SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=10, BF=12, AG bisects ∠FAB, the center of ⊙A is A, BA and CD are opposite sides of the ▱ ABCD. Find the length of line AE.
问题：如图所示，AB=10，BF=12，AG平分∠FAB，⊙A的圆心为A，AD和BC是平行四边形ABCD的一组对边。求直线AE的长度。
Construction (CDL):
Shape(AB,BH,HA)
Shape(AH,HF,FA)
Shape(GH,HB,ABG)
Shape(FH,HG,AGF)
Shape(AGF,GE,EF)
Shape(ABG,BE,EG)
Shape(FE,EC,CD,DF)
Collinear(AFD)
Collinear(BEC)
Collinear(AHGE)
Collinear(BHF)
Cocircular(A,BGF)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BF),12)
IsBisectorOfAngle(AG,FAB)
IsCentreOfCircle(A,A)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AE))
Answer: 16
Theorem sequence:
line_addition(1,BH,HF)
line_addition(1,AH,HE)
radius_of_circle_property_length_equal(1,AB,A)
radius_of_circle_property_length_equal(1,AF,A)
parallel_property_collinear_extend(3,AD,BC,F)
parallel_property_collinear_extend(3,CB,FA,E)
parallel_property_alternate_interior_angle(1,AF,BE)
isosceles_triangle_judgment_angle_equal(1,BEA)
parallelogram_judgment_parallel_and_equal(1,BEFA)
parallel_property_alternate_interior_angle(2,BA,EF)
isosceles_triangle_judgment_angle_equal(1,FAE)
kite_judgment_equal_and_equal(1,BEFA)
kite_judgment_equal_and_equal(1,ABEF)
kite_property_diagonal_perpendicular_bisection(1,ABEF,H)
kite_property_diagonal_perpendicular_bisection(1,BEFA,H)
right_triangle_judgment_angle(1,BHA)
right_triangle_property_pythagorean(1,BHA)
Problem: As shown in the diagram, ∠DEF=52°, the center of circle I is I, the tangent to ⊙I is AD, the tangent to circle I is AF, BD is the tangent to circle I, BE is the tangent to ⊙I, CE is the tangent to ⊙I, the tangent to circle I is CF. Find the measure of ∠FAD.
问题：如图所示，∠DEF=52°，⊙I的圆心为I，AD是圆O的切线，⊙O的切线为AF，BD是⊙O的切线，圆O的切线为BE，⊙O的切线为CE，CF是⊙O的切线。求∠FAD的大小。
Construction (CDL):
Shape(FA,AD,IFD)
Shape(DB,BE,IDE)
Shape(ED,ODE)
Shape(DE,EF,FI,ID)
Shape(DI,IF,IFD)
Shape(FE,IEF)
Shape(EC,CF,IEF)
Collinear(ADB)
Collinear(BEC)
Collinear(AFC)
Cocircular(I,DEF)
Conditions (CDL):
Equal(MeasureOfAngle(DEF),52)
IsCentreOfCircle(I,I)
IsTangentOfCircle(AD,I)
IsTangentOfCircle(AF,I)
IsTangentOfCircle(BD,I)
IsTangentOfCircle(BE,I)
IsTangentOfCircle(CE,I)
IsTangentOfCircle(CF,I)
Goal: Value(MeasureOfAngle(FAD))
Answer: 76
Theorem sequence:
arc_property_circumference_angle_external(1,IFD,E)
arc_property_center_angle(1,IFD,I)
tangent_of_circle_property_perpendicular(1,AF,I,I)
tangent_of_circle_property_perpendicular(2,AD,I,I)
quadrilateral_property_angle_sum(1,ADIF)
Problem: As shown in the diagram, CB=CD, ∠ABC=90°, ∠CAB=35°, ∠CDA=90°. Find the measure of ∠BCD.
问题：如图所示，CB=CD，∠ABC=90°，∠CAB=35°，∠CDA=90°。求∠BCD的大小。
Construction (CDL):
Shape(CD,DA,AC)
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(CB),LengthOfLine(CD))
Equal(MeasureOfAngle(ABC),90)
Equal(MeasureOfAngle(CAB),35)
Equal(MeasureOfAngle(CDA),90)
Goal: Value(MeasureOfAngle(BCD))
Answer: 110
Theorem sequence:
triangle_property_angle_sum(1,CAB)
mirror_congruent_triangle_judgment_hl(1,ABC,ACD)
angle_addition(1,BCA,ACD)
mirror_congruent_triangle_property_angle_equal(1,CAB,CDA)
Problem: As shown in the diagram, ∠FEA=90°, ∠FED=35°, CH is parallel to DE. Find the measure of ∠HCE.
问题：如图所示，∠FEA=90°，∠FED=35°，CH∥DE。求∠HCE的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DE,EC,CB)
Shape(DF,FE,ED)
Shape(HC,CE)
Shape(AC,CH)
Collinear(ABDF)
Collinear(BCH)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(FEA),90)
Equal(MeasureOfAngle(FED),35)
ParallelBetweenLine(CH,DE)
Goal: Value(MeasureOfAngle(HCE))
Answer: 55
Theorem sequence:
angle_addition(1,FED,DEC)
parallel_property_alternate_interior_angle(1,CH,DE)
Problem: As shown in the diagram, DE=14, D is the midpoint of segment OA, E is the midpoint of segment OB. Find the length of line AB.
问题：如图所示，DE=14，D平分线段OA，E平分线段OB。求直线AB的长度。
Construction (CDL):
Shape(OD,DE,EO)
Shape(DA,AB,BE,ED)
Collinear(ODA)
Collinear(OEB)
Conditions (CDL):
Equal(LengthOfLine(DE),14)
IsMidpointOfLine(D,OA)
IsMidpointOfLine(E,OB)
Goal: Value(LengthOfLine(AB))
Answer: 28
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,DE,OAB)
midsegment_of_triangle_property_length(1,DE,OAB)
Problem: As shown in the diagram, ∠CAM=45°, ∠DMA=75°, O is the center of circle O. Find the measure of ∠DBM.
问题：如图所示，∠CAM=45°，∠DMA=75°，O是圆O的圆心。求∠DBM的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(MC,CA,AM)
Shape(CM,MB,OBC)
Shape(BM,MD,DB)
Shape(DM,MA,OAD)
Shape(BD,ODB)
Collinear(AMB)
Collinear(CMD)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAM),45)
Equal(MeasureOfAngle(DMA),75)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBM))
Answer: 30
Theorem sequence:
adjacent_complementary_angle(1,DMA,AMC)
triangle_property_angle_sum(1,MCA)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_circumference_angle_external(1,OAD,B)
Problem: As shown in the diagram, ∠ACD=40°, ∠BCA=90°, BA∥CD. Find the measure of ∠ABC.
问题：如图所示，∠ACD=40°，∠BCA=90°，BA平行于CD。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),40)
Equal(MeasureOfAngle(BCA),90)
ParallelBetweenLine(BA,CD)
Goal: Value(MeasureOfAngle(ABC))
Answer: 50
Theorem sequence:
parallel_property_alternate_interior_angle(2,BA,CD)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠BDA=55°, BC is parallel to EF. Find the measure of ∠EGD.
问题：如图所示，∠BDA=55°，BC平行于EF。求∠EGD的大小。
Construction (CDL):
Shape(BD,DA)
Shape(AD,DC)
Shape(GD,DB)
Shape(CD,DG)
Shape(EG,GD)
Shape(DG,GF)
Shape(HG,GE)
Shape(FG,GH)
Collinear(ADGF)
Collinear(BDC)
Collinear(EGF)
Conditions (CDL):
Equal(MeasureOfAngle(BDA),55)
ParallelBetweenLine(BC,EF)
Goal: Value(MeasureOfAngle(EGD))
Answer: 55
Theorem sequence:
parallel_property_collinear_extend(3,BC,EF,D)
parallel_property_collinear_extend(3,FE,DB,G)
parallel_property_corresponding_angle(2,GE,DB,A)
Problem: As shown in the diagram, ∠CEA=90°, ∠HAB=136°, AB is parallel to CD, AB is parallel to ME, ME is parallel to CD. Find the measure of ∠ECD.
问题：如图所示，∠CEA=90°，∠HAB=136°，AB平行于CD，AB∥ME，ME平行于CD。求∠ECD的大小。
Construction (CDL):
Shape(HA,AB)
Shape(BA,AE)
Shape(CE,EM)
Shape(ME,EA)
Shape(EC,CD)
Collinear(HAE)
Conditions (CDL):
Equal(MeasureOfAngle(CEA),90)
Equal(MeasureOfAngle(HAB),136)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(AB,ME)
ParallelBetweenLine(ME,CD)
Goal: Value(MeasureOfAngle(ECD))
Answer: 46
Theorem sequence:
adjacent_complementary_angle(1,HAB,BAE)
parallel_property_alternate_interior_angle(1,AB,ME)
parallel_property_alternate_interior_angle(2,ME,CD)
angle_addition(1,CEM,MEA)
Problem: As shown in the diagram, AB=8, the radius of circle O is 5, the center of circle O is O, AH⊥OH. Find the length of line HC.
问题：如图所示，AB=8，⊙O的半径为5，O是⊙O的圆心，AH⊥OH。求直线HC的长度。
Construction (CDL):
Shape(AH,HO,OB,OBA)
Shape(OH,HB,BO)
Shape(CH,HA,OAC)
Shape(BH,HC,OCB)
Collinear(AHB)
Collinear(OHC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AH,OH)
Goal: Value(LengthOfLine(HC))
Answer: 2
Theorem sequence:
line_addition(1,AH,HB)
line_addition(1,OH,HC)
circle_property_chord_perpendicular_bisect_chord(1,O,OH,AB)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
right_triangle_judgment_angle(1,OHB)
right_triangle_property_pythagorean(1,OHB)
Problem: As shown in the diagram, ∠CBO=25°, the center of ⊙O is O, AD∥CO. Find the measure of ∠DAO.
问题：如图所示，∠CBO=25°，O是⊙O的圆心，AD平行于CO。求∠DAO的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(AO,OB,OBD,DA)
Shape(CO,OA,OAC)
Shape(BO,OC,CB)
Shape(BC,OCB)
Collinear(AOB)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),25)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AD,CO)
Goal: Value(MeasureOfAngle(DAO))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,BOC)
adjacent_complementary_angle(1,BOC,COA)
parallel_property_alternate_interior_angle(1,AD,CO)
Problem: As shown in the diagram, ∠ABP=120°, ∠RDA=50°, triangle EPR is mirror congruent to triangle CRP, AD is parallel to ER, EP is parallel to AB. Find the measure of ∠PCR.
问题：如图所示，∠ABP=120°，∠RDA=50°，△EPR镜像全等于△CRP，AD平行于ER，EP∥AB。求∠PCR的大小。
Construction (CDL):
Shape(AB,BP,PE,ER,RD,DA)
Shape(PR,RE,EP)
Shape(RP,PC,CR)
Collinear(DRC)
Collinear(BPC)
Conditions (CDL):
Equal(MeasureOfAngle(ABP),120)
Equal(MeasureOfAngle(RDA),50)
MirrorCongruentBetweenTriangle(EPR,CRP)
ParallelBetweenLine(AD,ER)
ParallelBetweenLine(EP,AB)
Goal: Value(MeasureOfAngle(PCR))
Answer: 95
Theorem sequence:
parallel_property_corresponding_angle(1,RE,DA,C)
parallel_property_corresponding_angle(2,BA,PE,C)
quadrilateral_property_angle_sum(1,REPC)
mirror_congruent_triangle_property_angle_equal(1,EPR,CRP)
Problem: As shown in the diagram, ∠EAF=25°, ∠FEA=40°, AB∥CD. Find the measure of ∠FCD.
问题：如图所示，∠EAF=25°，∠FEA=40°，AB∥CD。求∠FCD的大小。
Construction (CDL):
Shape(EA,AF,FE)
Shape(EF,FB)
Shape(BF,FC)
Shape(CF,FA)
Shape(FC,CD)
Collinear(AFB)
Collinear(EFC)
Conditions (CDL):
Equal(MeasureOfAngle(EAF),25)
Equal(MeasureOfAngle(FEA),40)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(FCD))
Answer: 65
Theorem sequence:
triangle_property_angle_sum(1,EAF)
vertical_angle(1,AFE,BFC)
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_ipsilateral_internal_angle(1,FB,CD)
Problem: As shown in the diagram, ∠CBA=40°, AB is parallel to CH, AB∥EF, CH is parallel to DF, CD⊥FD. Find the measure of ∠BCD.
问题：如图所示，∠CBA=40°，AB∥CH，AB∥EF，CH平行于DF，CD垂直于FD。求∠BCD的大小。
Construction (CDL):
Shape(ED,DC)
Shape(CD,DF)
Shape(HC,CD)
Shape(BC,CH)
Shape(CB,BA)
Collinear(EDF)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),40)
ParallelBetweenLine(AB,CH)
ParallelBetweenLine(AB,EF)
ParallelBetweenLine(CH,DF)
PerpendicularBetweenLine(CD,FD)
Goal: Value(MeasureOfAngle(BCD))
Answer: 130
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CH)
parallel_property_ipsilateral_internal_angle(1,CH,DF)
angle_addition(1,BCH,HCD)
Problem: As shown in the diagram, ∠HGB=100°, BA∥DC. Find the measure of ∠CHF.
问题：如图所示，∠HGB=100°，BA平行于DC。求∠CHF的大小。
Construction (CDL):
Shape(EG,GA)
Shape(AG,GH)
Shape(GH,HC)
Shape(CH,HF)
Shape(GH,HD)
Shape(DH,HG)
Shape(HG,GB)
Shape(BG,GE)
Collinear(EGHF)
Collinear(AGB)
Collinear(CHD)
Conditions (CDL):
Equal(MeasureOfAngle(HGB),100)
ParallelBetweenLine(BA,DC)
Goal: Value(MeasureOfAngle(CHF))
Answer: 80
Theorem sequence:
parallel_property_collinear_extend(3,BA,DC,G)
parallel_property_collinear_extend(3,CD,GB,H)
parallel_property_corresponding_angle(1,HD,GB,F)
adjacent_complementary_angle(1,CHF,FHD)
Problem: As shown in the diagram, ∠ACD=30°, the center of ⊙O is O, CA is the tangent to circle O, CD is the tangent to ⊙O. Find the measure of ∠DBA.
问题：如图所示，∠ACD=30°，⊙O的圆心为O，CA是圆O的切线，CD是圆O的切线。求∠DBA的大小。
Construction (CDL):
Shape(AC,CD,OAD)
Shape(DA,OAD)
Shape(AD,DO,OA)
Shape(BO,OD,DB)
Shape(BD,ODB)
Shape(AO,OB,OBA)
Collinear(AOB)
Cocircular(O,ADB)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CA,O)
IsTangentOfCircle(CD,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 75
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CA,O,O)
tangent_of_circle_property_perpendicular(2,CD,O,O)
quadrilateral_property_angle_sum(1,ACDO)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OB,O)
adjacent_complementary_angle(1,BOD,DOA)
isosceles_triangle_judgment_line_equal(1,ODB)
isosceles_triangle_property_angle_equal(1,ODB)
triangle_property_angle_sum(1,BOD)
Problem: As shown in the diagram, ∠BHE=56°, CD is parallel to EF. Find the measure of ∠AGD.
问题：如图所示，∠BHE=56°，CD平行于EF。求∠AGD的大小。
Construction (CDL):
Shape(CG,GA)
Shape(AG,GD)
Shape(HG,GC)
Shape(DG,GH)
Shape(EH,HG)
Shape(GH,HF)
Shape(BH,HE)
Shape(FH,HB)
Collinear(CGD)
Collinear(EHF)
Collinear(AGHB)
Conditions (CDL):
Equal(MeasureOfAngle(BHE),56)
ParallelBetweenLine(CD,EF)
Goal: Value(MeasureOfAngle(AGD))
Answer: 56
Theorem sequence:
parallel_property_collinear_extend(3,CD,EF,G)
parallel_property_collinear_extend(3,FE,GC,H)
parallel_property_corresponding_angle(1,HE,GC,B)
vertical_angle(1,AGD,HGC)
Problem: As shown in the diagram, ∠AGD=108°, CD is parallel to EF. Find the measure of ∠EHG.
问题：如图所示，∠AGD=108°，CD∥EF。求∠EHG的大小。
Construction (CDL):
Shape(CG,GA)
Shape(AG,GD)
Shape(HG,GC)
Shape(DG,GH)
Shape(EH,HG)
Shape(GH,HF)
Shape(BH,HE)
Shape(FH,HB)
Collinear(CGD)
Collinear(EHF)
Collinear(AGHB)
Conditions (CDL):
Equal(MeasureOfAngle(AGD),108)
ParallelBetweenLine(CD,EF)
Goal: Value(MeasureOfAngle(EHG))
Answer: 72
Theorem sequence:
parallel_property_collinear_extend(3,CD,EF,G)
parallel_property_collinear_extend(3,FE,DG,H)
parallel_property_corresponding_angle(1,GD,HF,A)
adjacent_complementary_angle(1,EHG,GHF)
Problem: As shown in the diagram, ∠BHE=55°, CD∥EF. Find the measure of ∠AGD.
问题：如图所示，∠BHE=55°，CD平行于EF。求∠AGD的大小。
Construction (CDL):
Shape(CG,GA)
Shape(AG,GD)
Shape(HG,GC)
Shape(DG,GH)
Shape(EH,HG)
Shape(GH,HF)
Shape(BH,HE)
Shape(FH,HB)
Collinear(CGD)
Collinear(EHF)
Collinear(AGHB)
Conditions (CDL):
Equal(MeasureOfAngle(BHE),55)
ParallelBetweenLine(CD,EF)
Goal: Value(MeasureOfAngle(AGD))
Answer: 55
Theorem sequence:
parallel_property_collinear_extend(3,CD,EF,G)
parallel_property_collinear_extend(3,FE,GC,H)
parallel_property_corresponding_angle(1,HE,GC,B)
vertical_angle(1,AGD,HGC)
Problem: As shown in the diagram, ∠CGA=100°, CD∥EF. Find the measure of ∠GHF.
问题：如图所示，∠CGA=100°，CD平行于EF。求∠GHF的大小。
Construction (CDL):
Shape(CG,GA)
Shape(AG,GD)
Shape(HG,GC)
Shape(DG,GH)
Shape(EH,HG)
Shape(GH,HF)
Shape(BH,HE)
Shape(FH,HB)
Collinear(CGD)
Collinear(EHF)
Collinear(AGHB)
Conditions (CDL):
Equal(MeasureOfAngle(CGA),100)
ParallelBetweenLine(CD,EF)
Goal: Value(MeasureOfAngle(GHF))
Answer: 80
Theorem sequence:
parallel_property_collinear_extend(3,CD,EF,G)
parallel_property_collinear_extend(3,FE,GC,H)
parallel_property_corresponding_angle(2,HE,GC,A)
adjacent_complementary_angle(1,EHG,GHF)
Problem: As shown in the diagram, ∠HGF=42°, EF is the angle bisector of ∠GEB, CD∥AB. Find the measure of ∠DFE.
问题：如图所示，∠HGF=42°，EF平分∠GEB，CD∥AB。求∠DFE的大小。
Construction (CDL):
Shape(CG,GH)
Shape(HG,GF)
Shape(GF,FI)
Shape(IF,FD)
Shape(DF,FE)
Shape(EF,FG,GE)
Shape(EG,GC)
Shape(AE,EG)
Shape(FE,EB)
Collinear(CGFD)
Collinear(AEB)
Collinear(HGE)
Collinear(IFE)
Conditions (CDL):
Equal(MeasureOfAngle(HGF),42)
IsBisectorOfAngle(EF,GEB)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(DFE))
Answer: 159
Theorem sequence:
parallel_property_collinear_extend(3,CD,AB,G)
parallel_property_collinear_extend(3,CD,AB,F)
parallel_property_collinear_extend(3,BA,DG,E)
parallel_property_collinear_extend(3,BA,DF,E)
angle_addition(1,GEF,FEB)
parallel_property_corresponding_angle(1,GD,EB,H)
parallel_property_corresponding_angle(1,FD,EB,I)
adjacent_complementary_angle(1,IFD,DFE)
Problem: As shown in the diagram, ∠ECD=40°, the center of circle O is O, CD is perpendicular to OD, OE⊥CE. Find the measure of ∠APB.
问题：如图所示，∠ECD=40°，O是⊙O的圆心，CD垂直于OD，OE垂直于CE。求∠APB的大小。
Construction (CDL):
Shape(AD,DC,OCA)
Shape(CD,DO,OE,EC)
Shape(CE,EB,OBC)
Shape(OD,DA,AP,PB,BE,EO)
Shape(PA,OAP)
Shape(BP,OPB)
Collinear(ADO)
Collinear(OEB)
Cocircular(O,CAPB)
Conditions (CDL):
Equal(MeasureOfAngle(ECD),40)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CD,OD)
PerpendicularBetweenLine(OE,CE)
Goal: Value(MeasureOfAngle(APB))
Answer: 70
Theorem sequence:
quadrilateral_property_angle_sum(1,CDOE)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,P)
Problem: As shown in the diagram, ∠BAE=60°, EA is parallel to ID, EF∥GH, ID∥CH, quadrilateral ABCD is a rectangle. Find the measure of ∠DCH.
问题：如图所示，∠BAE=60°，EA∥ID，EF∥GH，ID平行于CH，ABCD是矩形。求∠DCH的大小。
Construction (CDL):
Shape(BA,AE)
Shape(AB,BC,CD,DA)
Shape(CD,DI)
Shape(ID,DA)
Shape(FA,AD)
Shape(GC,CB)
Shape(DC,CH)
Collinear(EAF)
Collinear(GCH)
Conditions (CDL):
Equal(MeasureOfAngle(BAE),60)
ParallelBetweenLine(EA,ID)
ParallelBetweenLine(EF,GH)
ParallelBetweenLine(ID,CH)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(DCH))
Answer: 60
Theorem sequence:
angle_addition(1,DAB,BAE)
angle_addition(1,CDI,IDA)
parallel_property_ipsilateral_internal_angle(1,DI,AE)
parallel_property_alternate_interior_angle(2,ID,CH)
Problem: As shown in the diagram, ∠COE=50°, EF∥CO, EO⊥FO. Find the measure of ∠GFB.
问题：如图所示，∠COE=50°，EF∥CO，EO垂直于FO。求∠GFB的大小。
Construction (CDL):
Shape(CO,OE)
Shape(EO,OF,FE)
Shape(FO,OD)
Shape(HE,EF,FG,GH)
Shape(GF,FB)
Collinear(COD)
Collinear(AEFB)
Collinear(HEO)
Collinear(OFG)
Conditions (CDL):
Equal(MeasureOfAngle(COE),50)
ParallelBetweenLine(EF,CO)
PerpendicularBetweenLine(EO,FO)
Goal: Value(MeasureOfAngle(GFB))
Answer: 40
Theorem sequence:
parallel_property_alternate_interior_angle(1,EF,CO)
triangle_property_angle_sum(1,EOF)
vertical_angle(1,OFA,GFB)
Problem: As shown in the diagram, ∠ABC=58°, AD is parallel to BC, CA⊥BA. Find the measure of ∠DAC.
问题：如图所示，∠ABC=58°，AD∥BC，CA⊥BA。求∠DAC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DA,AC)
Shape(AC,CE)
Collinear(BCE)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),58)
ParallelBetweenLine(AD,BC)
PerpendicularBetweenLine(CA,BA)
Goal: Value(MeasureOfAngle(DAC))
Answer: 32
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
angle_addition(1,DAC,CAB)
Problem: As shown in the diagram, ∠ACE=50°, AE bisects ∠BAC, AB∥CD. Find the measure of ∠AED.
问题：如图所示，∠ACE=50°，AE平分∠BAC，AB平行于CD。求∠AED的大小。
Construction (CDL):
Shape(AC,CE,EA)
Shape(BA,AE)
Shape(AE,ED)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),50)
IsBisectorOfAngle(AE,BAC)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(AED))
Answer: 115
Theorem sequence:
parallel_property_collinear_extend(3,DC,BA,E)
parallel_property_alternate_interior_angle(1,EC,BA)
triangle_property_angle_sum(1,ACE)
adjacent_complementary_angle(1,CEA,AED)
Problem: As shown in the diagram, ∠FGD=110°, AB is parallel to CD. Find the measure of ∠AFE.
问题：如图所示，∠FGD=110°，AB∥CD。求∠AFE的大小。
Construction (CDL):
Shape(AF,FE)
Shape(EF,FB)
Shape(GF,FA)
Shape(BF,FG)
Shape(CG,GF)
Shape(FG,GD)
Shape(HG,GC)
Shape(DG,GH)
Collinear(AFB)
Collinear(CGD)
Collinear(EFGH)
Conditions (CDL):
Equal(MeasureOfAngle(FGD),110)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(AFE))
Answer: 70
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_collinear_extend(3,DC,BF,G)
parallel_property_corresponding_angle(1,FB,GD,E)
adjacent_complementary_angle(1,AFE,EFB)
Problem: As shown in the diagram, AB=40, BC=40, ∠CBF=20°, ∠EAB=40°, AE is parallel to FB. Find the length of line AC.
问题：如图所示，AB=40，BC=40，∠CBF=20°，∠EAB=40°，AE∥FB。求直线AC的长度。
Construction (CDL):
Shape(CB,BA,AC)
Shape(EA,AB)
Shape(FB,BA)
Shape(CB,BF)
Conditions (CDL):
Equal(LengthOfLine(AB),40)
Equal(LengthOfLine(BC),40)
Equal(MeasureOfAngle(CBF),20)
Equal(MeasureOfAngle(EAB),40)
ParallelBetweenLine(AE,FB)
Goal: Value(LengthOfLine(AC))
Answer: 40
Theorem sequence:
parallel_property_alternate_interior_angle(1,AE,FB)
angle_addition(1,CBF,FBA)
triangle_property_angle_sum(1,CBA)
isosceles_triangle_judgment_line_equal(1,BAC)
isosceles_triangle_property_angle_equal(1,BAC)
isosceles_triangle_judgment_angle_equal(1,CBA)
Problem: As shown in the diagram, ∠BCD=110°, ∠CBA=30°, AB∥CF, AB is parallel to DE, CF∥DE. Find the measure of ∠CDE.
问题：如图所示，∠BCD=110°，∠CBA=30°，AB平行于CF，AB平行于DE，CF∥DE。求∠CDE的大小。
Construction (CDL):
Shape(CB,BA)
Shape(BC,CF)
Shape(FC,CD)
Shape(CD,DE)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),110)
Equal(MeasureOfAngle(CBA),30)
ParallelBetweenLine(AB,CF)
ParallelBetweenLine(AB,DE)
ParallelBetweenLine(CF,DE)
Goal: Value(MeasureOfAngle(CDE))
Answer: 100
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CF)
angle_addition(1,BCF,FCD)
parallel_property_ipsilateral_internal_angle(1,CF,DE)
Problem: As shown in the diagram, ∠ABC=60°, ∠BCA=90°, ∠CAB=30°, ∠ICE=65°, FG∥DE. Find the measure of ∠BHF.
问题：如图所示，∠ABC=60°，∠BCA=90°，∠CAB=30°，∠ICE=65°，FG平行于DE。求∠BHF的大小。
Construction (CDL):
Shape(FH,HA)
Shape(AH,HI,IA)
Shape(AI,IG)
Shape(BH,HF)
Shape(IH,HB,BC,CI)
Shape(GI,IC)
Shape(DC,CB)
Shape(IC,CE)
Collinear(FHIG)
Collinear(DCE)
Collinear(AHB)
Collinear(AIC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),60)
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CAB),30)
Equal(MeasureOfAngle(ICE),65)
ParallelBetweenLine(FG,DE)
Goal: Value(MeasureOfAngle(BHF))
Answer: 35
Theorem sequence:
parallel_property_collinear_extend(3,FG,DE,I)
parallel_property_collinear_extend(3,ED,IF,C)
parallel_property_alternate_interior_angle(2,EC,IF)
quadrilateral_property_angle_sum(1,IHBC)
adjacent_complementary_angle(1,IHB,BHF)
Problem: As shown in the diagram, ∠CDB=50°, O is the center of circle O, DC is the tangent to ⊙O. Find the measure of ∠OAC.
问题：如图所示，∠CDB=50°，⊙O的圆心为O，⊙O的切线为DC。求∠OAC的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(OA,AC,CO)
Shape(BO,OC,OCB)
Shape(CD,DB,OCB)
Shape(AO,OB,OBA)
Collinear(AOBD)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 20
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DC,O,O)
triangle_property_angle_sum(1,OCD)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, triangle AOB is congruent to triangle COD, ∠BAO=110°, ∠COA=80°, ∠ODC=40°. Find the measure of ∠DOA.
问题：如图所示，△AOB与△COD是全等三角形，∠BAO=110°，∠COA=80°，∠ODC=40°。求∠DOA的大小。
Construction (CDL):
Shape(DC,CO,OD)
Shape(DO,OA)
Shape(AO,OB,BA)
Conditions (CDL):
CongruentBetweenTriangle(AOB,COD)
Equal(MeasureOfAngle(BAO),110)
Equal(MeasureOfAngle(COA),80)
Equal(MeasureOfAngle(ODC),40)
Goal: Value(MeasureOfAngle(DOA))
Answer: 50
Theorem sequence:
congruent_triangle_property_angle_equal(1,AOB,COD)
triangle_property_angle_sum(1,DCO)
angle_addition(1,COD,DOA)
Problem: As shown in the diagram, ∠BED=40°, EA is the angle bisector of ∠CEB, AB∥CD. Find the measure of ∠BAE.
问题：如图所示，∠BED=40°，EA平分∠CEB，AB∥CD。求∠BAE的大小。
Construction (CDL):
Shape(CE,EA)
Shape(AE,EB,BA)
Shape(BE,ED)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(BED),40)
IsBisectorOfAngle(EA,CEB)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BAE))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,CEB,BED)
angle_addition(1,CEA,AEB)
parallel_property_collinear_extend(3,DC,BA,E)
parallel_property_alternate_interior_angle(1,EC,BA)
Problem: As shown in the diagram, ∠BCD=120°, the center of ⊙O is O, the tangent to circle O is PD. Find the measure of ∠DPA.
问题：如图所示，∠BCD=120°，O是圆O的圆心，PD是圆O的切线。求∠DPA的大小。
Construction (CDL):
Shape(DP,PA,ODA)
Shape(AD,ODA)
Shape(DO,OB,BC,CD)
Shape(AO,OD,DA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(PAOB)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),120)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PD,O)
Goal: Value(MeasureOfAngle(DPA))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_internal(1,ODB,A)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
tangent_of_circle_property_perpendicular(1,PD,O,O)
angle_addition(1,ODA,ADP)
adjacent_complementary_angle(1,PAD,DAO)
triangle_property_angle_sum(1,ADP)
Problem: As shown in the diagram, ∠CBH=60°, ∠FDC=45°, ∠GAF=30°, ∠HEG=45°, DE is parallel to CB. Find the measure of ∠EHB.
问题：如图所示，∠CBH=60°，∠FDC=45°，∠GAF=30°，∠HEG=45°，DE平行于CB。求∠EHB的大小。
Construction (CDL):
Shape(DC,CF,FD)
Shape(AF,FG,GA)
Shape(FC,CH,HG,GF)
Shape(FH,HE,EG)
Shape(HC,CB,BH)
Shape(EH,HB)
Collinear(DFGE)
Collinear(AFC)
Collinear(AGHB)
Collinear(CHE)
Conditions (CDL):
Equal(MeasureOfAngle(CBH),60)
Equal(MeasureOfAngle(FDC),45)
Equal(MeasureOfAngle(GAF),30)
Equal(MeasureOfAngle(HEG),45)
ParallelBetweenLine(DE,CB)
Goal: Value(MeasureOfAngle(EHB))
Answer: 105
Theorem sequence:
parallel_property_alternate_interior_angle(2,DE,CB)
triangle_property_angle_sum(1,HCB)
adjacent_complementary_angle(1,EHB,BHC)
Problem: As shown in the diagram, ∠COA=100°, O is the center of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠COA=100°，O是⊙O的圆心。求∠ABC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(AO,OC,CD,DA)
Shape(AB,BC,CO,OA)
Shape(CB,OBC)
Shape(DC,OCD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)