem: As shown in the diagram, ∠ADB=2*x+2*y°, ∠BCA=4*y+x°, ∠DBC=x+y°, BD and CA are opposite sides of the ▱ BCAD. Find the value of x.
问题：如图所示，∠ADB=2*x+2*y°，∠BCA=4*y+x°，∠DBC=x+y°，BD和CA是▱BCAD的一组对边。求x的值。
Construction (CDL):
Shape(DB,BC,CA,AD)
Conditions (CDL):
Equal(MeasureOfAngle(ADB),2*x+2*y)
Equal(MeasureOfAngle(BCA),4*y+x)
Equal(MeasureOfAngle(DBC),x+y)
Parallelogram(BCAD)
Goal: Value(x)
Answer: 40
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,CADB)
parallel_property_ipsilateral_internal_angle(1,BD,CA)
Problem: As shown in the diagram, AB=12, AC=15, BD=3*x-2, CF=3*x+2, BC∥DF. Find the value of x.
问题：如图所示，AB=12，AC=15，BD=3*x-2，CF=3*x+2，BC∥DF。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DF,FC,CB)
Collinear(ABD)
Collinear(ACF)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(AC),15)
Equal(LengthOfLine(BD),3*x-2)
Equal(LengthOfLine(CF),3*x+2)
ParallelBetweenLine(BC,DF)
Goal: Value(x)
Answer: 6
Theorem sequence:
parallel_property_corresponding_angle(1,BC,DF,A)
similar_triangle_judgment_aa(1,CAB,FAD)
line_addition(1,AB,BD)
line_addition(1,AC,CF)
similar_triangle_property_line_ratio(1,CAB,FAD)
similar_triangle_property_line_ratio(1,BCA,DFA)
Problem: As shown in the diagram, the area of △ABC is 40, the area of △DEF is 30, AC=5, DF=x, triangle ABC is similar to triangle DEF.. Find the value of x.
问题：如图所示，三角形ABC的面积为40，三角形DEF的面积为30，AC=5，DF=x，三角形ABC与三角形DEF是相似三角形。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(AreaOfTriangle(ABC),40)
Equal(AreaOfTriangle(DEF),30)
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(DF),x)
SimilarBetweenTriangle(ABC,DEF)
Goal: Value(x)
Answer: 5*sqrt(3)/2
Theorem sequence:
similar_triangle_property_line_ratio(1,BCA,EFD)
similar_triangle_property_area_square_ratio(1,BCA,EFD)
Problem: As shown in the diagram, AD=y, CA=x, CB=24, ∠CAB=32°, AD is perpendicular to CD, BC⊥AC. Find the value of y.
问题：如图所示，AD=y，CA=x，CB=24，∠CAB=32°，AD垂直于CD，BC垂直于AC。求y的值。
Construction (CDL):
Shape(AD,DC,CA)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AD),y)
Equal(LengthOfLine(CA),x)
Equal(LengthOfLine(CB),24)
Equal(MeasureOfAngle(CAB),32)
PerpendicularBetweenLine(AD,CD)
PerpendicularBetweenLine(BC,AC)
Goal: Value(y)
Answer: 24*sin(29*pi/90)**2/sin(8*pi/45)
Theorem sequence:
adjacent_complementary_angle(1,ADC,CDB)
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,ABC)
sine_theorem(1,CDB)
sine_theorem(1,DCA)
Problem: As shown in the diagram, AC=HeightOfTriangle(ACE), BF=5, BF=HeightOfTriangle(BFD), BF=DF, DB=5*sqrt(2), EA=HeightOfQuadrilateral(EADB), EB=10, EC=5, EC=AC, AC is perpendicular to EC, AD is perpendicular to BD, BE is perpendicular to AE, BF⊥DF, DB⊥EB, EA⊥DA. Find the area of quadrilateral EADB minus the areas of triangle BFD and triangle EAC.
问题：如图所示，AC=HeightOfTriangle(ACE)，BF=5，BF=HeightOfTriangle(BFD)，BF=DF，DB=5*sqrt(2)，EA=HeightOfQuadrilateral(EADB)，EB=10，EC=5，EC=AC，AC垂直于EC，AD垂直于BD，BE垂直于AE，BF⊥DF，DB垂直于EB，EA⊥DA。求四边形EADB面积减去三角形BFD面积和三角形EAC面积。
Construction (CDL):
Shape(AC,CE,EA)
Shape(BF,FD,DB)
Shape(EC,CA,AD,DF,FB,BE)
Conditions (CDL):
Equal(LengthOfLine(AC),HeightOfTriangle(ACE))
Equal(LengthOfLine(BF),5)
Equal(LengthOfLine(BF),HeightOfTriangle(BFD))
Equal(LengthOfLine(BF),LengthOfLine(DF))
Equal(LengthOfLine(DB),5*sqrt(2))
Equal(LengthOfLine(EA),HeightOfQuadrilateral(EADB))
Equal(LengthOfLine(EB),10)
Equal(LengthOfLine(EC),5)
Equal(LengthOfLine(EC),LengthOfLine(AC))
PerpendicularBetweenLine(AC,EC)
PerpendicularBetweenLine(AD,BD)
PerpendicularBetweenLine(BE,AE)
PerpendicularBetweenLine(BF,DF)
PerpendicularBetweenLine(DB,EB)
PerpendicularBetweenLine(EA,DA)
Goal: Value(Sub(AreaOfQuadrilateral(EADB),Add(AreaOfTriangle(BFD),AreaOfTriangle(EAC))))
Answer: -25+50*sqrt(2)
Theorem sequence:
parallel_judgment_per_per(1,EB,AD)
parallel_judgment_per_per(1,AE,DB)
parallelogram_judgment_parallel_and_parallel(1,EADB)
parallelogram_property_opposite_line_equal(1,EADB)
parallelogram_property_opposite_line_equal(1,ADBE)
triangle_area_formula_common(1,ACE)
triangle_area_formula_common(1,BFD)
parallelogram_area_formula_sine(1,EADB)
Problem: As shown in the diagram, ∠CAG=38°, ABDC is a rectangle. Find the measure of ∠DGB.
问题：如图所示，∠CAG=38°，ABDC是矩形。求∠DGB的大小。
Construction (CDL):
Shape(BG,GA,AB)
Shape(GB,BD,DG)
Shape(GD,DC,CG)
Shape(AG,GC,CA)
Collinear(AGD)
Collinear(BGC)
Conditions (CDL):
Equal(MeasureOfAngle(CAG),38)
Rectangle(ABDC)
Goal: Value(MeasureOfAngle(DGB))
Answer: 104
Theorem sequence:
rectangle_property_diagonal_equal(1,ABDC)
parallelogram_property_diagonal_bisection(1,ABDC,G)
parallelogram_property_diagonal_bisection(1,BDCA,G)
line_addition(1,AG,GD)
line_addition(1,BG,GC)
isosceles_triangle_judgment_line_equal(1,GAB)
isosceles_triangle_judgment_line_equal(1,GDC)
isosceles_triangle_judgment_line_equal(1,GCA)
isosceles_triangle_property_angle_equal(1,GAB)
isosceles_triangle_property_angle_equal(1,GDC)
isosceles_triangle_property_angle_equal(1,GCA)
triangle_property_angle_sum(1,GBD)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,ABD)
triangle_property_angle_sum(1,BDC)
triangle_property_angle_sum(1,DCA)
Problem: As shown in the diagram, ∠GAH=28°, the center of circle B is B. Find the measure of ⌒BHG.
问题：如图所示，∠GAH=28°，圆B的圆心为B。求弧BHG的角度。
Construction (CDL):
Shape(AG,BGA)
Shape(HA,BAH)
Shape(GA,AH,BHG)
Cocircular(B,AHG)
Conditions (CDL):
Equal(MeasureOfAngle(GAH),28)
IsCentreOfCircle(B,B)
Goal: Value(MeasureOfArc(BHG))
Answer: 56
Theorem sequence:
arc_property_circumference_angle_external(1,BHG,A)
Problem: As shown in the diagram, JK=3*y-2, LK=5*y-8, ∠PMJ=3*x-6°, JM is the altitude of △ JPL. Find the value of x.
问题：如图所示，JK=3*y-2，LK=5*y-8，∠PMJ=3*x-6°，JM是三角形JPL的高。求x的值。
Construction (CDL):
Shape(LK,KA,AM,ML)
Shape(KJ,JA,AK)
Shape(AJ,JP,PA)
Shape(MA,AP,PM)
Collinear(JKL)
Collinear(JAM)
Collinear(KAP)
Collinear(LMP)
Conditions (CDL):
Equal(LengthOfLine(JK),3*y-2)
Equal(LengthOfLine(LK),5*y-8)
Equal(MeasureOfAngle(PMJ),3*x-6)
IsAltitudeOfTriangle(JM,JPL)
Goal: Value(x)
Answer: 32
Theorem sequence:
Problem: As shown in the diagram, BC=4*x-17, DF=2*x-1, ∠CBF=3*y+5°, ∠DFB=5*y-13°, DB and FC are opposite sides of the ▱ BDFC. Find the value of x.
问题：如图所示，BC=4*x-17，DF=2*x-1，∠CBF=3*y+5°，∠DFB=5*y-13°，BDFC是平行四边形。求x的值。
Construction (CDL):
Shape(BF,FC,CB)
Shape(BD,DF,FB)
Conditions (CDL):
Equal(LengthOfLine(BC),4*x-17)
Equal(LengthOfLine(DF),2*x-1)
Equal(MeasureOfAngle(CBF),3*y+5)
Equal(MeasureOfAngle(DFB),5*y-13)
Parallelogram(BDFC)
Goal: Value(x)
Answer: 8
Theorem sequence:
parallelogram_property_opposite_line_equal(1,DFCB)
Problem: As shown in the diagram, AB=25, AC=26, BC=14, ∠BCA=x°. Find the value of x.
问题：如图所示，AB=25，AC=26，BC=14，∠BCA=x°。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),25)
Equal(LengthOfLine(AC),26)
Equal(LengthOfLine(BC),14)
Equal(MeasureOfAngle(BCA),x)
Goal: Value(x)
Answer: 180*acos(19/56)/pi
Theorem sequence:
cosine_theorem(1,CAB)
Problem: As shown in the diagram, ZJ=12, ZP=20, ∠YXP=31°, ∠ZYP=27°, the center of the inscribed circle of △XZY is P, PJ is perpendicular to YJ, PL is perpendicular to ZL, YK⊥PK. Find the measure of ∠LZP.
问题：如图所示，ZJ=12，ZP=20，∠YXP=31°，∠ZYP=27°，P是三角形XZY的内心，PJ⊥YJ，PL⊥ZL，YK垂直于PK。求∠LZP的大小。
Construction (CDL):
Shape(XL,LP,PX)
Shape(PL,LZ,ZP)
Shape(PZ,ZJ,JP)
Shape(PJ,JY,YP)
Shape(PY,YK,KP)
Shape(PK,KX,XP)
Collinear(XLZ)
Collinear(ZJY)
Collinear(XKY)
Collinear(LPY)
Collinear(ZPK)
Collinear(XPJ)
Conditions (CDL):
Equal(LengthOfLine(ZJ),12)
Equal(LengthOfLine(ZP),20)
Equal(MeasureOfAngle(YXP),31)
Equal(MeasureOfAngle(ZYP),27)
IsIncenterOfTriangle(P,XZY)
PerpendicularBetweenLine(PJ,YJ)
PerpendicularBetweenLine(PL,ZL)
PerpendicularBetweenLine(YK,PK)
Goal: Value(MeasureOfAngle(LZP))
Answer: 63/2
Theorem sequence:
triangle_property_angle_sum(1,ZYL)
angle_addition(1,LZP,PZJ)
Problem: As shown in the diagram, PQ=721, PS=756, QR=547, RS=593, ∠QPS=58°. Find the measure of ∠SRQ.
问题：如图所示，PQ=721，PS=756，QR=547，RS=593，∠QPS=58°。求∠SRQ的大小。
Construction (CDL):
Shape(PS,SQ,QP)
Shape(QS,SR,RQ)
Conditions (CDL):
Equal(LengthOfLine(PQ),721)
Equal(LengthOfLine(PS),756)
Equal(LengthOfLine(QR),547)
Equal(LengthOfLine(RS),593)
Equal(MeasureOfAngle(QPS),58)
Goal: Value(MeasureOfAngle(SRQ))
Answer: 180*acos(-440519/648742+545076*cos(29*pi/90)/324371)/pi
Theorem sequence:
cosine_theorem(1,PSQ)
cosine_theorem(1,RQS)
Problem: As shown in the diagram, JP=9, KR=RJ, KS=SL, LT=TJ, RP=3.5. Find the length of line PS.
问题：如图所示，JP=9，KR=RJ，KS=SL，LT=TJ，RP=3.5。求直线PS的长度。
Construction (CDL):
Shape(PK,KR,RP)
Shape(PR,RJ,JP)
Shape(PJ,JT,TP)
Shape(PT,TL,LP)
Shape(PL,LS,SP)
Shape(PS,SK,KP)
Collinear(JRK)
Collinear(JPS)
Collinear(JTL)
Collinear(KSL)
Collinear(KPT)
Collinear(LPR)
Conditions (CDL):
Equal(LengthOfLine(JP),9)
Equal(LengthOfLine(KR),LengthOfLine(RJ))
Equal(LengthOfLine(KS),LengthOfLine(SL))
Equal(LengthOfLine(LT),LengthOfLine(TJ))
Equal(LengthOfLine(RP),3.5)
Goal: Value(LengthOfLine(PS))
Answer: 9/2
Theorem sequence:
median_of_triangle_judgment(1,LR,LKJ)
median_of_triangle_judgment(1,KT,KJL)
centroid_of_triangle_judgment_intersection(1,P,JLK,T,R)
centroid_of_triangle_property_line_ratio(1,P,JLK,S)
Problem: As shown in the diagram, AB=h, BC=8, ∠ACB=45°, BA⊥CA. Find the value of h.
问题：如图所示，AB=h，BC=8，∠ACB=45°，BA垂直于CA。求h的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),h)
Equal(LengthOfLine(BC),8)
Equal(MeasureOfAngle(ACB),45)
PerpendicularBetweenLine(BA,CA)
Goal: Value(h)
Answer: 4*sqrt(2)
Theorem sequence:
sine_theorem(1,BAC)
Problem: As shown in the diagram, AB=8, ∠BAC=121°, A is the center of ⊙A. Find the area of the sector ACB.
问题：如图所示，AB=8，∠BAC=121°，圆A的圆心为A。求扇形ACB的面积。
Construction (CDL):
Shape(AB,ABC,CA)
Shape(AC,ACB,BA)
Cocircular(A,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(MeasureOfAngle(BAC),121)
IsCentreOfCircle(A,A)
Goal: Value(AreaOfSector(ACB))
Answer: 968*pi/45
Theorem sequence:
radius_of_circle_property_length_equal(1,AB,A)
arc_property_center_angle(1,ACB,A)
sector_area_formula(1,ACB)
Problem: As shown in the diagram, AB=x, AC=8, BC=y, ∠CBA=30°, BA⊥CA. Find the value of y.
问题：如图所示，AB=x，AC=8，BC=y，∠CBA=30°，BA⊥CA。求y的值。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(BC),y)
Equal(MeasureOfAngle(CBA),30)
PerpendicularBetweenLine(BA,CA)
Goal: Value(y)
Answer: 16
Theorem sequence:
sine_theorem(1,CBA)
Problem: As shown in the diagram, AX=XB, DY=YC, ∠ADY=78°, quadrilateral ADCB is a isosceles trapezoid. Find the measure of ∠CBX.
问题：如图所示，AX=XB，DY=YC，∠ADY=78°，AD和CB是等腰梯形ADCB的腰。求∠CBX的大小。
Construction (CDL):
Shape(BX,XY,YC,CB)
Shape(XA,AD,DY,YX)
Collinear(BXA)
Collinear(CYD)
Conditions (CDL):
Equal(LengthOfLine(AX),LengthOfLine(XB))
Equal(LengthOfLine(DY),LengthOfLine(YC))
Equal(MeasureOfAngle(ADY),78)
IsoscelesTrapezoid(ADCB)
Goal: Value(MeasureOfAngle(CBX))
Answer: 102
Theorem sequence:
isosceles_trapezoid_property_angle_equal(1,ADCB)
isosceles_trapezoid_property_angle_equal(1,CBAD)
quadrilateral_property_angle_sum(1,CBAD)
Problem: As shown in the diagram, EF=6, FL=7, GJ=11, HE=4, JH=12, LG=6, x=Add(LengthOfLine(LG). Find the value of x.
问题：如图所示，EF=6，FL=7，GJ=11，HE=4，JH=12，LG=6，x=Add(LengthOfLine(LG)。求x的值。
Construction (CDL):
Shape(LG,GJ,JH,HE,EF,FL)
Conditions (CDL):
Equal(LengthOfLine(EF),6)
Equal(LengthOfLine(FL),7)
Equal(LengthOfLine(GJ),11)
Equal(LengthOfLine(HE),4)
Equal(LengthOfLine(JH),12)
Equal(LengthOfLine(LG),6)
Equal(x,Add(LengthOfLine(LG),LengthOfLine(GJ),LengthOfLine(JH),LengthOfLine(HE),LengthOfLine(EF),LengthOfLine(LF)))
Goal: Value(x)
Answer: 46
Theorem sequence:
Problem: As shown in the diagram, the measure of ⌒AJH is 114, A is the center of ⊙A, GB is perpendicular to HB, JH is perpendicular to GH. Find the measure of ∠FHG.
问题：如图所示，弧AJH的角度为114，圆A的圆心为A，GB垂直于HB，JH垂直于GH。求∠FHG的大小。
Construction (CDL):
Shape(AGF,FB,BG)
Shape(AFJ,JB,BF)
Shape(AJH,HJ)
Shape(BA,AH,HB)
Shape(BH,HG,GB)
Shape(AHG,GH)
Shape(HA,AJ,JH)
Collinear(GBAJ)
Collinear(FBH)
Cocircular(A,GFJH)
Conditions (CDL):
Equal(MeasureOfArc(A,JH),114)
IsCentreOfCircle(A,A)
PerpendicularBetweenLine(GB,HB)
PerpendicularBetweenLine(JH,GH)
Goal: Value(MeasureOfAngle(FHG))
Answer: 33
Theorem sequence:
arc_property_center_angle(1,AJH,A)
mirror_similar_triangle_judgment_aa(1,HGB,JHG)
mirror_similar_triangle_property_angle_equal(1,HGB,JHG)
radius_of_circle_property_length_equal(1,AJ,A)
radius_of_circle_property_length_equal(1,AH,A)
isosceles_triangle_judgment_line_equal(1,AJH)
isosceles_triangle_property_angle_equal(1,AJH)
triangle_property_angle_sum(1,AJH)
Problem: As shown in the diagram, AB=10/3*x+4, AC=5*x-1, BC=7*x-7, ∠ABC=∠CAB, ∠BCA=∠ABC. Find the value of x.
问题：如图所示，AB=10/3*x+4，AC=5*x-1，BC=7*x-7，∠ABC=∠CAB，∠BCA=∠ABC。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),10/3*x+4)
Equal(LengthOfLine(AC),5*x-1)
Equal(LengthOfLine(BC),7*x-7)
Equal(MeasureOfAngle(ABC),MeasureOfAngle(CAB))
Equal(MeasureOfAngle(BCA),MeasureOfAngle(ABC))
Goal: Value(x)
Answer: 3
Theorem sequence:
isosceles_triangle_judgment_angle_equal(1,ABC)
Problem: As shown in the diagram, ∠DCA=32°, ∠DGA=47°, ∠FDG=70°. Find the measure of ∠CAF.
问题：如图所示，∠DCA=32°，∠DGA=47°，∠FDG=70°。求∠CAF的大小。
Construction (CDL):
Shape(DG,GF,FD)
Shape(FC,CA,AF)
Collinear(DFC)
Collinear(GFA)
Conditions (CDL):
Equal(MeasureOfAngle(DCA),32)
Equal(MeasureOfAngle(DGA),47)
Equal(MeasureOfAngle(FDG),70)
Goal: Value(MeasureOfAngle(CAF))
Answer: 85
Theorem sequence:
triangle_property_angle_sum(1,GFD)
vertical_angle(1,GFD,AFC)
triangle_property_angle_sum(1,AFC)
Problem: As shown in the diagram, the diameter of circle A is 10, ∠BAD=80°, ∠CAL=64°, ∠DAC=112°, ∠EAB=64°, ∠LAE=40°, the center of circle A is A. Find the area of the sector ADB.
问题：如图所示，圆A的直径为10，∠BAD=80°，∠CAL=64°，∠DAC=112°，∠EAB=64°，∠LAE=40°，A是⊙A的圆心。求扇形ADB的面积。
Construction (CDL):
Shape(ACD,DA,AC)
Shape(ADB,BA,AD)
Shape(ABE,EA,AB)
Shape(AEL,LA,AE)
Shape(ALC,CA,AL)
Cocircular(A,CDBEL)
Conditions (CDL):
Equal(DiameterOfCircle(A),10)
Equal(MeasureOfAngle(BAD),80)
Equal(MeasureOfAngle(CAL),64)
Equal(MeasureOfAngle(DAC),112)
Equal(MeasureOfAngle(EAB),64)
Equal(MeasureOfAngle(LAE),40)
IsCentreOfCircle(A,A)
Goal: Value(AreaOfSector(ADB))
Answer: 50*pi/9
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,A)
arc_property_center_angle(1,ADB,A)
sector_area_formula(1,ADB)
Problem: As shown in the diagram, XY=6, YZ=3, WX is perpendicular to YX, XYZW is a rectangle. Find the area of quadrilateral XYZW.
问题：如图所示，XY=6，YZ=3，WX垂直于YX，四边形XYZW是长方形。求四边形XYZW的面积。
Construction (CDL):
Shape(WX,XY,YZ,ZW)
Conditions (CDL):
Equal(LengthOfLine(XY),6)
Equal(LengthOfLine(YZ),3)
PerpendicularBetweenLine(WX,YX)
Rectangle(XYZW)
Goal: Value(AreaOfQuadrilateral(XYZW))
Answer: 18
Theorem sequence:
parallelogram_area_formula_sine(1,XYZW)
Problem: As shown in the diagram, AC=2/3*x-4, AC=CE, CE=1/3*x+2, DB=7/3*y+8, EB=5*y, ∠CBE=∠ADB. Find the value of x.
问题：如图所示，AC=2/3*x-4，AC=CE，CE=1/3*x+2，DB=7/3*y+8，EB=5*y，∠CBE=∠ADB。求x的值。
Construction (CDL):
Shape(EC,CB,BE)
Shape(CA,AD,DB,BC)
Collinear(ECA)
Collinear(EBD)
Conditions (CDL):
Equal(LengthOfLine(AC),2/3*x-4)
Equal(LengthOfLine(AC),LengthOfLine(CE))
Equal(LengthOfLine(CE),1/3*x+2)
Equal(LengthOfLine(DB),7/3*y+8)
Equal(LengthOfLine(EB),5*y)
Equal(MeasureOfAngle(CBE),MeasureOfAngle(ADB))
Goal: Value(x)
Answer: 18
Theorem sequence:
Problem: As shown in the diagram, ∠KLM=57°, ∠LKJ=28°, ∠MKL=71°. Find the measure of ∠JLK.
问题：如图所示，∠KLM=57°，∠LKJ=28°，∠MKL=71°。求∠JLK的大小。
Construction (CDL):
Shape(KJ,JL,LK)
Shape(KL,LM,MK)
Collinear(JLM)
Conditions (CDL):
Equal(MeasureOfAngle(KLM),57)
Equal(MeasureOfAngle(LKJ),28)
Equal(MeasureOfAngle(MKL),71)
Goal: Value(MeasureOfAngle(JLK))
Answer: 123
Theorem sequence:
adjacent_complementary_angle(1,JLK,KLM)
Problem: As shown in the diagram, YX=4, ZX=4, ∠XYZ=60°, ∠YZX=60°. Find the length of line YZ.
问题：如图所示，YX=4，ZX=4，∠XYZ=60°，∠YZX=60°。求直线YZ的长度。
Construction (CDL):
Shape(XY,YZ,ZX)
Conditions (CDL):
Equal(LengthOfLine(YX),4)
Equal(LengthOfLine(ZX),4)
Equal(MeasureOfAngle(XYZ),60)
Equal(MeasureOfAngle(YZX),60)
Goal: Value(LengthOfLine(YZ))
Answer: 4
Theorem sequence:
triangle_property_angle_sum(1,XYZ)
isosceles_triangle_judgment_angle_equal(1,YZX)
Problem: As shown in the diagram, the area of quadrilateral RQTS is 72, RT=12, RQTS is a rhombus. Find the length of line QS.
问题：如图所示，RQTS的面积为72，RT=12，RQTS是菱形。求直线QS的长度。
Construction (CDL):
Shape(RQ,QB,BR)
Shape(BQ,QT,TB)
Shape(BT,TS,SB)
Shape(RB,BS,SR)
Collinear(RBT)
Collinear(QBS)
Conditions (CDL):
Equal(AreaOfQuadrilateral(RQTS),72)
Equal(LengthOfLine(RT),12)
Rhombus(RQTS)
Goal: Value(LengthOfLine(QS))
Answer: 12
Theorem sequence:
kite_area_formula_diagonal(1,RQTS)
Problem: As shown in the diagram, ∠DEG=x°, ∠DHG=20°, ∠GHF=150°, H is the center of ⊙H. Find the value of x.
问题：如图所示，∠DEG=x°，∠DHG=20°，∠GHF=150°，圆H的圆心为H。求x的值。
Construction (CDL):
Shape(HIF,FH,HI)
Shape(IE,EF,HIF)
Shape(HF,FG,GH)
Shape(HFG,GF)
Shape(DH,HG,GD)
Shape(IH,HD,HDI)
Collinear(DHIE)
Collinear(GFE)
Cocircular(H,DIFG)
Conditions (CDL):
Equal(MeasureOfAngle(DEG),x)
Equal(MeasureOfAngle(DHG),20)
Equal(MeasureOfAngle(GHF),150)
IsCentreOfCircle(H,H)
Goal: Value(x)
Answer: 5
Theorem sequence:
arc_property_center_angle(1,HGD,H)
flat_angle(1,DHI)
angle_addition(1,DHG,GHF)
angle_addition(1,DHF,FHI)
arc_property_center_angle(1,HIF,H)
circle_property_circular_power_segment_and_segment_angle(1,EID,EFG,H)
Problem: As shown in the diagram, ∠BCD=3*x+25°, ∠CDB=4*x+5°, ∠DBC=5*x-30°. Find the value of x.
问题：如图所示，∠BCD=3*x+25°，∠CDB=4*x+5°，∠DBC=5*x-30°。求x的值。
Construction (CDL):
Shape(DB,BC,CD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),3*x+25)
Equal(MeasureOfAngle(CDB),4*x+5)
Equal(MeasureOfAngle(DBC),5*x-30)
Goal: Value(x)
Answer: 15
Theorem sequence:
triangle_property_angle_sum(1,CDB)
Problem: As shown in the diagram, AC=6, CB=x, ∠ABC=17°, CA⊥BA. Find the value of x.
问题：如图所示，AC=6，CB=x，∠ABC=17°，CA⊥BA。求x的值。
Construction (CDL):
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(CB),x)
Equal(MeasureOfAngle(ABC),17)
PerpendicularBetweenLine(CA,BA)
Goal: Value(x)
Answer: 6/sin(17*pi/180)
Theorem sequence:
sine_theorem(1,CAB)
Problem: As shown in the diagram, AD=8, AE=12, EC=18, BC is parallel to DE. Find the length of line AB.
问题：如图所示，AD=8，AE=12，EC=18，BC平行于DE。求直线AB的长度。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EC,CB,BD)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),8)
Equal(LengthOfLine(AE),12)
Equal(LengthOfLine(EC),18)
ParallelBetweenLine(BC,DE)
Goal: Value(LengthOfLine(AB))
Answer: 20
Theorem sequence:
parallel_property_corresponding_angle(1,ED,CB,A)
similar_triangle_judgment_aa(1,DAE,BAC)
line_addition(1,AE,EC)
similar_triangle_property_line_ratio(1,DAE,BAC)
similar_triangle_property_line_ratio(1,EDA,CBA)
Problem: As shown in the diagram, ∠TSR=42°, the center of ⊙S is S. Find the measure of arc STK.
问题：如图所示，∠TSR=42°，⊙S的圆心为S。求⌒STK的角度。
Construction (CDL):
Shape(STK,KS,ST)
Shape(SKE,ES,SK)
Shape(SER,RS,SE)
Shape(SRT,TS,SR)
Collinear(RSK)
Collinear(TSE)
Cocircular(S,RTKE)
Conditions (CDL):
Equal(MeasureOfAngle(TSR),42)
IsCentreOfCircle(S,S)
Goal: Value(MeasureOfArc(STK))
Answer: 138
Theorem sequence:
flat_angle(1,KSR)
angle_addition(1,KST,TSR)
arc_property_center_angle(1,STK,S)