SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠DBA=60°, DA bisects ∠CDB, AB∥CD, AD⊥ND. Find the measure of ∠NDE.
问题：如图所示，∠DBA=60°，DA是∠CDB的角平分线，AB平行于CD，AD垂直于ND。求∠NDE的大小。
Construction (CDL):
Shape(AD,DB,BA)
Shape(BD,DN)
Shape(ND,DE)
Shape(CD,DA)
Collinear(CDE)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),60)
IsBisectorOfAngle(DA,CDB)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(AD,ND)
Goal: Value(MeasureOfAngle(NDE))
Answer: 30
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,DC,BA)
adjacent_complementary_angle(1,CDB,BDE)
angle_addition(1,CDA,ADB)
angle_addition(1,ADB,BDN)
angle_addition(1,BDN,NDE)
Problem: As shown in the diagram, ∠CBA=45°, ∠CDE=35°, AB∥CD. Find the measure of ∠BED.
问题：如图所示，∠CBA=45°，∠CDE=35°，AB平行于CD。求∠BED的大小。
Construction (CDL):
Shape(CB,BA)
Shape(EC,CD,DE)
Shape(BE,ED)
Collinear(CEB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),45)
Equal(MeasureOfAngle(CDE),35)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BED))
Answer: 80
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
triangle_property_angle_sum(1,ECD)
adjacent_complementary_angle(1,BED,DEC)
Problem: As shown in the diagram, AB=24, AC=32, BA is perpendicular to CA. Find the length of line BC.
问题：如图所示，AB=24，AC=32，BA垂直于CA。求直线BC的长度。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),24)
Equal(LengthOfLine(AC),32)
PerpendicularBetweenLine(BA,CA)
Goal: Value(LengthOfLine(BC))
Answer: 40
Theorem sequence:
right_triangle_judgment_angle(1,BAC)
right_triangle_property_pythagorean(1,BAC)
Problem: As shown in the diagram, triangle BAC is congruent to triangle EAF, ∠EFC=32°, triangle BAC is a right triangle. Find the measure of ∠AFE.
问题：如图所示，三角形BAC全等于三角形EAF，∠EFC=32°，△BAC为直角△。求∠AFE的大小。
Construction (CDL):
Shape(BA,AC,CB)
Shape(EA,AF,FE)
Shape(EF,FC,CE)
Collinear(CEA)
Collinear(BAF)
Conditions (CDL):
CongruentBetweenTriangle(BAC,EAF)
Equal(MeasureOfAngle(EFC),32)
RightTriangle(BAC)
Goal: Value(MeasureOfAngle(AFE))
Answer: 13
Theorem sequence:
congruent_triangle_property_line_equal(1,BAC,EAF)
isosceles_triangle_judgment_line_equal(1,AFC)
adjacent_complementary_angle(1,BAC,CAF)
isosceles_triangle_property_angle_equal(1,AFC)
triangle_property_angle_sum(1,AFC)
angle_addition(1,AFE,EFC)
Problem: As shown in the diagram, ∠HEB=60°, EB is parallel to FD. Find the measure of ∠HFD.
问题：如图所示，∠HEB=60°，EB∥FD。求∠HFD的大小。
Construction (CDL):
Shape(AE,EH)
Shape(HE,EB)
Shape(FE,EA)
Shape(BE,EF)
Shape(CF,FE)
Shape(EF,FD)
Shape(IF,FC)
Shape(DF,FI)
Collinear(AEB)
Collinear(CFD)
Collinear(HEFI)
Conditions (CDL):
Equal(MeasureOfAngle(HEB),60)
ParallelBetweenLine(EB,FD)
Goal: Value(MeasureOfAngle(HFD))
Answer: 60
Theorem sequence:
parallel_property_corresponding_angle(1,EB,FD,H)
Problem: As shown in the diagram, AB=3, BE=50, EC=30, DE∥AB. Find the length of line DE.
问题：如图所示，AB=3，BE=50，EC=30，DE∥AB。求直线DE的长度。
Construction (CDL):
Shape(AB,BE,ED,DA)
Shape(DE,EC,CD)
Collinear(BEC)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(BE),50)
Equal(LengthOfLine(EC),30)
ParallelBetweenLine(DE,AB)
Goal: Value(LengthOfLine(DE))
Answer: 9/8
Theorem sequence:
parallel_property_corresponding_angle(2,BA,ED,C)
similar_triangle_judgment_aa(1,DEC,ABC)
similar_triangle_property_line_ratio(1,DEC,ABC)
similar_triangle_property_line_ratio(1,CDE,CAB)
line_addition(1,BE,EC)
Problem: As shown in the diagram, ∠CEA=90°, ∠ECD=36°, AF∥CD. Find the measure of ∠BAE.
问题：如图所示，∠CEA=90°，∠ECD=36°，AF平行于CD。求∠BAE的大小。
Construction (CDL):
Shape(AE,EF,FA)
Shape(BF,FE)
Shape(EC,CD)
Shape(CE,EA)
Collinear(AFB)
Collinear(FEC)
Conditions (CDL):
Equal(MeasureOfAngle(CEA),90)
Equal(MeasureOfAngle(ECD),36)
ParallelBetweenLine(AF,CD)
Goal: Value(MeasureOfAngle(BAE))
Answer: 54
Theorem sequence:
parallel_property_alternate_interior_angle(2,AF,CD)
adjacent_complementary_angle(1,CEA,AEF)
triangle_property_angle_sum(1,AEF)
Problem: As shown in the diagram, ∠ECA=70°, ∠ECD=∠DCB, ∠ACB is the right angle of the right triangle ACB. Find the measure of ∠ACD.
问题：如图所示，∠ECA=70°，∠ECD=∠DCB，∠ACB是直角三角形ACB的直角。求∠ACD的大小。
Construction (CDL):
Shape(AF,FD,DA)
Shape(DF,FC,CD)
Shape(FE,EC,CF)
Shape(DC,CB,BD)
Collinear(ADB)
Collinear(AFC)
Conditions (CDL):
Equal(MeasureOfAngle(ECA),70)
Equal(MeasureOfAngle(ECD),MeasureOfAngle(DCB))
RightTriangle(ACB)
Goal: Value(MeasureOfAngle(ACD))
Answer: 10
Theorem sequence:
angle_addition(1,FCD,DCB)
angle_addition(1,ECF,FCD)
Problem: As shown in the diagram, ∠GED=104°, ED∥BF. Find the measure of ∠BFE.
问题：如图所示，∠GED=104°，ED平行于BF。求∠BFE的大小。
Construction (CDL):
Shape(AB,BF,FE,EA)
Shape(EF,FC,CD,DE)
Shape(GE,ED)
Collinear(AED)
Collinear(BFC)
Collinear(GEF)
Conditions (CDL):
Equal(MeasureOfAngle(GED),104)
ParallelBetweenLine(ED,BF)
Goal: Value(MeasureOfAngle(BFE))
Answer: 76
Theorem sequence:
adjacent_complementary_angle(1,GED,DEF)
parallel_property_alternate_interior_angle(1,ED,BF)
Problem: As shown in the diagram, ∠OPC=35°, OC is parallel to PD, PC is perpendicular to OC. Find the measure of ∠DPA.
问题：如图所示，∠OPC=35°，OC平行于PD，PC垂直于OC。求∠DPA的大小。
Construction (CDL):
Shape(OP,PC,CO)
Shape(DP,PA)
Collinear(OPA)
Conditions (CDL):
Equal(MeasureOfAngle(OPC),35)
ParallelBetweenLine(OC,PD)
PerpendicularBetweenLine(PC,OC)
Goal: Value(MeasureOfAngle(DPA))
Answer: 55
Theorem sequence:
triangle_property_angle_sum(1,OPC)
parallel_property_corresponding_angle(2,OC,PD,A)
Problem: As shown in the diagram, ∠AFD=65°, CF is parallel to EB. Find the measure of ∠EBF.
问题：如图所示，∠AFD=65°，CF平行于EB。求∠EBF的大小。
Construction (CDL):
Shape(CF,FA)
Shape(AF,FD)
Shape(BF,FC)
Shape(EB,BF)
Collinear(CFD)
Collinear(AFB)
Conditions (CDL):
Equal(MeasureOfAngle(AFD),65)
ParallelBetweenLine(CF,EB)
Goal: Value(MeasureOfAngle(EBF))
Answer: 115
Theorem sequence:
vertical_angle(1,AFD,BFC)
parallel_property_ipsilateral_internal_angle(1,BE,FC)
Problem: As shown in the diagram, ∠AGF=76°, ∠FED=32°, ∠MBG=∠HPF, FH bisects ∠EFG, DE∥FP. Find the measure of ∠PFH.
问题：如图所示，∠AGF=76°，∠FED=32°，∠MBG=∠HPF，FH是∠EFG的角平分线，DE∥FP。求∠PFH的大小。
Construction (CDL):
Shape(EF,FP,PC,CE)
Shape(FE,ED)
Shape(FH,HP,PF)
Shape(FG,GB,BH,HF)
Shape(AG,GF)
Shape(MB,BG)
Collinear(AGB)
Collinear(DEC)
Collinear(CPHBM)
Conditions (CDL):
Equal(MeasureOfAngle(AGF),76)
Equal(MeasureOfAngle(FED),32)
Equal(MeasureOfAngle(MBG),MeasureOfAngle(HPF))
IsBisectorOfAngle(FH,EFG)
ParallelBetweenLine(DE,FP)
Goal: Value(MeasureOfAngle(PFH))
Answer: 22
Theorem sequence:
parallel_judgment_corresponding_angle(1,BG,PF,M)
parallel_property_collinear_extend(2,BG,PF,A)
parallel_property_alternate_interior_angle(1,GA,PF)
parallel_property_alternate_interior_angle(2,DE,FP)
angle_addition(1,PFH,HFG)
angle_addition(1,EFP,PFH)
Problem: As shown in the diagram, ∠DCB=40°, BC is the angle bisector of ∠ABD, CD∥AB. Find the measure of ∠BDC.
问题：如图所示，∠DCB=40°，BC是∠ABD的角平分线，CD平行于AB。求∠BDC的大小。
Construction (CDL):
Shape(CB,BD,DC)
Shape(AB,BC)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),40)
IsBisectorOfAngle(BC,ABD)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(BDC))
Answer: 100
Theorem sequence:
parallel_property_alternate_interior_angle(1,CD,AB)
angle_addition(1,ABC,CBD)
parallel_property_ipsilateral_internal_angle(1,BA,DC)
Problem: As shown in the diagram, ∠BAE=100°, ∠ECD=120°, AB∥EH, EH is parallel to CD. Find the measure of ∠CEF.
问题：如图所示，∠BAE=100°，∠ECD=120°，AB平行于EH，EH∥CD。求∠CEF的大小。
Construction (CDL):
Shape(BA,AE)
Shape(AE,EH)
Shape(HE,EC)
Shape(EC,CD)
Shape(CE,EF)
Collinear(AEF)
Conditions (CDL):
Equal(MeasureOfAngle(BAE),100)
Equal(MeasureOfAngle(ECD),120)
ParallelBetweenLine(AB,EH)
ParallelBetweenLine(EH,CD)
Goal: Value(MeasureOfAngle(CEF))
Answer: 40
Theorem sequence:
parallel_property_corresponding_angle(2,AB,EH,F)
parallel_property_ipsilateral_internal_angle(1,EH,CD)
angle_addition(1,HEC,CEF)
Problem: As shown in the diagram, ∠AOC=80°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠AOC=80°，⊙O的圆心为O，AB是圆O的直径。求∠ABC的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),80)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠BAC=23°, the center of ⊙O is O, the tangent to ⊙O is DC. Find the measure of ∠CDB.
问题：如图所示，∠BAC=23°，O是圆O的圆心，DC是⊙O的切线。求∠CDB的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(OA,AC,CO)
Shape(OC,CB,BO)
Shape(BC,OCB)
Shape(CA,OAC)
Shape(CD,DB,OCB)
Collinear(AOBD)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),23)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 44
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
tangent_of_circle_property_perpendicular(1,DC,O,O)
triangle_property_angle_sum(1,OCD)
Problem: As shown in the diagram, ∠EMB=88°, ∠MCD=60°, MH is parallel to CD. Find the measure of ∠HMB.
问题：如图所示，∠EMB=88°，∠MCD=60°，MH∥CD。求∠HMB的大小。
Construction (CDL):
Shape(EM,MH)
Shape(HM,MB)
Shape(BM,MC)
Shape(MC,CD)
Collinear(EMC)
Conditions (CDL):
Equal(MeasureOfAngle(EMB),88)
Equal(MeasureOfAngle(MCD),60)
ParallelBetweenLine(MH,CD)
Goal: Value(MeasureOfAngle(HMB))
Answer: 28
Theorem sequence:
parallel_property_corresponding_angle(1,MH,CD,E)
angle_addition(1,EMH,HMB)
Problem: As shown in the diagram, ∠GBC=70°, AB is parallel to DE. Find the measure of ∠DEB.
问题：如图所示，∠GBC=70°，AB平行于DE。求∠DEB的大小。
Construction (CDL):
Shape(AB,BG)
Shape(GB,BC)
Shape(EB,BA)
Shape(CB,BE)
Shape(DE,EB)
Shape(BE,EF)
Shape(HE,ED)
Shape(FE,EH)
Collinear(ABC)
Collinear(DEF)
Collinear(GBEH)
Conditions (CDL):
Equal(MeasureOfAngle(GBC),70)
ParallelBetweenLine(AB,DE)
Goal: Value(MeasureOfAngle(DEB))
Answer: 110
Theorem sequence:
vertical_angle(1,GBC,EBA)
parallel_property_ipsilateral_internal_angle(1,ED,BA)
Problem: As shown in the diagram, ∠HED=110°, ED is parallel to BF. Find the measure of ∠BFE.
问题：如图所示，∠HED=110°，ED∥BF。求∠BFE的大小。
Construction (CDL):
Shape(AE,EH)
Shape(HE,ED)
Shape(FE,EA)
Shape(DE,EF)
Shape(BF,FE)
Shape(EF,FC)
Shape(GF,FB)
Shape(CF,FG)
Collinear(AED)
Collinear(BFC)
Collinear(HEFC)
Conditions (CDL):
Equal(MeasureOfAngle(HED),110)
ParallelBetweenLine(ED,BF)
Goal: Value(MeasureOfAngle(BFE))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,HED,DEF)
parallel_property_alternate_interior_angle(1,ED,BF)
Problem: As shown in the diagram, ∠BCD=45°, ∠EAB=90°, ∠FBA=63°. Find the measure of ∠EDH.
问题：如图所示，∠BCD=45°，∠EAB=90°，∠FBA=63°。求∠EDH的大小。
Construction (CDL):
Shape(AB,BD,DE,EA)
Shape(BC,CD,DB)
Shape(FB,BA)
Shape(ED,DH)
Shape(CB,BF)
Shape(HD,DC)
Collinear(FBDH)
Collinear(ABC)
Collinear(EDC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),45)
Equal(MeasureOfAngle(EAB),90)
Equal(MeasureOfAngle(FBA),63)
Goal: Value(MeasureOfAngle(EDH))
Answer: 72
Theorem sequence:
vertical_angle(1,FBA,DBC)
vertical_angle(1,EDH,CDF)
triangle_property_angle_sum(1,BCD)
Problem: As shown in the diagram, BC=24, BD=12, DE=40, BC is parallel to DE, AD⊥ED. Find the length of line AB.
问题：如图所示，BC=24，BD=12，DE=40，BC平行于DE，AD⊥ED。求直线AB的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DE,EC,CB)
Collinear(ACE)
Collinear(ABD)
Conditions (CDL):
Equal(LengthOfLine(BC),24)
Equal(LengthOfLine(BD),12)
Equal(LengthOfLine(DE),40)
ParallelBetweenLine(BC,DE)
PerpendicularBetweenLine(AD,ED)
Goal: Value(LengthOfLine(AB))
Answer: 18
Theorem sequence:
parallel_property_corresponding_angle(1,BC,DE,A)
parallel_property_corresponding_angle(2,ED,CB,A)
similar_triangle_judgment_aa(1,ABC,ADE)
similar_triangle_property_line_ratio(1,ABC,ADE)
similar_triangle_property_line_ratio(1,CAB,EAD)
line_addition(1,AB,BD)
Problem: As shown in the diagram, ∠EBF=40°, BF is parallel to CD, DB⊥CB. Find the measure of ∠CDB.
问题：如图所示，∠EBF=40°，BF∥CD，DB垂直于CB。求∠CDB的大小。
Construction (CDL):
Shape(EB,BF)
Shape(FB,BD)
Shape(BC,CD,DB)
Shape(BD,DH)
Collinear(EBC)
Collinear(CDH)
Conditions (CDL):
Equal(MeasureOfAngle(EBF),40)
ParallelBetweenLine(BF,CD)
PerpendicularBetweenLine(DB,CB)
Goal: Value(MeasureOfAngle(CDB))
Answer: 50
Theorem sequence:
angle_addition(1,EDF,FBD)
adjacent_complementary_angle(1,EBD,DBC)
parallel_property_alternate_interior_angle(1,BF,CD)
Problem: As shown in the diagram, ∠ABD=90°, ∠ABF=40°, BF∥GC. Find the measure of ∠GCB.
问题：如图所示，∠ABD=90°，∠ABF=40°，BF平行于GC。求∠GCB的大小。
Construction (CDL):
Shape(AB,BF,FA)
Shape(FB,BC,CE,EF)
Shape(EC,CD,DE)
Shape(GC,CB)
Shape(DC,CG)
Collinear(GCE)
Collinear(BCD)
Collinear(AFED)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),90)
Equal(MeasureOfAngle(ABF),40)
ParallelBetweenLine(BF,GC)
Goal: Value(MeasureOfAngle(GCB))
Answer: 50
Theorem sequence:
angle_addition(1,ABF,FBC)
parallel_property_alternate_interior_angle(1,BF,GC)
Problem: As shown in the diagram, AB=8, C is the midpoint of segment AB, M is the midpoint of segment AC, N bisects segment CB. Find the length of line MN.
问题：如图所示，AB=8，C平分线段AB，M平分线段AC，N平分线段CB。求直线MN的长度。
Construction (CDL):
Collinear(AMCNB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
IsMidpointOfLine(C,AB)
IsMidpointOfLine(M,AC)
IsMidpointOfLine(N,CB)
Goal: Value(LengthOfLine(MN))
Answer: 4
Theorem sequence:
line_addition(1,AM,MC)
line_addition(1,AM,MN)
line_addition(1,AC,CN)
line_addition(1,AN,NB)
Problem: As shown in the diagram, ∠CBD=60°, CF is parallel to AE, EF is perpendicular to CF. Find the measure of ∠BEF.
问题：如图所示，∠CBD=60°，CF∥AE，EF⊥CF。求∠BEF的大小。
Construction (CDL):
Shape(CB,BD)
Shape(DB,BF)
Shape(BE,EF,FB)
Shape(EB,BC)
Shape(AE,EB)
Collinear(DBE)
Collinear(CBF)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),60)
ParallelBetweenLine(CF,AE)
PerpendicularBetweenLine(EF,CF)
Goal: Value(MeasureOfAngle(BEF))
Answer: 30
Theorem sequence:
vertical_angle(1,CBD,FBE)
triangle_property_angle_sum(1,BEF)
Problem: As shown in the diagram, ∠ECF=90°, ∠FBQ=50°, AB∥MC. Find the measure of ∠MCE.
问题：如图所示，∠ECF=90°，∠FBQ=50°，AB平行于MC。求∠MCE的大小。
Construction (CDL):
Shape(PA,AE)
Shape(EA,AB)
Shape(AB,BF)
Shape(FB,BQ)
Shape(CA,AP)
Shape(AC,CB,BA)
Shape(QB,BC)
Shape(MC,CA)
Shape(BC,CN)
Collinear(PABQ)
Collinear(MCN)
Collinear(EAC)
Collinear(FBC)
Conditions (CDL):
Equal(MeasureOfAngle(ECF),90)
Equal(MeasureOfAngle(FBQ),50)
ParallelBetweenLine(AB,MC)
Goal: Value(MeasureOfAngle(MCE))
Answer: 40
Theorem sequence:
vertical_angle(1,FBQ,CBP)
triangle_property_angle_sum(1,ACB)
parallel_property_alternate_interior_angle(1,AB,MC)
Problem: As shown in the diagram, ∠EFD=120°, EB∥FD. Find the measure of ∠AEH.
问题：如图所示，∠EFD=120°，EB∥FD。求∠AEH的大小。
Construction (CDL):
Shape(AE,EH)
Shape(HE,EB)
Shape(FE,EA)
Shape(BE,EF)
Shape(CF,FE)
Shape(EF,FD)
Collinear(AEB)
Collinear(CFD)
Collinear(HEF)
Conditions (CDL):
Equal(MeasureOfAngle(EFD),120)
ParallelBetweenLine(EB,FD)
Goal: Value(MeasureOfAngle(AEH))
Answer: 60
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,EB,FD)
vertical_angle(1,AEH,BEF)
Problem: As shown in the diagram, ∠DBA=85°, ∠DEF=30°, AB is parallel to DF. Find the measure of ∠CFG.
问题：如图所示，∠DBA=85°，∠DEF=30°，AB平行于DF。求∠CFG的大小。
Construction (CDL):
Shape(DB,BA)
Shape(BD,DF,FC,CB)
Shape(DE,EF,FD)
Shape(CF,FG)
Shape(GF,FE)
Collinear(ABC)
Collinear(DFG)
Collinear(BDE)
Collinear(CFE)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),85)
Equal(MeasureOfAngle(DEF),30)
ParallelBetweenLine(AB,DF)
Goal: Value(MeasureOfAngle(CFG))
Answer: 55
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,DF)
adjacent_complementary_angle(1,GFE,EFD)
adjacent_complementary_angle(1,BDF,FDE)
adjacent_complementary_angle(1,CFG,GFE)
triangle_property_angle_sum(1,DEF)
Problem: As shown in the diagram, ∠DOB=22°, OB bisects ∠DOC, BO is perpendicular to AO. Find the measure of ∠COA.
问题：如图所示，∠DOB=22°，OB平分∠DOC，BO垂直于AO。求∠COA的大小。
Construction (CDL):
Shape(DO,OB)
Shape(BO,OC)
Shape(CO,OA)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),22)
IsBisectorOfAngle(OB,DOC)
PerpendicularBetweenLine(BO,AO)
Goal: Value(MeasureOfAngle(COA))
Answer: 68
Theorem sequence:
angle_addition(1,BOC,COA)
Problem: As shown in the diagram, ∠DEF=52°, I is the center of circle I, AD is the tangent to circle I, AF is the tangent to ⊙I, CE is the tangent to circle I. Find the measure of ∠CAB.
问题：如图所示，∠DEF=52°，⊙I的圆心为I，圆O的切线为AD，圆O的切线为AF，圆O的切线为CE。求∠CAB的大小。
Construction (CDL):
Shape(DB,BE,IDE)
Shape(ED,IDE)
Shape(DE,EF,FI,ID)
Shape(DI,IF,IFD)
Shape(FE,IEF)
Shape(EC,CF,IEF)
Shape(FA,AD,IFD)
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(CE,I)
Goal: Value(MeasureOfAngle(CAB))
Answer: 76
Theorem sequence:
arc_property_circumference_angle_external(1,IFD,E)
arc_property_center_angle(1,IFD,I)
tangent_of_circle_property_perpendicular(2,AD,I,I)
tangent_of_circle_property_perpendicular(1,AF,I,I)
quadrilateral_property_angle_sum(1,ADIF)
Problem: As shown in the diagram, ∠CDB=110°, the center of ⊙O is O, the diameter of ⊙O is AB, DB is the tangent to ⊙O, the tangent to circle O is DC. Find the measure of ∠BAC.
问题：如图所示，∠CDB=110°，O是圆O的圆心，AB是圆O的直径，DB是圆O的切线，DC是⊙O的切线。求∠BAC的大小。
Construction (CDL):
Shape(AC,CB,BA)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(CD,DB,OCB)
Shape(AB,OBA)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),110)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DB,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 35
Theorem sequence:
tangent_of_circle_property_length_equal(1,DC,DB,O)
isosceles_triangle_judgment_line_equal(1,DBC)
isosceles_triangle_property_angle_equal(1,DBC)
triangle_property_angle_sum(1,ACB)
triangle_property_angle_sum(1,BCD)
tangent_of_circle_property_perpendicular(2,DB,O,O)
angle_addition(1,DBC,CBA)
diameter_of_circle_property_right_angle(1,ACB,O)
Problem: As shown in the diagram, ∠OAB=35°, O is the center of ⊙O, PA is the tangent to circle O, PB is the tangent to circle O. Find the measure of ∠APB.
问题：如图所示，∠OAB=35°，⊙O的圆心为O，⊙O的切线为PA，⊙O的切线为PB。求∠APB的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(BA,OAB)
Shape(AB,BO,OA)
Shape(OB,OBC,CO)
Shape(AC,OCA)
Collinear(AOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),35)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(APB))
Answer: 70
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
tangent_of_circle_property_perpendicular(2,PB,O,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,ABO)
quadrilateral_property_angle_sum(1,APBO)
Problem: As shown in the diagram, ∠ABC=45°, ∠ADC=60°, DC∥BE. Find the measure of ∠CBE.
问题：如图所示，∠ABC=45°，∠ADC=60°，DC平行于BE。求∠CBE的大小。
Construction (CDL):
Shape(AD,DC,CA)
Shape(DB,BC,CD)
Shape(CB,BE)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),45)
Equal(MeasureOfAngle(ADC),60)
ParallelBetweenLine(DC,BE)
Goal: Value(MeasureOfAngle(CBE))
Answer: 15
Theorem sequence:
parallel_property_corresponding_angle(1,DC,BE,A)
angle_addition(1,DBC,CBE)
Problem: As shown in the diagram, Add(MeasureOfAngle(HBA)=MeasureOfAngle(DCB)), ∠EDF=121°, BC∥HD. Find the measure of ∠GHB.
问题：如图所示，Add(MeasureOfAngle(HBA)=MeasureOfAngle(DCB))，∠EDF=121°，BC∥HD。求∠GHB的大小。
Construction (CDL):
Shape(HB,BA)
Shape(BH,HD,DC,CB)
Shape(GH,HB)
Shape(CD,DE)
Shape(ED,DF)
Shape(DH,HI)
Shape(FD,DH)
Shape(IH,HG)
Collinear(ABC)
Collinear(GHDE)
Collinear(BHI)
Collinear(CDF)
Conditions (CDL):
Equal(Add(MeasureOfAngle(HBA),MeasureOfAngle(DCB)),90)
Equal(MeasureOfAngle(EDF),121)
ParallelBetweenLine(BC,HD)
Goal: Value(MeasureOfAngle(GHB))
Answer: 149
Theorem sequence:
parallel_property_collinear_extend(1,BC,HD,A)
parallel_property_collinear_extend(2,DH,BA,G)
vertical_angle(1,GDC,EDF)
parallel_property_ipsilateral_internal_angle(1,DH,CB)
parallel_property_ipsilateral_internal_angle(1,HG,BA)
Problem: As shown in the diagram, ∠BCA=50°, CF is the angle bisector of ∠ACM, BA∥CF. Find the measure of ∠ABC.
问题：如图所示，∠BCA=50°，CF平分∠ACM，BA平行于CF。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CF)
Shape(FC,CM)
Collinear(BCM)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
IsBisectorOfAngle(CF,ACM)
ParallelBetweenLine(BA,CF)
Goal: Value(MeasureOfAngle(ABC))
Answer: 65
Theorem sequence:
adjacent_complementary_angle(1,BCA,ACM)
angle_addition(1,ACF,FCM)
parallel_property_corresponding_angle(2,BA,CF,M)
Problem: As shown in the diagram, ∠BAP=40°, O is the center of circle O, PA is the tangent to circle O. Find the measure of ∠BOA.
问题：如图所示，∠BAP=40°，O是⊙O的圆心，PA是圆O的切线。求∠BOA的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(BA,OAB)
Shape(AB,BO,OA)
Shape(OB,OBA,AO)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(BAP),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
angle_addition(1,OAB,BAP)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,ABO)
Problem: As shown in the diagram, ∠CAP=35°, the center of circle O is O, the tangent to ⊙O is PC. Find the measure of ∠OPC.
问题：如图所示，∠CAP=35°，圆O的圆心为O，圆O的切线为PC。求∠OPC的大小。
Construction (CDL):
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(CO,OB,OBC)
Shape(BA,OAB)
Shape(BP,PC,OBC)
Collinear(AOBP)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAP),35)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PC,O)
Goal: Value(MeasureOfAngle(OPC))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
tangent_of_circle_property_perpendicular(2,PC,O,O)
triangle_property_angle_sum(1,COP)
Problem: As shown in the diagram, ∠CAB=35°, O is the center of circle O, the tangent to circle O is DC. Find the measure of ∠ODC.
问题：如图所示，∠CAB=35°，O是圆O的圆心，圆O的切线为DC。求∠ODC的大小。
Construction (CDL):
Shape(OC,OCA,CE,EO)
Shape(OE,EC,CO)
Shape(CE,EB,OBC)
Shape(EA,AB,BE)
Shape(BA,OAB)
Shape(BD,DC,OBC)
Collinear(OEBD)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(ODC))
Answer: 20
Theorem sequence:
tangent_of_circle_property_perpendicular(2,DC,O,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
triangle_property_angle_sum(1,COD)
Problem: As shown in the diagram, BC=4, ∠ABC=60°, BC⊥AC. Find the length of line AB.
问题：如图所示，BC=4，∠ABC=60°，BC垂直于AC。求直线AB的长度。
Construction (CDL):
Shape(CA,AD,BCD)
Shape(DB,BC,BCD)
Shape(BD,BDC,CB)
Collinear(ADB)
Cocircular(B,DC)
Conditions (CDL):
Equal(LengthOfLine(BC),4)
Equal(MeasureOfAngle(ABC),60)
PerpendicularBetweenLine(BC,AC)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
sine_theorem(1,ABC)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, ∠BAM=30°, O is the center of ⊙O, MA is the tangent to circle O. Find the measure of ∠ABO.
问题：如图所示，∠BAM=30°，圆O的圆心为O，圆O的切线为MA。求∠ABO的大小。
Construction (CDL):
Shape(AM,MB,OAB)
Shape(BA,OAB)
Shape(BO,OA,AB)
Shape(OC,CA,AO)
Shape(CB,OBC)
Shape(AC,OCA)
Collinear(BOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BAM),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(MA,O)
Goal: Value(MeasureOfAngle(ABO))
Answer: 60
Theorem sequence:
tangent_of_circle_property_perpendicular(1,MA,O,O)
angle_addition(1,OAB,BAM)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
Problem: As shown in the diagram, ∠ABC=100°, ∠CAB=50°, AC is parallel to BE. Find the measure of ∠CBE.
问题：如图所示，∠ABC=100°，∠CAB=50°，AC∥BE。求∠CBE的大小。
Construction (CDL):
Shape(CA,AB,BC)
Shape(CB,BE,EC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),100)
Equal(MeasureOfAngle(CAB),50)
ParallelBetweenLine(AC,BE)
Goal: Value(MeasureOfAngle(CBE))
Answer: 30
Theorem sequence:
triangle_property_angle_sum(1,CAB)
parallel_property_alternate_interior_angle(2,AC,BE)