SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AD=9, CD=x, CE=6, A is the center of circle A, CD is the tangent to circle A. Find the value of x.
问题：如图所示，AD=9，CD=x，CE=6，圆A的圆心为A，CD是圆O的切线。求x的值。
Construction (CDL):
Shape(DC,CE,ADE)
Shape(AE,AED)
Shape(ADE,EA,AD)
Collinear(CEA)
Cocircular(A,DE)
Conditions (CDL):
Equal(LengthOfLine(AD),9)
Equal(LengthOfLine(CD),x)
Equal(LengthOfLine(CE),6)
IsCentreOfCircle(A,A)
IsTangentOfCircle(CD,A)
Goal: Value(x)
Answer: 12
Theorem sequence:
radius_of_circle_property_length_equal(1,AD,A)
radius_of_circle_property_length_equal(1,AE,A)
line_addition(1,CE,EA)
tangent_of_circle_property_perpendicular(1,CD,A,A)
right_triangle_judgment_angle(1,ADC)
right_triangle_property_pythagorean(1,ADC)
Problem: As shown in the diagram, ∠ALH=110°, ∠GAH=x°, ∠JAK=97°, the center of ⊙A is A. Find the value of x.
问题：如图所示，∠ALH=110°，∠GAH=x°，∠JAK=97°，圆A的圆心为A。求x的值。
Construction (CDL):
Shape(AK,KL,LA)
Shape(AL,LH,HA)
Shape(KA,AH,AHK)
Shape(JL,LK,AKJ)
Shape(HL,LJ,AJH)
Collinear(GALJ)
Collinear(KLH)
Cocircular(A,JHGK)
Conditions (CDL):
Equal(MeasureOfAngle(ALH),110)
Equal(MeasureOfAngle(GAH),x)
Equal(MeasureOfAngle(JAK),97)
IsCentreOfCircle(A,A)
Goal: Value(x)
Answer: 123
Theorem sequence:
radius_of_circle_property_length_equal(1,AK,A)
radius_of_circle_property_length_equal(1,AH,A)
flat_angle(1,KLH)
angle_addition(1,KLA,ALH)
triangle_property_angle_sum(1,AKL)
isosceles_triangle_judgment_line_equal(1,AKH)
isosceles_triangle_property_angle_equal(1,AKH)
triangle_property_angle_sum(1,ALH)
flat_angle(1,GAL)
angle_addition(1,GAH,HAL)
Problem: As shown in the diagram, WZ=YZ, YX=12, XZ is the perpendicular bisector of WY. Find the length of line XW.
问题：如图所示，WZ=YZ，YX=12，XZ是WY的垂直平分线。求直线XW的长度。
Construction (CDL):
Shape(XW,WZ,ZX)
Shape(XZ,ZY,YX)
Collinear(WZY)
Collinear(BXZA)
Conditions (CDL):
Equal(LengthOfLine(WZ),LengthOfLine(YZ))
Equal(LengthOfLine(YX),12)
IsPerpendicularBisectorOfLine(XZ,WY)
Goal: Value(LengthOfLine(XW))
Answer: 12
Theorem sequence:
mirror_congruent_triangle_judgment_sas(1,ZXW,ZYX)
mirror_congruent_triangle_property_line_equal(1,ZXW,ZYX)
Problem: As shown in the diagram, ∠ARS=71°, the measure of arc BAS is 106, the center of ⊙B is B. Find the measure of ⌒BSQ.
问题：如图所示，∠ARS=71°，弧BAS的角度为106，B是圆B的圆心。求⌒BSQ的角度。
Construction (CDL):
Shape(AR,RS,BAS)
Shape(QA,BAQ)
Shape(AQ,BQA)
Collinear(CQAR)
Cocircular(B,SQA)
Conditions (CDL):
Equal(MeasureOfAngle(ARS),71)
Equal(MeasureOfArc(BAS),106)
IsCentreOfCircle(B,B)
Goal: Value(MeasureOfArc(BSQ))
Answer: 248
Theorem sequence:
circle_property_circular_power_tangent_and_segment_angle(2,RS,RAQ,B)
Problem: As shown in the diagram, ∠ACE=28°, ∠DBE=35°, ∠EDB=75°. Find the measure of ∠BED.
问题：如图所示，∠ACE=28°，∠DBE=35°，∠EDB=75°。求∠BED的大小。
Construction (CDL):
Shape(DB,BE,ED)
Shape(EA,AC,CE)
Collinear(DEA)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),28)
Equal(MeasureOfAngle(DBE),35)
Equal(MeasureOfAngle(EDB),75)
Goal: Value(MeasureOfAngle(BED))
Answer: 70
Theorem sequence:
triangle_property_angle_sum(1,DBE)
Problem: As shown in the diagram, TW=13, ZV=1, the center of circle T is T, TV⊥XV. Find the length of line XY.
问题：如图所示，TW=13，ZV=1，T是⊙T的圆心，TV垂直于XV。求直线XY的长度。
Construction (CDL):
Shape(ZV,VY,TYZ)
Shape(XV,VZ,TZX)
Shape(TV,VX,XT)
Shape(YV,VT,TY)
Shape(YT,TX,TXY)
Collinear(XVY)
Collinear(ZVTW)
Cocircular(T,YZXW)
Conditions (CDL):
Equal(LengthOfLine(TW),13)
Equal(LengthOfLine(ZV),1)
IsCentreOfCircle(T,T)
PerpendicularBetweenLine(TV,XV)
Goal: Value(LengthOfLine(XY))
Answer: 10
Theorem sequence:
radius_of_circle_property_length_equal(1,TW,T)
radius_of_circle_property_length_equal(1,TZ,T)
radius_of_circle_property_length_equal(1,TX,T)
radius_of_circle_property_length_equal(1,TY,T)
line_addition(1,ZV,VT)
flat_angle(1,YVX)
angle_addition(1,YVT,TVX)
right_triangle_judgment_angle(1,TVX)
right_triangle_judgment_angle(1,YVT)
right_triangle_property_pythagorean(1,TVX)
right_triangle_property_pythagorean(1,YVT)
line_addition(1,XV,VY)
Problem: As shown in the diagram, ST=UT, SY=YZ, UJ=9, VJ=3, ZT=18, ZV=UV. Find the length of line SV.
问题：如图所示，ST=UT，SY=YZ，UJ=9，VJ=3，ZT=18，ZV=UV。求直线SV的长度。
Construction (CDL):
Shape(SY,YJ,JS)
Shape(YZ,ZJ,JY)
Shape(JZ,ZV,VJ)
Shape(JV,VU,UJ)
Shape(TJ,JU,UT)
Shape(SJ,JT,TS)
Collinear(SYZ)
Collinear(STU)
Collinear(ZVU)
Collinear(SJV)
Collinear(ZJT)
Collinear(YJU)
Conditions (CDL):
Equal(LengthOfLine(ST),LengthOfLine(UT))
Equal(LengthOfLine(SY),LengthOfLine(YZ))
Equal(LengthOfLine(UJ),9)
Equal(LengthOfLine(VJ),3)
Equal(LengthOfLine(ZT),18)
Equal(LengthOfLine(ZV),LengthOfLine(UV))
Goal: Value(LengthOfLine(SV))
Answer: 9
Theorem sequence:
median_of_triangle_judgment(1,SV,SZU)
median_of_triangle_judgment(1,UY,USZ)
centroid_of_triangle_judgment_intersection(1,J,ZUS,V,Y)
centroid_of_triangle_property_line_ratio(1,J,SZU,V)
line_addition(1,SJ,JV)
Problem: As shown in the diagram, FG=18, FK=15, GH=42, JK=a. Find the value of a.
问题：如图所示，FG=18，FK=15，GH=42，JK=a。求a的值。
Construction (CDL):
Shape(GF,FK,AGK)
Shape(GH,AHG)
Shape(HG,AGK,KJ,AJH)
Shape(JK,AKJ)
Collinear(FGH)
Collinear(FKJ)
Cocircular(A,HGKJ)
Conditions (CDL):
Equal(LengthOfLine(FG),18)
Equal(LengthOfLine(FK),15)
Equal(LengthOfLine(GH),42)
Equal(LengthOfLine(JK),a)
Goal: Value(a)
Answer: 57
Theorem sequence:
line_addition(1,FG,GH)
line_addition(1,FK,KJ)
circle_property_circular_power_segment_and_segment_line(1,FGH,FKJ,A)
Problem: As shown in the diagram, ∠BCF=∠FBC, ∠DAG=136°, ∠DFG=47°, ∠EFD=69°, ∠FEB=63°. Find the measure of ∠CFB.
问题：如图所示，∠BCF=∠FBC，∠DAG=136°，∠DFG=47°，∠EFD=69°，∠FEB=63°。求∠CFB的大小。
Construction (CDL):
Shape(EB,BF,FE)
Shape(EF,FD,DE)
Shape(DF,FA,AD)
Shape(DA,AG,GD)
Shape(BC,CF,FB)
Collinear(EFC)
Collinear(BFAG)
Conditions (CDL):
Equal(MeasureOfAngle(BCF),MeasureOfAngle(FBC))
Equal(MeasureOfAngle(DAG),136)
Equal(MeasureOfAngle(DFG),47)
Equal(MeasureOfAngle(EFD),69)
Equal(MeasureOfAngle(FEB),63)
Goal: Value(MeasureOfAngle(CFB))
Answer: 116
Theorem sequence:
angle_addition(1,EFD,DFA)
vertical_angle(1,EFA,CFB)
Problem: As shown in the diagram, ∠ACD=7*y+2°, ∠BAC=6*x+14°, ∠CDB=8*x-8°, ∠DBA=6*y+16°, DC and BA are opposite sides of the ▱ DBAC. Find the value of y.
问题：如图所示，∠ACD=7*y+2°，∠BAC=6*x+14°，∠CDB=8*x-8°，∠DBA=6*y+16°，DBAC是▱。求y的值。
Construction (CDL):
Shape(DB,BA,AC,CD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),7*y+2)
Equal(MeasureOfAngle(BAC),6*x+14)
Equal(MeasureOfAngle(CDB),8*x-8)
Equal(MeasureOfAngle(DBA),6*y+16)
Parallelogram(DBAC)
Goal: Value(y)
Answer: 14
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,CDBA)
Problem: As shown in the diagram, ∠WZY=a+2°, ∠XWZ=1/2*a+8°, ∠YXW=a°, ∠ZYX=a-28°. Find the measure of ∠WZY.
问题：如图所示，∠WZY=a+2°，∠XWZ=1/2*a+8°，∠YXW=a°，∠ZYX=a-28°。求∠WZY的大小。
Construction (CDL):
Shape(XW,WZ,ZY,YX)
Conditions (CDL):
Equal(MeasureOfAngle(WZY),a+2)
Equal(MeasureOfAngle(XWZ),1/2*a+8)
Equal(MeasureOfAngle(YXW),a)
Equal(MeasureOfAngle(ZYX),a-28)
Goal: Value(MeasureOfAngle(WZY))
Answer: 110
Theorem sequence:
quadrilateral_property_angle_sum(1,XWZY)
Problem: As shown in the diagram, ∠XWY=49°, the measure of arc BXY is 96, the center of ⊙B is B. Find the measure of arc BYZ.
问题：如图所示，∠XWY=49°，弧BXY的角度为96，⊙B的圆心为B。求弧BYZ的角度。
Construction (CDL):
Shape(XW,WY,BXY)
Shape(ZX,BXZ)
Shape(XZ,BZX)
Collinear(CZXW)
Collinear(AYW)
Cocircular(B,YZX)
Conditions (CDL):
Equal(MeasureOfAngle(XWY),49)
Equal(MeasureOfArc(BXY),96)
IsCentreOfCircle(B,B)
Goal: Value(MeasureOfArc(BYZ))
Answer: 194
Theorem sequence:
circle_property_circular_power_tangent_and_segment_angle(2,WY,WXZ,B)
Problem: As shown in the diagram, RL=5, RT=9, WS=6, TS∥LW. Find the length of line RW.
问题：如图所示，RL=5，RT=9，WS=6，TS平行于LW。求直线RW的长度。
Construction (CDL):
Shape(LR,RW,WL)
Shape(TL,LW,WS,ST)
Collinear(RLT)
Collinear(RWS)
Conditions (CDL):
Equal(LengthOfLine(RL),5)
Equal(LengthOfLine(RT),9)
Equal(LengthOfLine(WS),6)
ParallelBetweenLine(TS,LW)
Goal: Value(LengthOfLine(RW))
Answer: 15/2
Theorem sequence:
line_addition(1,SW,WR)
parallel_property_corresponding_angle(2,TS,LW,R)
similar_triangle_judgment_aa(1,WLR,STR)
similar_triangle_property_line_ratio(1,WLR,STR)
similar_triangle_property_line_ratio(1,LRW,TRS)
Problem: As shown in the diagram, AG=26, BG=10, G is the center of circle G, GB⊥AB. Find the length of line AC.
问题：如图所示，AG=26，BG=10，⊙G的圆心为G，GB⊥AB。求直线AC的长度。
Construction (CDL):
Shape(AG,GB,BA)
Shape(BG,GC,CB)
Shape(GCA,AC)
Shape(GAC,CG,GA)
Collinear(ABC)
Cocircular(G,AC)
Conditions (CDL):
Equal(LengthOfLine(AG),26)
Equal(LengthOfLine(BG),10)
IsCentreOfCircle(G,G)
PerpendicularBetweenLine(GB,AB)
Goal: Value(LengthOfLine(AC))
Answer: 48
Theorem sequence:
radius_of_circle_property_length_equal(1,GA,G)
radius_of_circle_property_length_equal(1,GC,G)
flat_angle(1,CBA)
angle_addition(1,CBG,GBA)
right_triangle_judgment_angle(1,GBA)
right_triangle_property_pythagorean(1,GBA)
mirror_congruent_triangle_judgment_hl(2,GBA,GCB)
mirror_congruent_triangle_property_line_equal(1,GBA,GCB)
line_addition(1,AB,BC)
Problem: As shown in the diagram, AB=40, LA=27, NL=36, AN and BC are opposite sides of the ▱ NABC, NL is perpendicular to AL. Find the area of quadrilateral NABC.
问题：如图所示，AB=40，LA=27，NL=36，四边形NABC是平行四边形，NL⊥AL。求NABC的面积。
Construction (CDL):
Shape(NA,AB,BC,CN)
Shape(NL,LA,AN)
Collinear(LAB)
Conditions (CDL):
Equal(LengthOfLine(AB),40)
Equal(LengthOfLine(LA),27)
Equal(LengthOfLine(NL),36)
Parallelogram(NABC)
PerpendicularBetweenLine(NL,AL)
Goal: Value(AreaOfQuadrilateral(NABC))
Answer: 1440
Theorem sequence:
altitude_of_quadrilateral_judgment_left_vertex(3,NL,NABC)
parallelogram_area_formula_common(1,NABC)
Problem: As shown in the diagram, AD=13, BA=3*y-8, CB=2*x+7, DC=10, ∠ADF=59°, ∠BFC=49°, ∠FBA=20°, DA and CB are opposite sides of the ▱ ADCB. Find the measure of ∠DCA.
问题：如图所示，AD=13，BA=3*y-8，CB=2*x+7，DC=10，∠ADF=59°，∠BFC=49°，∠FBA=20°，AB和DC是▱ADCB的一组对边。求∠DCA的大小。
Construction (CDL):
Shape(AD,DF,FA)
Shape(FD,DC,CF)
Shape(FC,CB,BF)
Shape(AF,FB,BA)
Collinear(AFC)
Collinear(BFD)
Conditions (CDL):
Equal(LengthOfLine(AD),13)
Equal(LengthOfLine(BA),3*y-8)
Equal(LengthOfLine(CB),2*x+7)
Equal(LengthOfLine(DC),10)
Equal(MeasureOfAngle(ADF),59)
Equal(MeasureOfAngle(BFC),49)
Equal(MeasureOfAngle(FBA),20)
Parallelogram(ADCB)
Goal: Value(MeasureOfAngle(DCA))
Answer: 29
Theorem sequence:
flat_angle(1,AFC)
angle_addition(1,AFB,BFC)
triangle_property_angle_sum(1,AFB)
parallel_property_alternate_interior_angle(1,AB,DC)
Problem: As shown in the diagram, AC=CB, CD=AD, ∠ADC=92°. Find the measure of ∠CAD.
问题：如图所示，AC=CB，CD=AD，∠ADC=92°。求∠CAD的大小。
Construction (CDL):
Shape(AD,DC,CA)
Shape(AC,CB,BA)
Collinear(DCB)
Conditions (CDL):
Equal(LengthOfLine(AC),LengthOfLine(CB))
Equal(LengthOfLine(CD),LengthOfLine(AD))
Equal(MeasureOfAngle(ADC),92)
Goal: Value(MeasureOfAngle(CAD))
Answer: 44
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,DCA)
triangle_property_angle_sum(1,DCA)
Problem: As shown in the diagram, AB=x, AC=24, ∠ABC=61°, BC is perpendicular to AC. Find the value of x.
问题：如图所示，AB=x，AC=24，∠ABC=61°，BC垂直于AC。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),x)
Equal(LengthOfLine(AC),24)
Equal(MeasureOfAngle(ABC),61)
PerpendicularBetweenLine(BC,AC)
Goal: Value(x)
Answer: 24/sin(61*pi/180)
Theorem sequence:
sine_theorem(1,ABC)
Problem: As shown in the diagram, AD=17, CA=21, DE=8, ADBC is a ▱, CA is perpendicular to EA, DE is perpendicular to AE. Find the area of BCAD.
问题：如图所示，AD=17，CA=21，DE=8，DA和BC是▱ADBC的一组对边，CA⊥EA，DE⊥AE。求BCAD的面积。
Construction (CDL):
Shape(BC,CA,AE,EB)
Shape(EA,AD,DE)
Collinear(BED)
Conditions (CDL):
Equal(LengthOfLine(AD),17)
Equal(LengthOfLine(CA),21)
Equal(LengthOfLine(DE),8)
Parallelogram(ADBC)
PerpendicularBetweenLine(CA,EA)
PerpendicularBetweenLine(DE,AE)
Goal: Value(AreaOfQuadrilateral(BCAD))
Answer: 315
Theorem sequence:
right_triangle_judgment_angle(1,DEA)
right_triangle_property_pythagorean(1,DEA)
altitude_of_quadrilateral_judgment_left_vertex(1,AE,ADBC)
parallelogram_property_opposite_line_equal(1,DBCA)
parallelogram_area_formula_common(1,ADBC)
Problem: As shown in the diagram, ∠DOB=∠EOD, ∠EOD=∠FOE, ∠FOE=∠AOF, O is the center of circle O, CO is perpendicular to AO. Find the measure of ⌒OEA.
问题：如图所示，∠DOB=∠EOD，∠EOD=∠FOE，∠FOE=∠AOF，O是圆O的圆心，CO⊥AO。求弧OEA的角度。
Construction (CDL):
Shape(DO,OB,OBD)
Shape(EO,OD,ODE)
Shape(FO,OE,OEF)
Shape(OFA,AO,OF)
Shape(OA,OAC,CO)
Shape(OC,OCB,BO)
Collinear(AOB)
Collinear(EOC)
Cocircular(O,BDEFAC)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),MeasureOfAngle(EOD))
Equal(MeasureOfAngle(EOD),MeasureOfAngle(FOE))
Equal(MeasureOfAngle(FOE),MeasureOfAngle(AOF))
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CO,AO)
Goal: Value(MeasureOfArc(OEA))
Answer: 90
Theorem sequence:
flat_angle(1,COE)
angle_addition(1,COA,AOE)
arc_property_center_angle(1,OEA,O)
Problem: As shown in the diagram, ∠ADC=x°, ∠BAD=3*x°, CB is perpendicular to AB, DC is perpendicular to BC. Find the measure of ∠DCB.
问题：如图所示，∠ADC=x°，∠BAD=3*x°，CB⊥AB，DC⊥BC。求∠DCB的大小。
Construction (CDL):
Shape(AD,DC,CB,BA)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),x)
Equal(MeasureOfAngle(BAD),3*x)
PerpendicularBetweenLine(CB,AB)
PerpendicularBetweenLine(DC,BC)
Goal: Value(MeasureOfAngle(DCB))
Answer: 90
Theorem sequence:
Problem: As shown in the diagram, AE=x, BE=6, EC=3, ED=4, the center of circle F is F. Find the value of x.
问题：如图所示，AE=x，BE=6，EC=3，ED=4，F是⊙F的圆心。求x的值。
Construction (CDL):
Shape(EA,FAC,CE)
Shape(EC,FCB,BE)
Shape(EB,FBD,DE)
Shape(FDA,AE,ED)
Collinear(AEB)
Collinear(CED)
Cocircular(F,BDAC)
Conditions (CDL):
Equal(LengthOfLine(AE),x)
Equal(LengthOfLine(BE),6)
Equal(LengthOfLine(EC),3)
Equal(LengthOfLine(ED),4)
IsCentreOfCircle(F,F)
Goal: Value(x)
Answer: 2
Theorem sequence:
circle_property_circular_power_chord_and_chord(1,BEA,DEC,F)
Problem: As shown in the diagram, ∠BFC=6*x°, ∠CFD=3*x°, ∠EFA=12*y-10°, BF⊥DF. Find the value of x.
问题：如图所示，∠BFC=6*x°，∠CFD=3*x°，∠EFA=12*y-10°，BF垂直于DF。求x的值。
Construction (CDL):
Shape(BF,FC)
Shape(CF,FD)
Collinear(BFE)
Collinear(AFD)
Conditions (CDL):
Equal(MeasureOfAngle(BFC),6*x)
Equal(MeasureOfAngle(CFD),3*x)
Equal(MeasureOfAngle(EFA),12*y-10)
PerpendicularBetweenLine(BF,DF)
Goal: Value(x)
Answer: 10
Theorem sequence:
angle_addition(1,BFC,CFD)
Problem: As shown in the diagram, ∠ACB=2*x°, ∠BAC=x°, ∠CBA=3*x°. Find the value of x.
问题：如图所示，∠ACB=2*x°，∠BAC=x°，∠CBA=3*x°。求x的值。
Construction (CDL):
Shape(CB,BA,AC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),2*x)
Equal(MeasureOfAngle(BAC),x)
Equal(MeasureOfAngle(CBA),3*x)
Goal: Value(x)
Answer: 30
Theorem sequence:
triangle_property_angle_sum(1,CBA)
Problem: As shown in the diagram, ∠CAD=38°, AB is perpendicular to DB, ABDC is a rectangle. Find the measure of ∠DAB.
问题：如图所示，∠CAD=38°，AB⊥DB，ABDC是长方形。求∠DAB的大小。
Construction (CDL):
Shape(AB,BG,GA)
Shape(GB,BD,DG)
Shape(GD,DC,CG)
Shape(AG,GC,CA)
Collinear(AGD)
Collinear(CGB)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),38)
PerpendicularBetweenLine(AB,DB)
Rectangle(ABDC)
Goal: Value(MeasureOfAngle(DAB))
Answer: 52
Theorem sequence:
angle_addition(1,CAG,GAB)
Problem: As shown in the diagram, AC=16, BC=10, ∠ABC=x°, BC⊥AC. Find the value of x.
问题：如图所示，AC=16，BC=10，∠ABC=x°，BC垂直于AC。求x的值。
Construction (CDL):
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(AC),16)
Equal(LengthOfLine(BC),10)
Equal(MeasureOfAngle(ABC),x)
PerpendicularBetweenLine(BC,AC)
Goal: Value(x)
Answer: 180*acos(5*sqrt(89)/89)/pi
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
cosine_theorem(1,BCA)
Problem: As shown in the diagram, ∠YZW=60°, ∠ZWX=95°, the center of circle V is V. Find the measure of ∠WXY.
问题：如图所示，∠YZW=60°，∠ZWX=95°，V是圆V的圆心。求∠WXY的大小。
Construction (CDL):
Shape(VZW,WZ)
Shape(VWX,XW)
Shape(VXY,YX)
Shape(VYZ,ZY)
Shape(ZW,WV,VY,YZ)
Shape(VW,WX,XY,YV)
Cocircular(V,ZWXY)
Conditions (CDL):
Equal(MeasureOfAngle(YZW),60)
Equal(MeasureOfAngle(ZWX),95)
IsCentreOfCircle(V,V)
Goal: Value(MeasureOfAngle(WXY))
Answer: 120
Theorem sequence:
arc_property_circumference_angle_external(1,VWY,Z)
arc_property_circumference_angle_internal(1,VWY,X)
Problem: As shown in the diagram, BA=2, CD=5, DE=x+5, EA=x-1, CD∥AB. Find the length of line DE.
问题：如图所示，BA=2，CD=5，DE=x+5，EA=x-1，CD平行于AB。求直线DE的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(ED,DC,CE)
Collinear(AED)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(BA),2)
Equal(LengthOfLine(CD),5)
Equal(LengthOfLine(DE),x+5)
Equal(LengthOfLine(EA),x-1)
ParallelBetweenLine(CD,AB)
Goal: Value(LengthOfLine(DE))
Answer: 10
Theorem sequence:
parallel_property_alternate_interior_angle(2,CD,AB)
vertical_angle(1,BEA,CED)
similar_triangle_judgment_aa(1,BEA,CED)
similar_triangle_property_line_ratio(1,BEA,CED)
similar_triangle_property_line_ratio(1,EAB,EDC)
Problem: As shown in the diagram, PX=4, WX=6, quadrilateral XWZY is a kite. Find the length of line WP.
问题：如图所示，PX=4，WX=6，四边形XWZY是风筝形。求直线WP的长度。
Construction (CDL):
Shape(XW,WP,PX)
Shape(PW,WZ,ZP)
Shape(PZ,ZY,YP)
Shape(XP,PY,YX)
Collinear(XPZ)
Collinear(WPY)
Conditions (CDL):
Equal(LengthOfLine(PX),4)
Equal(LengthOfLine(WX),6)
Kite(XWZY)
Goal: Value(LengthOfLine(WP))
Answer: 2*sqrt(5)
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,XWZY,P)
right_triangle_judgment_angle(1,WPX)
right_triangle_property_pythagorean(1,WPX)
Problem: As shown in the diagram, AB=15, AC=9, BC=12, BC is perpendicular to AC. Find the area of △ABC.
问题：如图所示，AB=15，AC=9，BC=12，BC垂直于AC。求三角形ABC的面积。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),15)
Equal(LengthOfLine(AC),9)
Equal(LengthOfLine(BC),12)
PerpendicularBetweenLine(BC,AC)
Goal: Value(AreaOfTriangle(ABC))
Answer: 54
Theorem sequence:
triangle_area_formula_sine(1,CAB)
Problem: As shown in the diagram, AB=y, AC=x, BC=10*sqrt(3), ∠CBA=30°, AC is perpendicular to BC. Find the value of x.
问题：如图所示，AB=y，AC=x，BC=10*sqrt(3)，∠CBA=30°，AC⊥BC。求x的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BC),10*sqrt(3))
Equal(MeasureOfAngle(CBA),30)
PerpendicularBetweenLine(AC,BC)
Goal: Value(x)
Answer: 10
Theorem sequence:
triangle_property_angle_sum(1,BAC)
sine_theorem(1,CBA)
Problem: As shown in the diagram, BC=17, DE=8, DE=AE, DBAC is a rhombus. Find the area of DBAC.
问题：如图所示，BC=17，DE=8，DE=AE，DBAC是菱形。求DBAC的面积。
Construction (CDL):
Shape(DB,BE,ED)
Shape(BA,AE,EB)
Shape(EA,AC,CE)
Shape(DE,EC,CD)
Collinear(BEC)
Collinear(DEA)
Conditions (CDL):
Equal(LengthOfLine(BC),17)
Equal(LengthOfLine(DE),8)
Equal(LengthOfLine(DE),LengthOfLine(AE))
Rhombus(DBAC)
Goal: Value(AreaOfQuadrilateral(DBAC))
Answer: 136
Theorem sequence:
line_addition(1,DE,EA)
kite_area_formula_diagonal(1,DBAC)
Problem: As shown in the diagram, AB=4, A is the center of ⊙A. Find the area of the ⊙A.
问题：如图所示，AB=4，⊙A的圆心为A。求⊙A的面积。
Construction (CDL):
Shape(AB)
Cocircular(A,B)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
IsCentreOfCircle(A,A)
Goal: Value(AreaOfCircle(A))
Answer: 16*pi
Theorem sequence:
radius_of_circle_property_length_equal(1,AB,A)
circle_area_formula(1,A)
Problem: As shown in the diagram, AB=2*x+3, BC=x+7, quadrilateral ADCB is a rhombus. Find the length of line CD.
问题：如图所示，AB=2*x+3，BC=x+7，四边形ADCB是菱形。求直线CD的长度。
Construction (CDL):
Shape(AD,DC,CA)
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),2*x+3)
Equal(LengthOfLine(BC),x+7)
Rhombus(ADCB)
Goal: Value(LengthOfLine(CD))
Answer: 11
Theorem sequence:
Problem: As shown in the diagram, TU=x, VT=11, VU=7, T is the center of circle T, the tangent to circle T is VU. Find the value of x.
问题：如图所示，TU=x，VT=11，VU=7，⊙T的圆心为T，VU是⊙O的切线。求x的值。
Construction (CDL):
Shape(BV,VU,TBU)
Shape(TBU,UT,TB)
Shape(TU,TUB,BT)
Collinear(VBT)
Cocircular(T,BU)
Conditions (CDL):
Equal(LengthOfLine(TU),x)
Equal(LengthOfLine(VT),11)
Equal(LengthOfLine(VU),7)
IsCentreOfCircle(T,T)
IsTangentOfCircle(VU,T)
Goal: Value(x)
Answer: 6*sqrt(2)
Theorem sequence:
tangent_of_circle_property_perpendicular(2,VU,T,T)
right_triangle_judgment_angle(1,VUT)
right_triangle_property_pythagorean(1,VUT)
Problem: As shown in the diagram, AR=RC, AT=16, CT=TB, MC=7, RM=4, SA=SB. Find the length of line MS.
问题：如图所示，AR=RC，AT=16，CT=TB，MC=7，RM=4，SA=SB。求直线MS的长度。
Construction (CDL):
Shape(AR,RM,MA)
Shape(RC,CM,MR)
Shape(CT,TM,MC)
Shape(MT,TB,BM)
Shape(MB,BS,SM)
Shape(AM,MS,SA)
Collinear(ARC)
Collinear(CTB)
Collinear(BSA)
Collinear(RMB)
Collinear(CMS)
Collinear(TMA)
Conditions (CDL):
Equal(LengthOfLine(AR),LengthOfLine(RC))
Equal(LengthOfLine(AT),16)
Equal(LengthOfLine(CT),LengthOfLine(TB))
Equal(LengthOfLine(MC),7)
Equal(LengthOfLine(RM),4)
Equal(LengthOfLine(SA),LengthOfLine(SB))
Goal: Value(LengthOfLine(MS))
Answer: 7/2
Theorem sequence:
median_of_triangle_judgment(1,CS,CBA)
median_of_triangle_judgment(1,AT,ACB)
centroid_of_triangle_judgment_intersection(1,M,BAC,S,T)
centroid_of_triangle_property_line_ratio(1,M,CBA,S)
Problem: As shown in the diagram, JK=10, PM=6, the measure of ⌒PJK is 134, P is the center of ⊙P, PQ is perpendicular to KQ. Find the measure of arc PJL.
问题：如图所示，JK=10，PM=6，⌒PJK的角度为134，⊙P的圆心为P，PQ垂直于KQ。求弧PJL的角度。
Construction (CDL):
Shape(JQ,QP,PM,PMJ)
Shape(PJL,LQ,QJ)
Shape(PJK,KQ,QL)
Shape(MP,PQ,QK,PKM)
Collinear(JQK)
Collinear(MPQL)
Cocircular(P,MJLK)
Conditions (CDL):
Equal(LengthOfLine(JK),10)
Equal(LengthOfLine(PM),6)
Equal(MeasureOfArc(PJK),134)
IsCentreOfCircle(P,P)
PerpendicularBetweenLine(PQ,KQ)
Goal: Value(MeasureOfArc(PJL))
Answer: 67
Theorem sequence:
adjacent_complementary_angle(1,JQP,PQK)
circle_property_chord_perpendicular_bisect_arc(1,PJK,PQL)
congruent_arc_judgment_length_equal(1,PJL,PLK)
congruent_arc_property_measure_equal(1,PJL,PLK)
arc_addition_measure(1,PJL,PLK)
Problem: As shown in the diagram, ∠YXW=30°, WY⊥ZY, WYZX is a rectangle. Find the measure of ∠WZX.
问题：如图所示，∠YXW=30°，WY垂直于ZY，WYZX是矩形。求∠WZX的大小。
Construction (CDL):
Shape(WY,YA,AW)
Shape(AY,YZ,ZA)
Shape(AZ,ZX,XA)
Shape(WA,AX,XW)
Collinear(WAZ)
Collinear(YAX)
Conditions (CDL):
Equal(MeasureOfAngle(YXW),30)
PerpendicularBetweenLine(WY,ZY)
Rectangle(WYZX)
Goal: Value(MeasureOfAngle(WZX))
Answer: 60
Theorem sequence:
rectangle_property_diagonal_equal(1,WYZX)
parallelogram_property_opposite_line_equal(1,YZXW)
mirror_congruent_triangle_judgment_sss(1,WYX,YZW)
mirror_congruent_triangle_property_angle_equal(1,XWY,ZWY)
angle_addition(1,YZA,AZX)
Problem: As shown in the diagram, BC=6, CB=6, ∠DAB=Mul(MeasureOfAngle(CDA),2)°, DABC is a rhombus. Find the measure of ∠BDA.
问题：如图所示，BC=6，CB=6，∠DAB=Mul(MeasureOfAngle(CDA),2)°，四边形DABC是菱形。求∠BDA的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EA,AB,BE)
Shape(EB,BC,CE)
Shape(DE,EC,CD)
Collinear(DEB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(BC),6)
Equal(LengthOfLine(CB),6)
Equal(MeasureOfAngle(DAB),Mul(MeasureOfAngle(CDA),2))
Rhombus(DABC)
Goal: Value(MeasureOfAngle(BDA))
Answer: 30
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,DC,AB)
parallel_property_ipsilateral_internal_angle(1,BA,CD)
parallel_property_ipsilateral_internal_angle(1,AD,BC)
angle_addition(1,CDE,EDA)
isosceles_triangle_judgment_line_equal(1,CDB)
isosceles_triangle_property_angle_equal(1,CDB)
triangle_property_angle_sum(1,CDB)
Problem: As shown in the diagram, ∠JKL=y°, ∠LJK=4*x-13°, ∠WZY=44°, ∠YWZ=71°, △JKL is mirror similar to △WZY.. Find the value of x.
问题：如图所示，∠JKL=y°，∠LJK=4*x-13°，∠WZY=44°，∠YWZ=71°，三角形JKL与三角形WZY是镜像相似三角形。求x的值。
Construction (CDL):
Shape(JK,KL,LJ)
Shape(WZ,ZY,YW)
Conditions (CDL):
Equal(MeasureOfAngle(JKL),y)
Equal(MeasureOfAngle(LJK),4*x-13)
Equal(MeasureOfAngle(WZY),44)
Equal(MeasureOfAngle(YWZ),71)
MirrorSimilarBetweenTriangle(JKL,WZY)
Goal: Value(x)
Answer: 21
Theorem sequence:
mirror_similar_triangle_property_angle_equal(1,JKL,WZY)
Problem: As shown in the diagram, AB=15, AD=x, DF=8, FC=12, the center of ⊙E is E. Find the value of x.
问题：如图所示，AB=15，AD=x，DF=8，FC=12，E是圆E的圆心。求x的值。
Construction (CDL):
Shape(EAF,FD,DA)
Shape(EAB,BA)
Shape(EBC,CF,EFA,AB)
Shape(ECF,FC)
Collinear(BAD)
Collinear(CFD)
Cocircular(E,ABCF)
Conditions (CDL):
Equal(LengthOfLine(AB),15)
Equal(LengthOfLine(AD),x)
Equal(LengthOfLine(DF),8)
Equal(LengthOfLine(FC),12)
IsCentreOfCircle(E,E)
Goal: Value(x)
Answer: -15/2+sqrt(865)/2
Theorem sequence:
line_addition(1,BA,AD)
line_addition(1,CF,FD)
circle_property_circular_power_segment_and_segment_line(1,DAB,DFC,E)
Problem: As shown in the diagram, ∠BAC=42°, ∠CBA=72°, ∠CDB=38°. Find the measure of ∠ACB.
问题：如图所示，∠BAC=42°，∠CBA=72°，∠CDB=38°。求∠ACB的大小。
Construction (CDL):
Shape(AC,CB,BA)
Shape(BC,CD,DB)
Collinear(ABD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),42)
Equal(MeasureOfAngle(CBA),72)
Equal(MeasureOfAngle(CDB),38)
Goal: Value(MeasureOfAngle(ACB))
Answer: 66
Theorem sequence:
triangle_property_angle_sum(1,ACB)