SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=20, AC=16, BD=17, CD=x, ∠DAC=∠BAD. Find the value of x.
问题：如图所示，AB=20，AC=16，BD=17，CD=x，∠DAC=∠BAD。求x的值。
Construction (CDL):
Shape(CD,DA,AC)
Shape(AD,DB,BA)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(AB),20)
Equal(LengthOfLine(AC),16)
Equal(LengthOfLine(BD),17)
Equal(LengthOfLine(CD),x)
Equal(MeasureOfAngle(DAC),MeasureOfAngle(BAD))
Goal: Value(x)
Answer: 68/5
Theorem sequence:
bisector_of_angle_judgment_angle_equal(1,AD,BAC)
bisector_of_angle_property_line_ratio(1,AD,BAC)
Problem: As shown in the diagram, ∠BGC=40°, ∠DGF=53°, CB⊥GB, FG⊥CG, GF is perpendicular to DF. Find the measure of ∠BAF.
问题：如图所示，∠BGC=40°，∠DGF=53°，CB⊥GB，FG垂直于CG，GF⊥DF。求∠BAF的大小。
Construction (CDL):
Shape(DG,GF,FD)
Shape(FG,GB,BA,AF)
Shape(BG,GC,CB)
Shape(AB,BC,CA)
Collinear(DFA)
Collinear(GBA)
Conditions (CDL):
Equal(MeasureOfAngle(BGC),40)
Equal(MeasureOfAngle(DGF),53)
PerpendicularBetweenLine(CB,GB)
PerpendicularBetweenLine(FG,CG)
PerpendicularBetweenLine(GF,DF)
Goal: Value(MeasureOfAngle(BAF))
Answer: 40
Theorem sequence:
angle_addition(1,FGB,BGC)
adjacent_complementary_angle(1,AFG,GFD)
triangle_property_angle_sum(1,FGA)
Problem: As shown in the diagram, GD=2*x-2, HD=2*x+4, HG=7, JH=10, ∠GDH=∠JHG. Find the length of line DH.
问题：如图所示，GD=2*x-2，HD=2*x+4，HG=7，JH=10，∠GDH=∠JHG。求直线DH的长度。
Construction (CDL):
Shape(GD,DH,HG)
Shape(HD,DJ,JH)
Collinear(GDJ)
Conditions (CDL):
Equal(LengthOfLine(GD),2*x-2)
Equal(LengthOfLine(HD),2*x+4)
Equal(LengthOfLine(HG),7)
Equal(LengthOfLine(JH),10)
Equal(MeasureOfAngle(GDH),MeasureOfAngle(JHG))
Goal: Value(LengthOfLine(DH))
Answer: 20
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,HGD,JHG)
mirror_similar_triangle_property_line_ratio(1,HGD,JHG)
mirror_similar_triangle_property_line_ratio(1,GDH,GJH)
Problem: As shown in the diagram, JP=20, JQ=25, ∠PLQ=26°, ∠QKM=29°, the center of the inscribed circle of △JKL is Q, LP is perpendicular to QP, ML is perpendicular to QL, QN is perpendicular to KN. Find the length of line QP.
问题：如图所示，JP=20，JQ=25，∠PLQ=26°，∠QKM=29°，Q是△JKL内切圆的圆心，LP⊥QP，ML⊥QL，QN⊥KN。求直线QP的长度。
Construction (CDL):
Shape(JQ,QP,PJ)
Shape(JN,NQ,QJ)
Shape(NK,KQ,QN)
Shape(QK,KM,MQ)
Shape(QM,ML,LQ)
Shape(QL,LP,PQ)
Collinear(JPL)
Collinear(JNK)
Collinear(LMK)
Conditions (CDL):
Equal(LengthOfLine(JP),20)
Equal(LengthOfLine(JQ),25)
Equal(MeasureOfAngle(PLQ),26)
Equal(MeasureOfAngle(QKM),29)
IsIncenterOfTriangle(Q,JKL)
PerpendicularBetweenLine(LP,QP)
PerpendicularBetweenLine(ML,QM)
PerpendicularBetweenLine(QN,KN)
Goal: Value(LengthOfLine(QP))
Answer: 15
Theorem sequence:
adjacent_complementary_angle(1,LPQ,QPJ)
right_triangle_judgment_angle(1,QPJ)
right_triangle_property_pythagorean(1,QPJ)
Problem: As shown in the diagram, AB=x, AD=13, BD=7, BD⊥AD. Find the value of x.
问题：如图所示，AB=x，AD=13，BD=7，BD垂直于AD。求x的值。
Construction (CDL):
Shape(CB,BA,AC)
Shape(AB,BD,DA)
Conditions (CDL):
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(AD),13)
Equal(LengthOfLine(BD),7)
PerpendicularBetweenLine(BD,AD)
Goal: Value(x)
Answer: sqrt(218)
Theorem sequence:
right_triangle_judgment_angle(1,BDA)
right_triangle_property_pythagorean(1,BDA)
Problem: As shown in the diagram, JK=10, MP=6, the measure of ⌒PLJ is 134, P is the center of circle P, PQ⊥KQ. Find the length of line PQ.
问题：如图所示，JK=10，MP=6，弧PLJ的角度为134，P是圆P的圆心，PQ垂直于KQ。求直线PQ的长度。
Construction (CDL):
Shape(PMJ,JP,PM)
Shape(PJ,JQ,QP)
Shape(QJ,PJL,LQ)
Shape(QL,PLK,KQ)
Shape(PQ,QK,PKM,MP)
Collinear(LQPM)
Collinear(JQK)
Cocircular(P,JLKM)
Conditions (CDL):
Equal(LengthOfLine(JK),10)
Equal(LengthOfLine(MP),6)
Equal(MeasureOfArc(PLJ),134)
IsCentreOfCircle(P,P)
PerpendicularBetweenLine(PQ,KQ)
Goal: Value(LengthOfLine(PQ))
Answer: sqrt(11)
Theorem sequence:
radius_of_circle_property_length_equal(1,PJ,P)
radius_of_circle_property_length_equal(1,PM,P)
adjacent_complementary_angle(1,JQP,PQK)
circle_property_chord_perpendicular_bisect_chord(1,P,PQ,JK)
line_addition(1,JQ,QK)
right_triangle_judgment_angle(1,JQP)
right_triangle_property_pythagorean(1,JQP)
Problem: As shown in the diagram, XW=9, XZ=9, ZC=3, XC⊥ZC. Find the length of line WC.
问题：如图所示，XW=9，XZ=9，ZC=3，XC垂直于ZC。求直线WC的长度。
Construction (CDL):
Shape(XW,WC,CX)
Shape(XC,CZ,ZX)
Collinear(WCZ)
Conditions (CDL):
Equal(LengthOfLine(XW),9)
Equal(LengthOfLine(XZ),9)
Equal(LengthOfLine(ZC),3)
PerpendicularBetweenLine(XC,ZC)
Goal: Value(LengthOfLine(WC))
Answer: 3
Theorem sequence:
adjacent_complementary_angle(1,WCX,XCZ)
right_triangle_judgment_angle(1,XCZ)
right_triangle_judgment_angle(1,WCX)
right_triangle_property_pythagorean(1,XCZ)
right_triangle_property_pythagorean(1,WCX)
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 y.
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y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(y)
Answer: 3
Theorem sequence:
similar_triangle_judgment_aa(1,CBE,ADE)
similar_triangle_property_line_ratio(1,BEC,DEA)
similar_triangle_property_line_ratio(1,CBE,ADE)
line_addition(1,EC,CA)
line_addition(1,EB,BD)
Problem: As shown in the diagram, AC=y, AD=9, BC=z, CD=x, DB=16, AD⊥CD, BC is perpendicular to AC. Find the value of z.
问题：如图所示，AC=y，AD=9，BC=z，CD=x，DB=16，AD垂直于CD，BC⊥AC。求z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(z)
Answer: 20
Theorem sequence:
adjacent_complementary_angle(1,ADC,CDB)
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)
line_addition(1,AD,DB)
Problem: As shown in the diagram, AB=40, AD=CD, BC=22, ∠BAD=y°, ∠CBD=x°, AC is perpendicular to BC. Find the value of y.
问题：如图所示，AB=40，AD=CD，BC=22，∠BAD=y°，∠CBD=x°，AC垂直于BC。求y的值。
Construction (CDL):
Shape(AD,DB,BA)
Shape(BD,DC,CB)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AB),40)
Equal(LengthOfLine(AD),LengthOfLine(CD))
Equal(LengthOfLine(BC),22)
Equal(MeasureOfAngle(BAD),y)
Equal(MeasureOfAngle(CBD),x)
PerpendicularBetweenLine(AC,BC)
Goal: Value(y)
Answer: 180*asin(11/20)/pi
Theorem sequence:
sine_theorem(1,BAC)
Problem: As shown in the diagram, AB=x, AC=21, BC=y, ∠ABC=30°, ∠CAB=60°, BC is perpendicular to AC. Find the value of x.
问题：如图所示，AB=x，AC=21，BC=y，∠ABC=30°，∠CAB=60°，BC垂直于AC。求x的值。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(AC),21)
Equal(LengthOfLine(BC),y)
Equal(MeasureOfAngle(ABC),30)
Equal(MeasureOfAngle(CAB),60)
PerpendicularBetweenLine(BC,AC)
Goal: Value(x)
Answer: 42
Theorem sequence:
sine_theorem(1,ABC)
Problem: As shown in the diagram, ∠ZYX=110°, quadrilateral WZYX is a rhombus. Find the measure of ∠YWZ.
问题：如图所示，∠ZYX=110°，WZYX是菱形。求∠YWZ的大小。
Construction (CDL):
Shape(WZ,ZY,YW)
Shape(XW,WY,YX)
Conditions (CDL):
Equal(MeasureOfAngle(ZYX),110)
Rhombus(WZYX)
Goal: Value(MeasureOfAngle(YWZ))
Answer: 55
Theorem sequence:
kite_property_opposite_angle_equal(1,WZYX)
kite_property_opposite_angle_equal(1,ZYXW)
quadrilateral_property_angle_sum(1,WZYX)
sine_theorem(1,ZYW)
triangle_property_angle_sum(1,WZY)
Problem: As shown in the diagram, JV=3, ST=UT, SY=YZ, UJ=9, ZT=18, ZV=UV. Find the length of line YU.
问题：如图所示，JV=3，ST=UT，SY=YZ，UJ=9，ZT=18，ZV=UV。求直线YU的长度。
Construction (CDL):
Shape(SY,YJ,JS)
Shape(YZ,ZJ,JY)
Shape(JZ,ZV,VJ)
Shape(JV,VU,UJ)
Shape(JU,UT,TJ)
Shape(JT,TS,SJ)
Collinear(SYZ)
Collinear(ZVU)
Collinear(STU)
Collinear(SJV)
Collinear(ZJT)
Collinear(YJU)
Conditions (CDL):
Equal(LengthOfLine(JV),3)
Equal(LengthOfLine(ST),LengthOfLine(UT))
Equal(LengthOfLine(SY),LengthOfLine(YZ))
Equal(LengthOfLine(UJ),9)
Equal(LengthOfLine(ZT),18)
Equal(LengthOfLine(ZV),LengthOfLine(UV))
Goal: Value(LengthOfLine(YU))
Answer: 27/2
Theorem sequence:
median_of_triangle_judgment(1,ZT,ZUS)
median_of_triangle_judgment(1,UY,USZ)
centroid_of_triangle_judgment_intersection(1,J,SZU,Y,T)
centroid_of_triangle_property_line_ratio(1,J,USZ,Y)
line_addition(1,YJ,JU)
Problem: As shown in the diagram, ∠DHG=2*x+11°, ∠GFD=2*y°, ∠HGF=x-5°, DF and HG are opposite sides of the parallelogram FDHG. Find the value of x.
问题：如图所示，∠DHG=2*x+11°，∠GFD=2*y°，∠HGF=x-5°，DF和HG是▱FDHG的一组对边。求x的值。
Construction (CDL):
Shape(FD,DH,HG,GF)
Conditions (CDL):
Equal(MeasureOfAngle(DHG),2*x+11)
Equal(MeasureOfAngle(GFD),2*y)
Equal(MeasureOfAngle(HGF),x-5)
Parallelogram(FDHG)
Goal: Value(x)
Answer: 58
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,HD,GF)
Problem: As shown in the diagram, AD=5, ∠ACD=59°, CD⊥AD. Find the perimeter of △ACD.
问题：如图所示，AD=5，∠ACD=59°，CD⊥AD。求△ACD的周长。
Construction (CDL):
Shape(AC,CD,DA)
Conditions (CDL):
Equal(LengthOfLine(AD),5)
Equal(MeasureOfAngle(ACD),59)
PerpendicularBetweenLine(CD,AD)
Goal: Value(PerimeterOfTriangle(ACD))
Answer: 5*(1+sqrt(2)*sin(19*pi/45))/sin(59*pi/180)
Theorem sequence:
sine_theorem(1,ACD)
right_triangle_judgment_angle(1,CDA)
right_triangle_property_pythagorean(1,CDA)
triangle_perimeter_formula(1,ACD)
Problem: As shown in the diagram, AB=26, AD=12, DG=9/2, EF=8, GF=14, ∠AGF=108°, AGFE is similar to ADCB. Find the ratio of the perimeter of ADCB to the perimeter of AGFE.
问题：如图所示，AB=26，AD=12，DG=9/2，EF=8，GF=14，∠AGF=108°，四边形AGFE相似于四边形ADCB。求四边形ADCB的周长与AGFE的周长之比。
Construction (CDL):
Shape(AG,GF,FE,EA)
Shape(GD,DC,CB,BE,EF,FG)
Collinear(AGD)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),26)
Equal(LengthOfLine(AD),12)
Equal(LengthOfLine(DG),9/2)
Equal(LengthOfLine(EF),8)
Equal(LengthOfLine(GF),14)
Equal(MeasureOfAngle(AGF),108)
SimilarBetweenQuadrilateral(AGFE,ADCB)
Goal: Value(Div(PerimeterOfQuadrilateral(ADCB),PerimeterOfQuadrilateral(AGFE)))
Answer: 8/5
Theorem sequence:
line_addition(1,AG,GD)
similar_quadrilateral_property_line_ratio(1,AGFE,ADCB)
similar_quadrilateral_property_line_ratio(1,GFEA,DCBA)
similar_quadrilateral_property_line_ratio(1,FEAG,CBAD)
similar_quadrilateral_property_line_ratio(1,EAGF,BADC)
quadrilateral_perimeter_formula(1,AGFE)
quadrilateral_perimeter_formula(1,ADCB)
Problem: As shown in the diagram, ∠HGJ=20°, A is the center of ⊙A. Find the measure of ⌒AHG.
问题：如图所示，∠HGJ=20°，A是⊙A的圆心。求弧AHG的角度。
Construction (CDL):
Shape(AHG,GH)
Shape(HG,GA,AJ,AJH)
Shape(JA,AG,AGJ)
Collinear(GAJ)
Cocircular(A,GJH)
Conditions (CDL):
Equal(MeasureOfAngle(HGJ),20)
IsCentreOfCircle(A,A)
Goal: Value(MeasureOfArc(AHG))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_external(1,AJH,G)
arc_property_center_angle(1,AJG,A)
flat_angle(1,GAJ)
arc_addition_measure(1,AJH,AHG)
Problem: As shown in the diagram, AD=42, EA=18, EC=12, EG∥CD. Find the length of line DG.
问题：如图所示，AD=42，EA=18，EC=12，EG∥CD。求直线DG的长度。
Construction (CDL):
Shape(CD,DG,GE,EC)
Shape(EG,GA,AE)
Collinear(CEA)
Collinear(DGA)
Conditions (CDL):
Equal(LengthOfLine(AD),42)
Equal(LengthOfLine(EA),18)
Equal(LengthOfLine(EC),12)
ParallelBetweenLine(EG,CD)
Goal: Value(LengthOfLine(DG))
Answer: 84/5
Theorem sequence:
line_addition(1,CE,EA)
line_addition(1,DG,GA)
parallel_property_corresponding_angle(1,EG,CD,A)
similar_triangle_judgment_aa(1,GAE,DAC)
similar_triangle_property_line_ratio(1,GAE,DAC)
similar_triangle_property_line_ratio(1,EGA,CDA)
Problem: As shown in the diagram, ∠AMC=9*x-5°, ∠FCB=7*x+3°, GB is parallel to AE. Find the value of x.
问题：如图所示，∠AMC=9*x-5°，∠FCB=7*x+3°，GB∥AE。求x的值。
Construction (CDL):
Shape(DM,MA)
Shape(AM,MC)
Shape(MC,CG)
Shape(GC,CF)
Shape(FC,CB)
Shape(BC,CM)
Shape(CM,ME)
Shape(EM,MD)
Collinear(AME)
Collinear(GCB)
Collinear(DMCF)
Conditions (CDL):
Equal(MeasureOfAngle(AMC),9*x-5)
Equal(MeasureOfAngle(FCB),7*x+3)
ParallelBetweenLine(GB,AE)
Goal: Value(x)
Answer: 91/8
Theorem sequence:
parallel_property_collinear_extend(3,GB,AE,C)
parallel_property_collinear_extend(3,EA,CG,M)
parallel_property_ipsilateral_internal_angle(1,MA,CG)
vertical_angle(1,MCG,FCB)
Problem: As shown in the diagram, ∠HJF=130°, ∠IKG=145°, BA∥DI, MF∥BA. Find the measure of ∠KAJ.
问题：如图所示，∠HJF=130°，∠IKG=145°，BA∥DI，MF∥BA。求∠KAJ的大小。
Construction (CDL):
Shape(IK,KG)
Shape(GK,KD)
Shape(DK,KA)
Shape(KA,AB)
Shape(BA,AJ)
Shape(AJ,JM)
Shape(MJ,JH)
Shape(HJ,JF)
Shape(FJ,JA)
Shape(JA,AK)
Shape(AK,KI)
Collinear(GKA)
Collinear(DKI)
Collinear(AJH)
Collinear(MJF)
Conditions (CDL):
Equal(MeasureOfAngle(HJF),130)
Equal(MeasureOfAngle(IKG),145)
ParallelBetweenLine(BA,DI)
ParallelBetweenLine(MF,BA)
Goal: Value(MeasureOfAngle(KAJ))
Answer: 85
Theorem sequence:
angle_addition(1,KAB,BAJ)
vertical_angle(1,IKG,DKA)
vertical_angle(1,AJM,HJF)
parallel_property_collinear_extend(3,ID,AB,K)
parallel_property_collinear_extend(3,MF,BA,J)
parallel_property_ipsilateral_internal_angle(1,KD,AB)
parallel_property_ipsilateral_internal_angle(1,AB,JM)
Problem: As shown in the diagram, ∠PSR=x+10°, ∠QPS=x°, ∠RQP=2*x-16°, ∠SRQ=2*x°. Find the measure of ∠RQP.
问题：如图所示，∠PSR=x+10°，∠QPS=x°，∠RQP=2*x-16°，∠SRQ=2*x°。求∠RQP的大小。
Construction (CDL):
Shape(QP,PS,SR,RQ)
Conditions (CDL):
Equal(MeasureOfAngle(PSR),x+10)
Equal(MeasureOfAngle(QPS),x)
Equal(MeasureOfAngle(RQP),2*x-16)
Equal(MeasureOfAngle(SRQ),2*x)
Goal: Value(MeasureOfAngle(RQP))
Answer: 106
Theorem sequence:
quadrilateral_property_angle_sum(1,QPSR)
Problem: As shown in the diagram, AB=x-2, AD=x, DG=15/2, EB=5, BADC is similar to EAGF. Find the length of line AG.
问题：如图所示，AB=x-2，AD=x，DG=15/2，EB=5，四边形BADC相似于四边形EAGF。求直线AG的长度。
Construction (CDL):
Shape(BA,AD,DC,CB)
Shape(EB,BC,CD,DG,GF,FE)
Collinear(ABE)
Collinear(ADG)
Conditions (CDL):
Equal(LengthOfLine(AB),x-2)
Equal(LengthOfLine(AD),x)
Equal(LengthOfLine(DG),15/2)
Equal(LengthOfLine(EB),5)
SimilarBetweenQuadrilateral(BADC,EAGF)
Goal: Value(LengthOfLine(AG))
Answer: 27/2
Theorem sequence:
line_addition(1,EB,BA)
line_addition(1,AD,DG)
similar_quadrilateral_property_line_ratio(1,BADC,EAGF)
similar_quadrilateral_property_line_ratio(1,ADCB,AGFE)
Problem: As shown in the diagram, AB=21, ∠CDA=168°, the center of ⊙D is D, the diameter of ⊙D is AB. Find the length of arc DCB.
问题：如图所示，AB=21，∠CDA=168°，圆D的圆心为D，圆D的直径为AB。求⌒DCB的长度。
Construction (CDL):
Shape(DB,DBA,AD)
Shape(BD,DC,DCB)
Shape(DA,DAC,CD)
Collinear(ADB)
Cocircular(D,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),21)
Equal(MeasureOfAngle(CDA),168)
IsCentreOfCircle(D,D)
IsDiameterOfCircle(AB,D)
Goal: Value(LengthOfArc(DCB))
Answer: 7*pi/10
Theorem sequence:
adjacent_complementary_angle(1,BDC,CDA)
arc_property_center_angle(1,DCB,D)
diameter_of_circle_property_length_equal(1,AB,D)
circle_property_length_of_radius_and_diameter(1,D)
arc_length_formula(1,DCB)
Problem: As shown in the diagram, ∠AGB=30°, G is the center of ⊙G, CG⊥DG. Find the measure of arc GBF.
问题：如图所示，∠AGB=30°，G是圆G的圆心，CG⊥DG。求⌒GBF的角度。
Construction (CDL):
Shape(GBA,AG,GB)
Shape(GA,GAF,FG)
Shape(GF,GFD,DG)
Shape(GD,GDC,CG)
Shape(GC,GCB,BG)
Collinear(BGF)
Collinear(AGD)
Cocircular(G,AFDCB)
Conditions (CDL):
Equal(MeasureOfAngle(AGB),30)
IsCentreOfCircle(G,G)
PerpendicularBetweenLine(CG,DG)
Goal: Value(MeasureOfArc(GBF))
Answer: 180
Theorem sequence:
flat_angle(1,FGB)
arc_property_center_angle(1,GBF,G)
Problem: As shown in the diagram, AB=x, AC=12, AD=y, BD=z, CD=4, AD⊥CD, CA⊥BA. Find the value of z.
问题：如图所示，AB=x，AC=12，AD=y，BD=z，CD=4，AD垂直于CD，CA垂直于BA。求z的值。
Construction (CDL):
Shape(BD,DA,AB)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(AD),y)
Equal(LengthOfLine(BD),z)
Equal(LengthOfLine(CD),4)
PerpendicularBetweenLine(AD,CD)
PerpendicularBetweenLine(CA,BA)
Goal: Value(z)
Answer: 32
Theorem sequence:
adjacent_complementary_angle(1,BDA,ADC)
right_triangle_judgment_angle(1,BDA)
right_triangle_judgment_angle(1,ADC)
right_triangle_judgment_angle(1,CAB)
right_triangle_property_pythagorean(1,BDA)
right_triangle_property_pythagorean(1,ADC)
right_triangle_property_pythagorean(1,CAB)
line_addition(1,BD,DC)
Problem: As shown in the diagram, ∠ABX=5*y-6°, ∠BXC=2*x+24°, ∠CAB=3*x-17°, ∠XCA=y+58°, quadrilateral ABXC is a parallelogram. Find the value of x.
问题：如图所示，∠ABX=5*y-6°，∠BXC=2*x+24°，∠CAB=3*x-17°，∠XCA=y+58°，四边形ABXC是▱。求x的值。
Construction (CDL):
Shape(AB,BX,XC,CA)
Conditions (CDL):
Equal(MeasureOfAngle(ABX),5*y-6)
Equal(MeasureOfAngle(BXC),2*x+24)
Equal(MeasureOfAngle(CAB),3*x-17)
Equal(MeasureOfAngle(XCA),y+58)
Parallelogram(ABXC)
Goal: Value(x)
Answer: 41
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,ABXC)
Problem: As shown in the diagram, AE=4, △BCD is an equilateral △, A is the center of circle A, the tangent to circle A is BE, BF is the tangent to ⊙A. Find the area of △BCD minus the area of the circle A.
问题：如图所示，AE=4，△BCD为等边△，圆A的圆心为A，BE是⊙O的切线，BF是⊙O的切线。求三角形BCD的面积减去圆A的面积。
Construction (CDL):
Shape(BE,ANE,NB)
Shape(BN,AFN,FB)
Shape(AN,ANE,EA)
Shape(AF,AFN,NA)
Shape(AM,AMF,FA)
Shape(MD,DF,AMF)
Shape(AE,AEG,AGM,MA)
Shape(AGM,GD,DM)
Shape(AEG,EC,CG)
Collinear(BEC)
Collinear(CGD)
Collinear(BFD)
Collinear(BNA)
Collinear(EAMD)
Cocircular(A,NEGMF)
Conditions (CDL):
Equal(LengthOfLine(AE),4)
EquilateralTriangle(BCD)
IsCentreOfCircle(A,A)
IsTangentOfCircle(BE,A)
IsTangentOfCircle(BF,A)
Goal: Value(Sub(AreaOfTriangle(BCD),AreaOfCircle(A)))
Answer: -16*pi+48*sqrt(3)
Theorem sequence:
equilateral_triangle_property_angle(1,BCD)
tangent_of_circle_property_length_equal(1,BE,BF,A)
radius_of_circle_property_length_equal(1,AE,A)
radius_of_circle_property_length_equal(1,AF,A)
mirror_congruent_triangle_judgment_sss(1,BEA,BAF)
mirror_congruent_triangle_property_angle_equal(1,BEA,BAF)
angle_addition(1,FBA,ABE)
tangent_of_circle_property_perpendicular(2,BE,A,A)
triangle_property_angle_sum(1,ABE)
triangle_property_angle_sum(1,BED)
sine_theorem(1,EAB)
sine_theorem(1,BED)
triangle_area_formula_sine(1,BCD)
circle_area_formula(1,A)
Problem: As shown in the diagram, AB=21, AC=h, ∠CAB=45°, BC⊥AC. Find the value of h.
问题：如图所示，AB=21，AC=h，∠CAB=45°，BC⊥AC。求h的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),21)
Equal(LengthOfLine(AC),h)
Equal(MeasureOfAngle(CAB),45)
PerpendicularBetweenLine(BC,AC)
Goal: Value(h)
Answer: 21*sqrt(2)/2
Theorem sequence:
triangle_property_angle_sum(1,BCA)
sine_theorem(1,ABC)
Problem: As shown in the diagram, ∠GJD=4*x°, ∠HJM=8*x-12°, ∠KHA=6*y+10°, KO is parallel to DM. Find the value of x.
问题：如图所示，∠GJD=4*x°，∠HJM=8*x-12°，∠KHA=6*y+10°，KO∥DM。求x的值。
Construction (CDL):
Shape(AH,HO)
Shape(OH,HJ)
Shape(HJ,JM)
Shape(MJ,JG)
Shape(GJ,JD)
Shape(DJ,JH)
Shape(JH,HK)
Shape(KH,HA)
Collinear(KHO)
Collinear(DJM)
Collinear(AHJG)
Conditions (CDL):
Equal(MeasureOfAngle(GJD),4*x)
Equal(MeasureOfAngle(HJM),8*x-12)
Equal(MeasureOfAngle(KHA),6*y+10)
ParallelBetweenLine(KO,DM)
Goal: Value(x)
Answer: 3
Theorem sequence:
vertical_angle(1,AJM,GJD)
Problem: As shown in the diagram, AB=x, AC=y, BC=7, ∠CAB=30°, BC⊥AC. Find the value of y.
问题：如图所示，AB=x，AC=y，BC=7，∠CAB=30°，BC垂直于AC。求y的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(AC),y)
Equal(LengthOfLine(BC),7)
Equal(MeasureOfAngle(CAB),30)
PerpendicularBetweenLine(BC,AC)
Goal: Value(y)
Answer: 7*sqrt(3)
Theorem sequence:
triangle_property_angle_sum(1,BCA)
sine_theorem(1,BCA)
sine_theorem(1,ABC)
Problem: As shown in the diagram, AB=8, JB=13, KL=10, BA is parallel to LK. Find the length of line JL.
问题：如图所示，AB=8，JB=13，KL=10，BA∥LK。求直线JL的长度。
Construction (CDL):
Shape(JB,BA,AJ)
Shape(AB,BL,LK,KA)
Collinear(JBL)
Collinear(JAK)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(JB),13)
Equal(LengthOfLine(KL),10)
ParallelBetweenLine(BA,LK)
Goal: Value(LengthOfLine(JL))
Answer: 65/4
Theorem sequence:
parallel_property_corresponding_angle(1,BA,LK,J)
similar_triangle_judgment_aa(1,AJB,KJL)
similar_triangle_property_line_ratio(1,JBA,JLK)
similar_triangle_property_line_ratio(1,AJB,KJL)
Problem: As shown in the diagram, ∠ANE=30°, ∠CEB=110°, ∠ECA=∠AFN, ∠ENJ=130°. Find the measure of ∠ECA.
问题：如图所示，∠ANE=30°，∠CEB=110°，∠ECA=∠AFN，∠ENJ=130°。求∠ECA的大小。
Construction (CDL):
Shape(CA,AE,EC)
Shape(EA,AN,NE)
Shape(AF,FN,NA)
Shape(EN,NJ)
Shape(CE,EB)
Collinear(CAF)
Collinear(CEN)
Collinear(AEB)
Collinear(FNJ)
Conditions (CDL):
Equal(MeasureOfAngle(ANE),30)
Equal(MeasureOfAngle(CEB),110)
Equal(MeasureOfAngle(ECA),MeasureOfAngle(AFN))
Equal(MeasureOfAngle(ENJ),130)
Goal: Value(MeasureOfAngle(ECA))
Answer: 65
Theorem sequence:
adjacent_complementary_angle(1,FNE,ENJ)
triangle_property_angle_sum(1,CFN)
Problem: As shown in the diagram, ∠BED=29°, ∠EAD=17°, DB is perpendicular to AB. Find the measure of ∠EDB.
问题：如图所示，∠BED=29°，∠EAD=17°，DB垂直于AB。求∠EDB的大小。
Construction (CDL):
Shape(DB,BE,ED)
Shape(DE,EA,AD)
Collinear(BEA)
Conditions (CDL):
Equal(MeasureOfAngle(BED),29)
Equal(MeasureOfAngle(EAD),17)
PerpendicularBetweenLine(DB,AB)
Goal: Value(MeasureOfAngle(EDB))
Answer: 61
Theorem sequence:
triangle_property_angle_sum(1,BED)
Problem: As shown in the diagram, PQ=3, ∠RPS=130°, ∠TPQ=112°, the center of ⊙P is P. Find the length of ⌒PRS.
问题：如图所示，PQ=3，∠RPS=130°，∠TPQ=112°，P是圆P的圆心。求⌒PRS的长度。
Construction (CDL):
Shape(PR,PRQ,QP)
Shape(PQ,PQT,TP)
Shape(PT,PTS,SP)
Shape(PS,PSR,RP)
Collinear(RPT)
Cocircular(P,RQTS)
Conditions (CDL):
Equal(LengthOfLine(PQ),3)
Equal(MeasureOfAngle(RPS),130)
Equal(MeasureOfAngle(TPQ),112)
IsCentreOfCircle(P,P)
Goal: Value(LengthOfArc(PRS))
Answer: 23*pi/6
Theorem sequence:
arc_property_center_angle(1,PSR,P)
round_arc(1,PRS,PSR)
radius_of_circle_property_length_equal(1,PQ,P)
arc_length_formula(1,PRS)
Problem: As shown in the diagram, MN=10*x+3, PO=4*x+7, RS=9*x-1, RS is the midsegment of MPON, MN and PO are the parallel sides of trapezoid MPON. Find the length of line RS.
问题：如图所示，MN=10*x+3，PO=4*x+7，RS=9*x-1，四边形MPON的中位线为RS，MN和PO是梯形MPON的平行边。求直线RS的长度。
Construction (CDL):
Shape(MR,RS,SN,NM)
Shape(RP,PO,OS,SR)
Collinear(MRP)
Collinear(NSO)
Conditions (CDL):
Equal(LengthOfLine(MN),10*x+3)
Equal(LengthOfLine(PO),4*x+7)
Equal(LengthOfLine(RS),9*x-1)
IsMidsegmentOfQuadrilateral(RS,MPON)
Trapezoid(MPON)
Goal: Value(LengthOfLine(RS))
Answer: 26
Theorem sequence:
midsegment_of_quadrilateral_property_length(1,RS,MPON)
Problem: As shown in the diagram, ∠WZX=24°, ∠XWZ=23°, ∠YXZ=105°, ∠ZYX=∠XZY. Find the measure of ∠XZY.
问题：如图所示，∠WZX=24°，∠XWZ=23°，∠YXZ=105°，∠ZYX=∠XZY。求∠XZY的大小。
Construction (CDL):
Shape(WZ,ZX,XW)
Shape(XZ,ZY,YX)
Conditions (CDL):
Equal(MeasureOfAngle(WZX),24)
Equal(MeasureOfAngle(XWZ),23)
Equal(MeasureOfAngle(YXZ),105)
Equal(MeasureOfAngle(ZYX),MeasureOfAngle(XZY))
Goal: Value(MeasureOfAngle(XZY))
Answer: 75/2
Theorem sequence:
triangle_property_angle_sum(1,XZY)
Problem: As shown in the diagram, AB=2, BG=2, CE=5, DC=5, GF=2, HE=5, quadrilateral DHEC is a rectangle, FABG is a rectangle. Find the area of DHEC minus the area of quadrilateral FABG.
问题：如图所示，AB=2，BG=2，CE=5，DC=5，GF=2，HE=5，DHEC是矩形，四边形FABG是长方形。求DHEC的面积减去四边形FABG的面积。
Construction (CDL):
Shape(DF,FG,GB,BA,AH,HE,EC,CD)
Shape(FA,AB,BG,GF)
Collinear(DFAH)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(BG),2)
Equal(LengthOfLine(CE),5)
Equal(LengthOfLine(DC),5)
Equal(LengthOfLine(GF),2)
Equal(LengthOfLine(HE),5)
Rectangle(DHEC)
Rectangle(FABG)
Goal: Value(Sub(AreaOfQuadrilateral(DHEC),AreaOfQuadrilateral(FABG)))
Answer: 21
Theorem sequence:
parallelogram_area_formula_sine(1,HECD)
parallelogram_area_formula_sine(1,ABGF)
Problem: As shown in the diagram, ∠VWX=67°, ∠XWZ=∠WZX, ∠XZY=65°, ∠ZYW=58°. Find the measure of ∠XVW.
问题：如图所示，∠VWX=67°，∠XWZ=∠WZX，∠XZY=65°，∠ZYW=58°。求∠XVW的大小。
Construction (CDL):
Shape(VW,WX,XV)
Shape(XW,WZ,ZX)
Shape(XZ,ZY,YX)
Collinear(VXZ)
Collinear(WXY)
Conditions (CDL):
Equal(MeasureOfAngle(VWX),67)
Equal(MeasureOfAngle(XWZ),MeasureOfAngle(WZX))
Equal(MeasureOfAngle(XZY),65)
Equal(MeasureOfAngle(ZYW),58)
Goal: Value(MeasureOfAngle(XVW))
Answer: 56
Theorem sequence:
triangle_property_angle_sum(1,XZY)
vertical_angle(1,WXV,YXZ)
triangle_property_angle_sum(1,VWX)