SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AC=10, AD=13, ∠CBD=45°, DACB is a parallelogram, CE is perpendicular to DE. Find the area of ACBD.
问题：如图所示，AC=10，AD=13，∠CBD=45°，四边形DACB是▱，CE⊥DE。求ACBD的面积。
Construction (CDL):
Shape(DA,AC,CE,ED)
Shape(DE,EB,BD)
Collinear(CEB)
Conditions (CDL):
Equal(LengthOfLine(AC),10)
Equal(LengthOfLine(AD),13)
Equal(MeasureOfAngle(CBD),45)
Parallelogram(DACB)
PerpendicularBetweenLine(CE,DE)
Goal: Value(AreaOfQuadrilateral(ACBD))
Answer: 65*sqrt(2)
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,ACBD)
parallelogram_area_formula_sine(1,DACB)
Problem: As shown in the diagram, BC=10, HA=7, BH⊥AH. Find the area of triangle ABC.
问题：如图所示，BC=10，HA=7，BH⊥AH。求三角形ABC的面积。
Construction (CDL):
Shape(AB,BH,HA)
Shape(AH,HC,CA)
Collinear(BHC)
Conditions (CDL):
Equal(LengthOfLine(BC),10)
Equal(LengthOfLine(HA),7)
PerpendicularBetweenLine(BH,AH)
Goal: Value(AreaOfTriangle(ABC))
Answer: 35
Theorem sequence:
altitude_of_triangle_judgment(1,AH,ABC)
triangle_area_formula_common(1,ABC)
Problem: As shown in the diagram, ED=16, EF=12, FG=x, the center of ⊙G is G, ED is the tangent to circle G. Find the value of x.
问题：如图所示，ED=16，EF=12，FG=x，G是⊙G的圆心，⊙O的切线为ED。求x的值。
Construction (CDL):
Shape(DE,EF,GDF)
Shape(GDF,FG,GD)
Shape(GFD,DG,GF)
Collinear(EFG)
Cocircular(G,DF)
Conditions (CDL):
Equal(LengthOfLine(ED),16)
Equal(LengthOfLine(EF),12)
Equal(LengthOfLine(FG),x)
IsCentreOfCircle(G,G)
IsTangentOfCircle(ED,G)
Goal: Value(x)
Answer: 14/3
Theorem sequence:
line_addition(1,EF,FG)
tangent_of_circle_property_perpendicular(1,ED,G,G)
right_triangle_judgment_angle(1,GDE)
right_triangle_property_pythagorean(1,GDE)
radius_of_circle_property_length_equal(1,GF,G)
radius_of_circle_property_length_equal(1,GD,G)
Problem: As shown in the diagram, WN=5, triangle INW is an equilateral triangle, I is the center of circle I. Find the measure of ⌒INW.
问题：如图所示，WN=5，△INW为等边△，圆I的圆心为I。求弧INW的角度。
Construction (CDL):
Shape(WI,IN,NW)
Shape(INW,WN)
Shape(IWN,NI,IW)
Cocircular(I,WN)
Conditions (CDL):
Equal(LengthOfLine(WN),5)
EquilateralTriangle(INW)
IsCentreOfCircle(I,I)
Goal: Value(MeasureOfArc(INW))
Answer: 60
Theorem sequence:
equilateral_triangle_property_angle(1,INW)
arc_property_center_angle(1,INW,I)
Problem: As shown in the diagram, AC=22, BA=13, CB=x, CB is perpendicular to AB. Find the value of x.
问题：如图所示，AC=22，BA=13，CB=x，CB垂直于AB。求x的值。
Construction (CDL):
Shape(CB,BA,AC)
Conditions (CDL):
Equal(LengthOfLine(AC),22)
Equal(LengthOfLine(BA),13)
Equal(LengthOfLine(CB),x)
PerpendicularBetweenLine(CB,AB)
Goal: Value(x)
Answer: 3*sqrt(35)
Theorem sequence:
right_triangle_judgment_angle(1,CBA)
right_triangle_property_pythagorean(1,CBA)
Problem: As shown in the diagram, AD=24, BD=14, AC⊥BC, CD⊥AD. Find the length of line CD.
问题：如图所示，AD=24，BD=14，AC⊥BC，CD垂直于AD。求直线CD的长度。
Construction (CDL):
Shape(BD,DC,CB)
Shape(DA,AC,CD)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AD),24)
Equal(LengthOfLine(BD),14)
PerpendicularBetweenLine(AC,BC)
PerpendicularBetweenLine(CD,AD)
Goal: Value(LengthOfLine(CD))
Answer: 4*sqrt(21)
Theorem sequence:
adjacent_complementary_angle(1,BDC,CDA)
line_addition(1,AD,DB)
right_triangle_judgment_angle(1,BDC)
right_triangle_judgment_angle(1,CDA)
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,BDC)
right_triangle_property_pythagorean(1,CDA)
right_triangle_property_pythagorean(1,ACB)
Problem: As shown in the diagram, ∠HLI=59°, L is the center of ⊙L, IL⊥JL. Find the measure of arc LIJ.
问题：如图所示，∠HLI=59°，L是⊙L的圆心，IL垂直于JL。求弧LIJ的角度。
Construction (CDL):
Shape(LIH,HL,LI)
Shape(LHG,GL,LH)
Shape(LGK,KL,LG)
Shape(LKJ,JL,LK)
Shape(LJI,IL,LJ)
Collinear(ILG)
Collinear(HLK)
Cocircular(L,IHGKJ)
Conditions (CDL):
Equal(MeasureOfAngle(HLI),59)
IsCentreOfCircle(L,L)
PerpendicularBetweenLine(IL,JL)
Goal: Value(MeasureOfArc(LIJ))
Answer: 270
Theorem sequence:
round_angle(1,ILJ,JLI)
arc_property_center_angle(1,LIJ,L)
Problem: As shown in the diagram, the area of quadrilateral HGKJ is 188, HI=8, HJ=16, HI is perpendicular to KI, HJ and GK are the parallel sides of trapezoid HGKJ. Find the length of line GK.
问题：如图所示，HGKJ的面积为188，HI=8，HJ=16，HI垂直于KI，HG和KJ是梯形HGKJ的腰。求直线GK的长度。
Construction (CDL):
Shape(HG,GI,IH)
Shape(HI,IK,KJ,JH)
Collinear(GIK)
Conditions (CDL):
Equal(AreaOfQuadrilateral(HGKJ),188)
Equal(LengthOfLine(HI),8)
Equal(LengthOfLine(HJ),16)
PerpendicularBetweenLine(HI,KI)
Trapezoid(HGKJ)
Goal: Value(LengthOfLine(GK))
Answer: 31
Theorem sequence:
adjacent_complementary_angle(1,GIH,HIK)
altitude_of_quadrilateral_judgment_left_vertex(2,HI,HGKJ)
trapezoid_area_formula(1,HGKJ)
Problem: As shown in the diagram, ∠EAJ=10°, the measure of ⌒ICF is 95, the measure of arc IEJ is 15, the center of ⊙I is I. Find the measure of ∠GBD.
问题：如图所示，∠EAJ=10°，弧ICF的角度为95，⌒IEJ的角度为15，I是⊙I的圆心。求∠GBD的大小。
Construction (CDL):
Shape(EA,AJ,IEJ)
Shape(IEJ,JG,GH,HE)
Shape(IJG,GJ)
Shape(IGD,DH,HG)
Shape(BD,IGD,GB)
Shape(IDC,CD)
Shape(IFC,FH,HD,DC)
Shape(IFE,EH,HF)
Collinear(AEHD)
Collinear(AJG)
Collinear(BGHF)
Collinear(BDC)
Cocircular(I,EJGDCF)
Conditions (CDL):
Equal(MeasureOfAngle(EAJ),10)
Equal(MeasureOfArc(ICF),95)
Equal(MeasureOfArc(IEJ),15)
IsCentreOfCircle(I,I)
Goal: Value(MeasureOfAngle(GBD))
Answer: 30
Theorem sequence:
circle_property_circular_power_segment_and_segment_angle(1,AED,AJG,I)
circle_property_circular_power_segment_and_segment_angle(1,BGF,BDC,I)
Problem: As shown in the diagram, AB=2*x, AC=12, BC=20, DE=y+4, DF=12, EF=15, ∠ABC=∠EFD, ∠ACB=∠DEF, ∠BAC=∠FDE, △ACB is similar to △DEF.. Find the value of x.
问题：如图所示，AB=2*x，AC=12，BC=20，DE=y+4，DF=12，EF=15，∠ABC=∠EFD，∠ACB=∠DEF，∠BAC=∠FDE，△ACB相似于△DEF。求x的值。
Construction (CDL):
Shape(AC,CB,BA)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(LengthOfLine(AB),2*x)
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BC),20)
Equal(LengthOfLine(DE),y+4)
Equal(LengthOfLine(DF),12)
Equal(LengthOfLine(EF),15)
Equal(MeasureOfAngle(ABC),MeasureOfAngle(EFD))
Equal(MeasureOfAngle(ACB),MeasureOfAngle(DEF))
Equal(MeasureOfAngle(BAC),MeasureOfAngle(FDE))
SimilarBetweenTriangle(ACB,DEF)
Goal: Value(x)
Answer: 8
Theorem sequence:
similar_triangle_property_line_ratio(1,ACB,DEF)
similar_triangle_property_line_ratio(1,CBA,EFD)
Problem: As shown in the diagram, AD=10, BA=x-2, BC=5, ED=4, CD∥BE, EB∥DC. Find the length of line AB.
问题：如图所示，AD=10，BA=x-2，BC=5，ED=4，CD∥BE，EB∥DC。求直线AB的长度。
Construction (CDL):
Shape(AE,EB,BA)
Shape(ED,DC,CB,BE)
Collinear(AED)
Collinear(ABC)
Conditions (CDL):
Equal(LengthOfLine(AD),10)
Equal(LengthOfLine(BA),x-2)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(ED),4)
ParallelBetweenLine(CD,BE)
ParallelBetweenLine(EB,DC)
Goal: Value(LengthOfLine(AB))
Answer: 15/2
Theorem sequence:
parallel_property_corresponding_angle(1,EB,DC,A)
parallel_property_corresponding_angle(2,CD,BE,A)
similar_triangle_judgment_aa(1,AEB,ADC)
line_addition(1,AE,ED)
line_addition(1,AB,BC)
similar_triangle_property_line_ratio(1,EBA,DCA)
similar_triangle_property_line_ratio(1,BAE,CAD)
Problem: As shown in the diagram, BA=3*y-8, BC=2*x+7, DA=13, DC=10, ∠ADF=59°, ∠BFC=49°, ∠FBA=20°, ABCD is a parallelogram. Find the measure of ∠BAD.
问题：如图所示，BA=3*y-8，BC=2*x+7，DA=13，DC=10，∠ADF=59°，∠BFC=49°，∠FBA=20°，ABCD是平行四边形。求∠BAD的大小。
Construction (CDL):
Shape(FA,AD,DF)
Shape(FD,DC,CF)
Shape(FC,CB,BF)
Shape(FB,BA,AF)
Collinear(AFC)
Collinear(BFD)
Conditions (CDL):
Equal(LengthOfLine(BA),3*y-8)
Equal(LengthOfLine(BC),2*x+7)
Equal(LengthOfLine(DA),13)
Equal(LengthOfLine(DC),10)
Equal(MeasureOfAngle(ADF),59)
Equal(MeasureOfAngle(BFC),49)
Equal(MeasureOfAngle(FBA),20)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BAD))
Answer: 101
Theorem sequence:
triangle_property_angle_sum(1,ADB)
Problem: As shown in the diagram, AB=y, BD=x, CD=12, ∠BCD=55°, ∠DAB=47°, BD is perpendicular to AD. Find the value of y.
问题：如图所示，AB=y，BD=x，CD=12，∠BCD=55°，∠DAB=47°，BD⊥AD。求y的值。
Construction (CDL):
Shape(BC,CD,DB)
Shape(BD,DA,AB)
Collinear(CDA)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(BD),x)
Equal(LengthOfLine(CD),12)
Equal(MeasureOfAngle(BCD),55)
Equal(MeasureOfAngle(DAB),47)
PerpendicularBetweenLine(BD,AD)
Goal: Value(y)
Answer: 12/(sin(47*pi/180)*tan(7*pi/36))
Theorem sequence:
adjacent_complementary_angle(1,CDB,BDA)
triangle_property_angle_sum(1,BCD)
sine_theorem(1,DBC)
sine_theorem(1,BDA)
Problem: As shown in the diagram, ∠AED=x°, CE is perpendicular to AE. Find the value of x.
问题：如图所示，∠AED=x°，CE垂直于AE。求x的值。
Construction (CDL):
Shape(AE,ED)
Shape(CE,EA)
Collinear(DEC)
Conditions (CDL):
Equal(MeasureOfAngle(AED),x)
PerpendicularBetweenLine(CE,AE)
Goal: Value(x)
Answer: 90
Theorem sequence:
adjacent_complementary_angle(1,CEA,AED)
Problem: As shown in the diagram, ∠EAB=40°, AB is perpendicular to DB, ABDC is a rectangle. Find the measure of ∠EDC.
问题：如图所示，∠EAB=40°，AB⊥DB，ABDC是长方形。求∠EDC的大小。
Construction (CDL):
Shape(EA,AB,BE)
Shape(EB,BD,DE)
Shape(ED,DC,CE)
Shape(EC,CA,AE)
Collinear(AED)
Collinear(CEB)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),40)
PerpendicularBetweenLine(AB,DB)
Rectangle(ABDC)
Goal: Value(MeasureOfAngle(EDC))
Answer: 40
Theorem sequence:
parallel_property_alternate_interior_angle(2,BA,DC)
Problem: As shown in the diagram, JK=4, LK=x, XY=5, YZ=15, ∠LKJ=51°, ∠XYZ=51°, ∠ZXY=∠KJL. Find the length of line KL.
问题：如图所示，JK=4，LK=x，XY=5，YZ=15，∠LKJ=51°，∠XYZ=51°，∠ZXY=∠KJL。求直线KL的长度。
Construction (CDL):
Shape(XY,YZ,ZX)
Shape(JL,LK,KJ)
Conditions (CDL):
Equal(LengthOfLine(JK),4)
Equal(LengthOfLine(LK),x)
Equal(LengthOfLine(XY),5)
Equal(LengthOfLine(YZ),15)
Equal(MeasureOfAngle(LKJ),51)
Equal(MeasureOfAngle(XYZ),51)
Equal(MeasureOfAngle(ZXY),MeasureOfAngle(KJL))
Goal: Value(LengthOfLine(KL))
Answer: 12
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,ZXY,LKJ)
mirror_similar_triangle_property_line_ratio(1,ZXY,LKJ)
mirror_similar_triangle_property_line_ratio(1,XYZ,JLK)
Problem: As shown in the diagram, AD=8, BC=30, BD=y, CD=28, ∠ABC=∠DBA. Find the value of y.
问题：如图所示，AD=8，BC=30，BD=y，CD=28，∠ABC=∠DBA。求y的值。
Construction (CDL):
Shape(BC,CA,AB)
Shape(BA,AD,DB)
Collinear(CAD)
Conditions (CDL):
Equal(LengthOfLine(AD),8)
Equal(LengthOfLine(BC),30)
Equal(LengthOfLine(BD),y)
Equal(LengthOfLine(CD),28)
Equal(MeasureOfAngle(ABC),MeasureOfAngle(DBA))
Goal: Value(y)
Answer: 12
Theorem sequence:
line_addition(1,CA,AD)
bisector_of_angle_judgment_angle_equal(1,BA,DBC)
bisector_of_angle_property_line_ratio(1,BA,DBC)
Problem: As shown in the diagram, SR=6, US=5, the center of ⊙R is R, VS⊥US. Find the length of line TV.
问题：如图所示，SR=6，US=5，圆R的圆心为R，VS垂直于US。求直线TV的长度。
Construction (CDL):
Shape(RTU,US,ST)
Shape(RUV,VS,SU)
Shape(TS,SR,RT)
Shape(RS,SV,VR)
Shape(RVT,TR,RV)
Collinear(USR)
Collinear(TSV)
Cocircular(R,TUV)
Conditions (CDL):
Equal(LengthOfLine(SR),6)
Equal(LengthOfLine(US),5)
IsCentreOfCircle(R,R)
PerpendicularBetweenLine(VS,US)
Goal: Value(LengthOfLine(TV))
Answer: 2*sqrt(85)
Theorem sequence:
line_addition(1,US,SR)
line_addition(1,TS,SV)
radius_of_circle_property_length_equal(1,RU,R)
radius_of_circle_property_length_equal(1,RT,R)
radius_of_circle_property_length_equal(1,RV,R)
vertical_angle(1,VSU,TSR)
adjacent_complementary_angle(1,TSR,RSV)
right_triangle_judgment_angle(1,TSR)
right_triangle_judgment_angle(1,RSV)
right_triangle_property_pythagorean(1,TSR)
right_triangle_property_pythagorean(1,RSV)
Problem: As shown in the diagram, ∠AFD=50°, F is the center of ⊙F, CF is perpendicular to BF. Find the measure of ⌒FEA.
问题：如图所示，∠AFD=50°，⊙F的圆心为F，CF垂直于BF。求弧FEA的角度。
Construction (CDL):
Shape(FDA,AF,FD)
Shape(FAE,EF,FA)
Shape(FEB,BF,FE)
Shape(FBC,CF,FB)
Shape(FCD,DF,FC)
Collinear(DFE)
Collinear(AFB)
Cocircular(F,AEBCD)
Conditions (CDL):
Equal(MeasureOfAngle(AFD),50)
IsCentreOfCircle(F,F)
PerpendicularBetweenLine(CF,BF)
Goal: Value(MeasureOfArc(FEA))
Answer: 230
Theorem sequence:
vertical_angle(1,AFD,BFE)
flat_angle(1,AFB)
angle_addition(1,AFB,BFE)
arc_property_center_angle(1,FEA,F)
Problem: As shown in the diagram, the measure of arc EDF is 102, the center of ⊙E is E, the tangent to ⊙E is CF. Find the measure of ∠DFC.
问题：如图所示，⌒EDF的角度为102，E是圆E的圆心，CF是圆O的切线。求∠DFC的大小。
Construction (CDL):
Shape(BD,DE)
Shape(EF,FC)
Shape(ED,DF,FE)
Shape(EDF,FD)
Shape(EFD,DE,EF)
Collinear(BDF)
Cocircular(E,DF)
Conditions (CDL):
Equal(MeasureOfArc(EDF),102)
IsCentreOfCircle(E,E)
IsTangentOfCircle(CF,E)
Goal: Value(MeasureOfAngle(DFC))
Answer: 129
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CF,E,E)
radius_of_circle_property_length_equal(1,ED,E)
radius_of_circle_property_length_equal(1,EF,E)
arc_property_center_angle(1,EDF,E)
angle_addition(1,DFE,EFC)
triangle_property_angle_sum(1,EDF)
isosceles_triangle_judgment_line_equal(1,EDF)
isosceles_triangle_property_angle_equal(1,EDF)
Problem: As shown in the diagram, AM=4*x-6, AN=3*x-2, BM=24, CN=20, MN∥BC. Find the value of x.
问题：如图所示，AM=4*x-6，AN=3*x-2，BM=24，CN=20，MN∥BC。求x的值。
Construction (CDL):
Shape(AM,MN,NA)
Shape(MB,BC,CN,NM)
Collinear(AMB)
Collinear(ANC)
Conditions (CDL):
Equal(LengthOfLine(AM),4*x-6)
Equal(LengthOfLine(AN),3*x-2)
Equal(LengthOfLine(BM),24)
Equal(LengthOfLine(CN),20)
ParallelBetweenLine(MN,BC)
Goal: Value(x)
Answer: 9
Theorem sequence:
parallel_property_corresponding_angle(1,MN,BC,A)
similar_triangle_judgment_aa(1,NAM,CAB)
similar_triangle_property_line_ratio(1,MNA,BCA)
similar_triangle_property_line_ratio(1,NAM,CAB)
line_addition(1,AN,NC)
line_addition(1,AM,MB)
Problem: As shown in the diagram, ∠AJH=x°, ∠HGB=2*x°, GB is perpendicular to HB, JH⊥GH. Find the measure of ∠HGB.
问题：如图所示，∠AJH=x°，∠HGB=2*x°，GB垂直于HB，JH垂直于GH。求∠HGB的大小。
Construction (CDL):
Shape(AGF,FB,BG)
Shape(AFJ,JA,AB,BF)
Shape(AJH,HJ)
Shape(AHG,GH)
Shape(GB,BH,HG)
Shape(HB,BA,AJ,JH)
Collinear(FBH)
Collinear(GBAJ)
Cocircular(A,JHGF)
Conditions (CDL):
Equal(MeasureOfAngle(AJH),x)
Equal(MeasureOfAngle(HGB),2*x)
PerpendicularBetweenLine(GB,HB)
PerpendicularBetweenLine(JH,GH)
Goal: Value(MeasureOfAngle(HGB))
Answer: 60
Theorem sequence:
triangle_property_angle_sum(1,GJH)
Problem: As shown in the diagram, ∠EFG=x°, ∠FGC=55°, FE∥GC. Find the value of x.
问题：如图所示，∠EFG=x°，∠FGC=55°，FE平行于GC。求x的值。
Construction (CDL):
Shape(HG,GB)
Shape(BG,GF)
Shape(FG,FC)
Shape(CG,GH)
Shape(GF,FD)
Shape(DF,FA)
Shape(AF,FE)
Shape(EF,FG)
Collinear(HGFA)
Collinear(BGC)
Collinear(DFE)
Conditions (CDL):
Equal(MeasureOfAngle(EFG),x)
Equal(MeasureOfAngle(FGC),55)
ParallelBetweenLine(FE,GC)
Goal: Value(x)
Answer: 125
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,FE,GC)
Problem: As shown in the diagram, AB=y, AC=6, AD=4, BC=z, BD=x, BA⊥CA, CB is perpendicular to DB. Find the value of y.
问题：如图所示，AB=y，AC=6，AD=4，BC=z，BD=x，BA⊥CA，CB垂直于DB。求y的值。
Construction (CDL):
Shape(BD,DA,AB)
Shape(BA,AC,CB)
Collinear(DAC)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(AD),4)
Equal(LengthOfLine(BC),z)
Equal(LengthOfLine(BD),x)
PerpendicularBetweenLine(BA,CA)
PerpendicularBetweenLine(CB,DB)
Goal: Value(y)
Answer: 2*sqrt(6)
Theorem sequence:
line_addition(1,DA,AC)
adjacent_complementary_angle(1,DAB,BAC)
triangle_property_angle_sum(1,BDA)
triangle_property_angle_sum(1,BDC)
sine_theorem(1,BDA)
sine_theorem(1,DAB)
sine_theorem(1,DCB)
Problem: As shown in the diagram, CB=12, DB=20, ∠DBC=60°, DA and BC are opposite sides of the parallelogram DBCA, DE⊥CE. Find the area of quadrilateral DBCA.
问题：如图所示，CB=12，DB=20，∠DBC=60°，DBCA是平行四边形，DE垂直于CE。求DBCA的面积。
Construction (CDL):
Shape(DB,BE,ED)
Shape(DE,EC,CA,AD)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(CB),12)
Equal(LengthOfLine(DB),20)
Equal(MeasureOfAngle(DBC),60)
Parallelogram(DBCA)
PerpendicularBetweenLine(DE,CE)
Goal: Value(AreaOfQuadrilateral(DBCA))
Answer: 120*sqrt(3)
Theorem sequence:
parallelogram_area_formula_sine(1,DBCA)
Problem: As shown in the diagram, PQ=6-x, QR=6+x, SQ=3+x, TQ=3, RS is parallel to TP, SR∥PT. Find the length of line PQ.
问题：如图所示，PQ=6-x，QR=6+x，SQ=3+x，TQ=3，RS平行于TP，SR∥PT。求直线PQ的长度。
Construction (CDL):
Shape(QP,PT,TQ)
Shape(SQ,QR,RS)
Collinear(PQR)
Collinear(TQS)
Conditions (CDL):
Equal(LengthOfLine(PQ),6-x)
Equal(LengthOfLine(QR),6+x)
Equal(LengthOfLine(SQ),3+x)
Equal(LengthOfLine(TQ),3)
ParallelBetweenLine(RS,TP)
ParallelBetweenLine(SR,PT)
Goal: Value(LengthOfLine(PQ))
Answer: 6
Theorem sequence:
parallel_property_alternate_interior_angle(1,SR,PT)
parallel_property_alternate_interior_angle(2,SR,PT)
similar_triangle_judgment_aa(1,QPT,QRS)
similar_triangle_property_line_ratio(1,PTQ,RSQ)
similar_triangle_property_line_ratio(1,TQP,SQR)
Problem: As shown in the diagram, AE=x, AF=sqrt(174), CE=3*x+5, B is the center of ⊙B, AF is the tangent to circle B. Find the value of x.
问题：如图所示，AE=x，AF=sqrt(174)，CE=3*x+5，B是圆B的圆心，圆O的切线为AF。求x的值。
Construction (CDL):
Shape(BEF,BFC,CE)
Shape(EA,AF,BEF)
Shape(BCE,EC)
Collinear(AEC)
Cocircular(B,EFC)
Conditions (CDL):
Equal(LengthOfLine(AE),x)
Equal(LengthOfLine(AF),sqrt(174))
Equal(LengthOfLine(CE),3*x+5)
IsCentreOfCircle(B,B)
IsTangentOfCircle(AF,B)
Goal: Value(x)
Answer: 6
Theorem sequence:
line_addition(1,AE,EC)
circle_property_circular_power_tangent_and_segment_line(1,AF,AEC,B)
Problem: As shown in the diagram, QT=5*y, RT=x, ST=2*y+12, TP=5*x-28, QR and PS are opposite sides of the ▱ RQPS. Find the value of y.
问题：如图所示，QT=5*y，RT=x，ST=2*y+12，TP=5*x-28，QR和PS是▱RQPS的一组对边。求y的值。
Construction (CDL):
Shape(TR,RQ,QT)
Shape(TQ,QP,PT)
Shape(TP,PS,ST)
Shape(TS,SR,RT)
Collinear(RTP)
Collinear(STQ)
Conditions (CDL):
Equal(LengthOfLine(QT),5*y)
Equal(LengthOfLine(RT),x)
Equal(LengthOfLine(ST),2*y+12)
Equal(LengthOfLine(TP),5*x-28)
Parallelogram(RQPS)
Goal: Value(y)
Answer: 4
Theorem sequence:
parallelogram_property_diagonal_bisection(1,QPSR,T)
Problem: As shown in the diagram, AC=BC, DC=AD, ∠ADB=92°. Find the measure of ∠CBA.
问题：如图所示，AC=BC，DC=AD，∠ADB=92°。求∠CBA的大小。
Construction (CDL):
Shape(AD,DC,CA)
Shape(AC,CB,BA)
Collinear(DCB)
Conditions (CDL):
Equal(LengthOfLine(AC),LengthOfLine(BC))
Equal(LengthOfLine(DC),LengthOfLine(AD))
Equal(MeasureOfAngle(ADB),92)
Goal: Value(MeasureOfAngle(CBA))
Answer: 22
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_judgment_line_equal(1,CBA)
isosceles_triangle_property_angle_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,CBA)
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,ACB)
flat_angle(1,DCB)
angle_addition(1,DCA,ACB)
Problem: As shown in the diagram, the diameter of circle A is 10, the diameter of circle B is 30, the diameter of ⊙C is 10, AZ=CW, CW=2, A is the center of circle A, B is the center of ⊙B, the center of circle C is C. Find the length of line AC.
问题：如图所示，圆A的直径为10，圆B的直径为30，圆C的直径为10，AZ=CW，CW=2，A是⊙A的圆心，圆B的圆心为B，C是圆C的圆心。求直线AC的长度。
Construction (CDL):
Shape(AFE,EA,AZ,BFA)
Shape(AEG,BZG,ZA,AE)
Shape(AXF,BFZ,ZX)
Shape(AGX,BZG,XZ)
Shape(BGI,CYI,YB,BX,AGX)
Shape(BHF,AXF,XB,BY,CHY)
Shape(CHY,YW,BWH)
Shape(CYI,BIW,WY)
Shape(CID,DC,CW,BIW)
Shape(CDH,BWH,WC,CD)
Collinear(EAZXBYWCD)
Cocircular(A,XFEG)
Cocircular(B,ZGIWHF)
Cocircular(C,HYID)
Conditions (CDL):
Equal(DiameterOfCircle(A),10)
Equal(DiameterOfCircle(B),30)
Equal(DiameterOfCircle(C),10)
Equal(LengthOfLine(AZ),LengthOfLine(CW))
Equal(LengthOfLine(CW),2)
IsCentreOfCircle(A,A)
IsCentreOfCircle(B,B)
IsCentreOfCircle(C,C)
Goal: Value(LengthOfLine(AC))
Answer: 34
Theorem sequence:
line_addition(1,AZ,ZB)
line_addition(1,BW,WC)
circle_property_length_of_radius_and_diameter(1,B)
radius_of_circle_property_length_equal(1,BZ,B)
radius_of_circle_property_length_equal(1,BW,B)
line_addition(1,AB,BC)
Problem: As shown in the diagram, BX=s-7, WB=8, YB=6, ZB=2*t-6, YW and ZX are opposite sides of the parallelogram WYZX. Find the value of t.
问题：如图所示，BX=s-7，WB=8，YB=6，ZB=2*t-6，WX和YZ是平行四边形WYZX的一组对边。求t的值。
Construction (CDL):
Shape(BW,WY,YB)
Shape(BY,YZ,ZB)
Shape(BZ,ZX,XB)
Shape(BX,XW,WB)
Collinear(WBZ)
Collinear(XBY)
Conditions (CDL):
Equal(LengthOfLine(BX),s-7)
Equal(LengthOfLine(WB),8)
Equal(LengthOfLine(YB),6)
Equal(LengthOfLine(ZB),2*t-6)
Parallelogram(WYZX)
Goal: Value(t)
Answer: 7
Theorem sequence:
parallelogram_property_diagonal_bisection(1,WYZX,B)
Problem: As shown in the diagram, AD=3*x+11, BD=5*x-8, BD=AD, BE=2*y-1, CE=4*y-7, AC is parallel to DE. Find the value of x.
问题：如图所示，AD=3*x+11，BD=5*x-8，BD=AD，BE=2*y-1，CE=4*y-7，AC∥DE。求x的值。
Construction (CDL):
Shape(DA,AC,CE,ED)
Shape(BD,DE,EB)
Collinear(ADB)
Collinear(CEB)
Conditions (CDL):
Equal(LengthOfLine(AD),3*x+11)
Equal(LengthOfLine(BD),5*x-8)
Equal(LengthOfLine(BD),LengthOfLine(AD))
Equal(LengthOfLine(BE),2*y-1)
Equal(LengthOfLine(CE),4*y-7)
ParallelBetweenLine(AC,DE)
Goal: Value(x)
Answer: 19/2
Theorem sequence:
Problem: As shown in the diagram, RQ=5*x, RQ=SQ, RS=6*x-5, RS=RQ, SQ=3*x+10. Find the length of line QS.
问题：如图所示，RQ=5*x，RQ=SQ，RS=6*x-5，RS=RQ，SQ=3*x+10。求直线QS的长度。
Construction (CDL):
Shape(RQ,QS,SR)
Conditions (CDL):
Equal(LengthOfLine(RQ),5*x)
Equal(LengthOfLine(RQ),LengthOfLine(SQ))
Equal(LengthOfLine(RS),6*x-5)
Equal(LengthOfLine(RS),LengthOfLine(RQ))
Equal(LengthOfLine(SQ),3*x+10)
Goal: Value(LengthOfLine(QS))
Answer: 25
Theorem sequence:
Problem: As shown in the diagram, AF=10, AW=x, FW=14. Find the value of x.
问题：如图所示，AF=10，AW=x，FW=14。求x的值。
Construction (CDL):
Shape(FC,CA,AF)
Shape(EC,CF,FE)
Shape(BD,DC,CE,EB)
Shape(DW,WA,AC,CD)
Collinear(WAF)
Collinear(DCF)
Collinear(BEF)
Conditions (CDL):
Equal(LengthOfLine(AF),10)
Equal(LengthOfLine(AW),x)
Equal(LengthOfLine(FW),14)
Goal: Value(x)
Answer: 4
Theorem sequence:
line_addition(1,WA,AF)
Problem: As shown in the diagram, AB=10, AC=x, BC=y, ∠BAC=60°, AC⊥BC. Find the value of x.
问题：如图所示，AB=10，AC=x，BC=y，∠BAC=60°，AC垂直于BC。求x的值。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BC),y)
Equal(MeasureOfAngle(BAC),60)
PerpendicularBetweenLine(AC,BC)
Goal: Value(x)
Answer: 5
Theorem sequence:
triangle_property_angle_sum(1,BAC)
sine_theorem(1,ACB)
Problem: As shown in the diagram, MN=3*x-4, NQ=15.4, PN=2*y+5, PQ=11.1, RM=17.9, RP=20, RQ=3*z-3, ∠MRQ=38°, ∠NQP=83°, ∠QNM=33°, RM and PN are opposite sides of the ▱ MRPN. Find the measure of ∠PNM.
问题：如图所示，MN=3*x-4，NQ=15.4，PN=2*y+5，PQ=11.1，RM=17.9，RP=20，RQ=3*z-3，∠MRQ=38°，∠NQP=83°，∠QNM=33°，四边形MRPN是▱。求∠PNM的大小。
Construction (CDL):
Shape(MQ,QN,NM)
Shape(MR,RQ,QM)
Shape(QR,RP,PQ)
Shape(QP,PN,NQ)
Collinear(MQP)
Collinear(NQR)
Conditions (CDL):
Equal(LengthOfLine(MN),3*x-4)
Equal(LengthOfLine(NQ),15.4)
Equal(LengthOfLine(PN),2*y+5)
Equal(LengthOfLine(PQ),11.1)
Equal(LengthOfLine(RM),17.9)
Equal(LengthOfLine(RP),20)
Equal(LengthOfLine(RQ),3*z-3)
Equal(MeasureOfAngle(MRQ),38)
Equal(MeasureOfAngle(NQP),83)
Equal(MeasureOfAngle(QNM),33)
Parallelogram(MRPN)
Goal: Value(MeasureOfAngle(PNM))
Answer: 71
Theorem sequence:
parallel_property_alternate_interior_angle(1,RM,PN)
angle_addition(1,PNQ,QNM)
Problem: As shown in the diagram, ∠LMN=36°, the measure of arc BLN is 78, B is the center of circle B, ML is the tangent to ⊙B. Find the measure of ⌒BPL.
问题：如图所示，∠LMN=36°，⌒BLN的角度为78，⊙B的圆心为B，⊙O的切线为ML。求⌒BPL的角度。
Construction (CDL):
Shape(LM,MN,BLN)
Shape(BLN,NP,BPL)
Shape(BNP,PN)
Collinear(MNP)
Cocircular(B,LNP)
Conditions (CDL):
Equal(MeasureOfAngle(LMN),36)
Equal(MeasureOfArc(BLN),78)
IsCentreOfCircle(B,B)
IsTangentOfCircle(ML,B)
Goal: Value(MeasureOfArc(BPL))
Answer: 150
Theorem sequence:
circle_property_circular_power_tangent_and_segment_angle(1,ML,MNP,B)
Problem: As shown in the diagram, BA=9*x-1, CA=4*x+1, CA=BC, CB=5*x-1/2, ∠CBA and ∠BAC are the base angles of the isosceles triangle CBA. Find the length of line AC.
问题：如图所示，BA=9*x-1，CA=4*x+1，CA=BC，CB=5*x-1/2，CB和CA是等腰△CBA的腰。求直线AC的长度。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(BA),9*x-1)
Equal(LengthOfLine(CA),4*x+1)
Equal(LengthOfLine(CA),LengthOfLine(BC))
Equal(LengthOfLine(CB),5*x-1/2)
IsoscelesTriangle(CBA)
Goal: Value(LengthOfLine(AC))
Answer: 7
Theorem sequence:
Problem: As shown in the diagram, ∠ADC=65°, ∠BAD=x°, AB∥DC. Find the value of x.
问题：如图所示，∠ADC=65°，∠BAD=x°，AB平行于DC。求x的值。
Construction (CDL):
Shape(BA,AD,DC,CB)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),65)
Equal(MeasureOfAngle(BAD),x)
ParallelBetweenLine(AB,DC)
Goal: Value(x)
Answer: 115
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,DC)
Problem: As shown in the diagram, TY=4*x, WY=2*x+10, △XTY is mirror congruent to △XYW. Find the value of x.
问题：如图所示，TY=4*x，WY=2*x+10，△XTY镜像全等于△XYW。求x的值。
Construction (CDL):
Shape(XT,TY,YX)
Shape(XTY,YT)
Shape(XYW,WY)
Shape(XY,YW,WX)
Shape(XWT,TX,XW)
Conditions (CDL):
Equal(LengthOfLine(TY),4*x)
Equal(LengthOfLine(WY),2*x+10)
MirrorCongruentBetweenTriangle(XTY,XYW)
Goal: Value(x)
Answer: 5
Theorem sequence:
mirror_congruent_triangle_property_line_equal(1,XTY,XYW)