SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠GFD=50°, CD is parallel to AB. Find the measure of ∠AEF.
问题：如图所示，∠GFD=50°，CD∥AB。求∠AEF的大小。
Construction (CDL):
Shape(CF,FG)
Shape(GF,FD)
Shape(EF,FC)
Shape(DF,FE)
Shape(AE,EF)
Shape(FE,EB)
Shape(HE,EA)
Shape(BE,EH)
Collinear(CFD)
Collinear(AEB)
Collinear(GFEH)
Conditions (CDL):
Equal(MeasureOfAngle(GFD),50)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(AEF))
Answer: 130
Theorem sequence:
parallel_property_collinear_extend(3,CD,AB,F)
parallel_property_collinear_extend(3,BA,DF,E)
parallel_property_corresponding_angle(1,FD,EB,G)
adjacent_complementary_angle(1,AEF,FEB)
Problem: As shown in the diagram, ∠EFN=∠NCD, ∠FBN=70°, ∠MEF=∠CDM, ∠NCD=80°, MN is the angle bisector of ∠DME, NM bisects ∠FNC. Find the measure of ∠BNM.
问题：如图所示，∠EFN=∠NCD，∠FBN=70°，∠MEF=∠CDM，∠NCD=80°，MN是∠DME的角平分线，NM平分∠FNC。求∠BNM的大小。
Construction (CDL):
Shape(AE,EM,MA)
Shape(EF,FN,NM,ME)
Shape(MN,NC,CD,DM)
Shape(FB,BN,NF)
Collinear(AEFB)
Collinear(AMD)
Collinear(BNC)
Conditions (CDL):
Equal(MeasureOfAngle(EFN),MeasureOfAngle(NCD))
Equal(MeasureOfAngle(FBN),70)
Equal(MeasureOfAngle(MEF),MeasureOfAngle(CDM))
Equal(MeasureOfAngle(NCD),80)
IsBisectorOfAngle(MN,DME)
IsBisectorOfAngle(NM,FNC)
Goal: Value(MeasureOfAngle(BNM))
Answer: 95
Theorem sequence:
adjacent_complementary_angle(1,EFN,NFB)
triangle_property_angle_sum(1,NFB)
adjacent_complementary_angle(1,BNF,FNC)
angle_addition(1,FNM,MNC)
angle_addition(1,BNF,FNM)
Problem: As shown in the diagram, AB=7, AC=9, BC=12, triangle ABC is mirror congruent to triangle DBC. Find the length of line BD.
问题：如图所示，AB=7，AC=9，BC=12，三角形ABC与三角形DBC是镜像全等三角形。求直线BD的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(DOB)
Conditions (CDL):
Equal(LengthOfLine(AB),7)
Equal(LengthOfLine(AC),9)
Equal(LengthOfLine(BC),12)
MirrorCongruentBetweenTriangle(ABC,DBC)
Goal: Value(LengthOfLine(BD))
Answer: 9
Theorem sequence:
mirror_congruent_triangle_property_line_equal(1,BCA,CDB)
Problem: As shown in the diagram, triangle BAC is congruent to triangle DAE, ∠ACB=30°, ∠CBA=70°, ∠DAC=35°. Find the measure of ∠CAE.
问题：如图所示，三角形BAC与三角形DAE是全等三角形，∠ACB=30°，∠CBA=70°，∠DAC=35°。求∠CAE的大小。
Construction (CDL):
Shape(BA,AD,DB)
Shape(DA,AF,FD)
Shape(DF,FC,CD)
Shape(FA,AE,EF)
Collinear(BDC)
Collinear(AFC)
Collinear(DFE)
Conditions (CDL):
CongruentBetweenTriangle(BAC,DAE)
Equal(MeasureOfAngle(ACB),30)
Equal(MeasureOfAngle(CBA),70)
Equal(MeasureOfAngle(DAC),35)
Goal: Value(MeasureOfAngle(CAE))
Answer: 45
Theorem sequence:
triangle_property_angle_sum(1,BAC)
congruent_triangle_property_angle_equal(1,ACB,AED)
angle_addition(1,DAF,FAE)
Problem: As shown in the diagram, triangle ABC is congruent to triangle DEF, BC=5, BF=7. Find the length of line EC.
问题：如图所示，三角形ABC全等于三角形DEF，BC=5，BF=7。求直线EC的长度。
Construction (CDL):
Shape(AB,BE,EM,MA)
Shape(ME,EC,CM)
Shape(DM,MC,CF,FD)
Collinear(AMC)
Collinear(EMD)
Collinear(BECF)
Conditions (CDL):
CongruentBetweenTriangle(ABC,DEF)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(BF),7)
Goal: Value(LengthOfLine(EC))
Answer: 3
Theorem sequence:
line_addition(1,BC,CF)
congruent_triangle_property_line_equal(1,ABC,DEF)
line_addition(1,EC,CF)
Problem: As shown in the diagram, AB=6, AC=BC, AD is the angle bisector of ∠CAE, BC is perpendicular to AC, DE⊥BE. Find the perimeter of △EBD.
问题：如图所示，AB=6，AC=BC，AD平分∠CAE，BC垂直于AC，DE⊥BE。求三角形EBD的周长。
Construction (CDL):
Shape(CA,AD,DC)
Shape(DA,AE,ED)
Shape(DE,EB,BD)
Collinear(CDB)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AC),LengthOfLine(BC))
IsBisectorOfAngle(AD,CAE)
PerpendicularBetweenLine(BC,AC)
PerpendicularBetweenLine(DE,BE)
Goal: Value(PerimeterOfTriangle(EBD))
Answer: 6
Theorem sequence:
adjacent_complementary_angle(1,AED,DEB)
mirror_congruent_triangle_judgment_aas(3,DCA,DAE)
mirror_congruent_triangle_property_line_equal(1,ADC,AED)
mirror_congruent_triangle_property_line_equal(1,DCA,DAE)
line_addition(1,CD,DB)
line_addition(1,AE,EB)
triangle_perimeter_formula(1,DEB)
Problem: As shown in the diagram, Add(MeasureOfAngle(DCA)=20), AE is the angle bisector of ∠CAB, CD is perpendicular to FD. Find the measure of ∠EFD.
问题：如图所示，Add(MeasureOfAngle(DCA)=20)，AE平分∠CAB，CD⊥FD。求∠EFD的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(EF,FD,DE)
Shape(CD,DF)
Collinear(BEDC)
Collinear(AEF)
Conditions (CDL):
Equal(Add(MeasureOfAngle(DCA),20),MeasureOfAngle(ABE))
IsBisectorOfAngle(AE,CAB)
PerpendicularBetweenLine(CD,FD)
Goal: Value(MeasureOfAngle(EFD))
Answer: 10
Theorem sequence:
angle_addition(1,CAE,EAB)
triangle_property_angle_sum(1,ABC)
adjacent_complementary_angle(1,BEA,AEC)
triangle_property_angle_sum(1,ABE)
adjacent_complementary_angle(1,CDF,FDB)
adjacent_complementary_angle(1,AEC,CEF)
triangle_property_angle_sum(1,EFD)
Problem: As shown in the diagram, AB=6, AD=4, BD=5, triangle DAB is mirror congruent to triangle CAB. Find the length of line BC.
问题：如图所示，AB=6，AD=4，BD=5，△DAB镜像全等于△CAB。求直线BC的长度。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EA,AB,BE)
Shape(CE,EB,BC)
Collinear(DEB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AD),4)
Equal(LengthOfLine(BD),5)
MirrorCongruentBetweenTriangle(DAB,CAB)
Goal: Value(LengthOfLine(BC))
Answer: 4
Theorem sequence:
mirror_congruent_triangle_property_line_equal(1,BDA,ABC)
Problem: As shown in the diagram, AB=6, OD=4, O is the center of ⊙O, AD is perpendicular to OD. Find the length of line DC.
问题：如图所示，AB=6，OD=4，O是⊙O的圆心，AD垂直于OD。求直线DC的长度。
Construction (CDL):
Shape(OA,AD,DO)
Shape(OD,DB,BO)
Shape(CD,DA,OAC)
Shape(BD,DC,OCB)
Shape(AO,OB,OBA)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(OD),4)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
Goal: Value(LengthOfLine(DC))
Answer: 1
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OD,AB)
line_addition(1,AD,DB)
right_triangle_judgment_angle(1,ADO)
right_triangle_property_pythagorean(1,ADO)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
line_addition(1,OD,DC)
Problem: As shown in the diagram, ∠EAB=30°, O is the center of circle O, the diameter of ⊙O is BD. Find the measure of ∠DBC.
问题：如图所示，∠EAB=30°，O是圆O的圆心，BD是⊙O的直径。求∠DBC的大小。
Construction (CDL):
Shape(AB,BO,OE,EA)
Shape(EO,OB,BC,CE)
Shape(DE,EC,CD)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AE,ED,ODA)
Collinear(BOED)
Collinear(CEA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(MeasureOfAngle(DBC))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
diameter_of_circle_property_right_angle(1,BCD,O)
triangle_property_angle_sum(1,BCD)
Problem: As shown in the diagram, ∠CAB=120°, BO is the angle bisector of ∠ABC, CO bisects ∠BCA. Find the measure of ∠COB.
问题：如图所示，∠CAB=120°，BO平分∠ABC，CO平分∠BCA。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),120)
IsBisectorOfAngle(BO,ABC)
IsBisectorOfAngle(CO,BCA)
Goal: Value(MeasureOfAngle(COB))
Answer: 150
Theorem sequence:
triangle_property_angle_sum(1,ABC)
angle_addition(1,ABO,OBC)
angle_addition(1,BCO,OCA)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, AB=8, ∠ABC=∠CDA, AC is the angle bisector of ∠DAB. Find the length of line AD.
问题：如图所示，AB=8，∠ABC=∠CDA，AC平分∠DAB。求直线AD的长度。
Construction (CDL):
Shape(DA,AC,CD)
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(MeasureOfAngle(ABC),MeasureOfAngle(CDA))
IsBisectorOfAngle(AC,DAB)
Goal: Value(LengthOfLine(AD))
Answer: 8
Theorem sequence:
mirror_congruent_triangle_judgment_aas(3,CDA,CAB)
mirror_congruent_triangle_property_line_equal(1,CDA,CAB)
Problem: As shown in the diagram, ∠ABC=80°, ∠BCA=60°, BE bisects ∠ABC, CF is the angle bisector of ∠BCA. Find the measure of ∠EDC.
问题：如图所示，∠ABC=80°，∠BCA=60°，BE是∠ABC的角平分线，CF平分∠BCA。求∠EDC的大小。
Construction (CDL):
Shape(AF,FD,DE,EA)
Shape(FB,BD,DF)
Shape(DB,BC,CD)
Shape(DC,CE,ED)
Collinear(FDC)
Collinear(BDE)
Collinear(AFB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),80)
Equal(MeasureOfAngle(BCA),60)
IsBisectorOfAngle(BE,ABC)
IsBisectorOfAngle(CF,BCA)
Goal: Value(MeasureOfAngle(EDC))
Answer: 70
Theorem sequence:
angle_addition(1,FBD,DBC)
angle_addition(1,BCD,DCE)
triangle_property_angle_sum(1,DBC)
adjacent_complementary_angle(1,EDC,CDB)
Problem: As shown in the diagram, ∠DAB=65°, O is the center of circle O, DE is perpendicular to OE. Find the measure of ∠COA.
问题：如图所示，∠DAB=65°，⊙O的圆心为O，DE垂直于OE。求∠COA的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AE,ED)
Shape(CE,EA,OAC)
Shape(OE,EC,CO)
Shape(BO,OC,OCB)
Shape(DE,EO,OB,OBD)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),65)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(DE,OE)
Goal: Value(MeasureOfAngle(COA))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,AED,DEO)
triangle_property_angle_sum(1,DAE)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, ∠BCA=35°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=35°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OA,AE,EO)
Shape(BE,EA,OAB)
Shape(EB,BC,CE)
Shape(CB,OBC)
Shape(AO,OE,EC,OCA)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, Add(MeasureOfAngle(BAD)=MeasureOfAngle(CBA)), CO is the angle bisector of ∠DCB, DO is the angle bisector of ∠ADC. Find the measure of ∠COD.
问题：如图所示，Add(MeasureOfAngle(BAD)=MeasureOfAngle(CBA))，CO平分∠DCB，DO是∠ADC的角平分线。求∠COD的大小。
Construction (CDL):
Shape(AD,DO,OC,CB,BA)
Shape(OD,DC,CO)
Conditions (CDL):
Equal(Add(MeasureOfAngle(BAD),MeasureOfAngle(CBA)),200)
IsBisectorOfAngle(CO,DCB)
IsBisectorOfAngle(DO,ADC)
Goal: Value(MeasureOfAngle(COD))
Answer: 100
Theorem sequence:
quadrilateral_property_angle_sum(1,ADCB)
angle_addition(1,ADO,ODC)
angle_addition(1,DCO,OCB)
triangle_property_angle_sum(1,ODC)
Problem: As shown in the diagram, ∠ACB=100°, ∠CBD=∠DEC, ∠DAE=20°. Find the measure of ∠EDA.
问题：如图所示，∠ACB=100°，∠CBD=∠DEC，∠DAE=20°。求∠EDA的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EC,CD,DE)
Shape(DC,CB,BD)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),100)
Equal(MeasureOfAngle(CBD),MeasureOfAngle(DEC))
Equal(MeasureOfAngle(DAE),20)
Goal: Value(MeasureOfAngle(EDA))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,ACB)
adjacent_complementary_angle(1,AED,DEC)
triangle_property_angle_sum(1,AED)
Problem: As shown in the diagram, ∠ABC=48°, ∠DCE=70°, BE is the altitude of triangle BCA, AD bisects ∠EAB. Find the measure of ∠DFB.
问题：如图所示，∠ABC=48°，∠DCE=70°，BE是△BCA的高，AD是∠EAB的角平分线。求∠DFB的大小。
Construction (CDL):
Shape(CE,EF,FD,DC)
Shape(EA,AF,FE)
Shape(FA,AB,BF)
Shape(FB,BD,DF)
Collinear(EFB)
Collinear(AFD)
Collinear(CEA)
Collinear(CDB)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),48)
Equal(MeasureOfAngle(DCE),70)
IsAltitudeOfTriangle(BE,BCA)
IsBisectorOfAngle(AD,EAB)
Goal: Value(MeasureOfAngle(DFB))
Answer: 59
Theorem sequence:
triangle_property_angle_sum(1,ABC)
angle_addition(1,EAF,FAB)
triangle_property_angle_sum(1,ABE)
triangle_property_angle_sum(1,ABF)
adjacent_complementary_angle(1,DFB,BFA)
Problem: As shown in the diagram, AB=10, BC=18, DE is the perpendicular bisector of CA. Find the perimeter of triangle ABD.
问题：如图所示，AB=10，BC=18，DE垂直平分AB。求三角形ABD的周长。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),18)
IsPerpendicularBisectorOfLine(DE,CA)
Goal: Value(PerimeterOfTriangle(ABD))
Answer: 28
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,CA)
line_addition(1,BD,DC)
triangle_perimeter_formula(1,ABD)
Problem: As shown in the diagram, ∠ADE=120°. Find the measure of ∠ABC.
问题：如图所示，∠ADE=120°。求∠ABC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(AD,DE)
Collinear(CDE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),120)
Goal: Value(MeasureOfAngle(ABC))
Answer: 120
Theorem sequence:
adjacent_complementary_angle(1,CDA,ADE)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, ∠EAF=50°, BE bisects ∠ABC, CF is the angle bisector of ∠BCA. Find the measure of ∠CDB.
问题：如图所示，∠EAF=50°，BE是∠ABC的角平分线，CF平分∠BCA。求∠CDB的大小。
Construction (CDL):
Shape(FB,BD,DF)
Shape(DB,BC,CD)
Shape(DC,CE,ED)
Shape(DE,EA,AF)
Collinear(AFB)
Collinear(AEC)
Collinear(BDE)
Collinear(CDF)
Conditions (CDL):
Equal(MeasureOfAngle(EAF),50)
IsBisectorOfAngle(BE,ABC)
IsBisectorOfAngle(CF,BCA)
Goal: Value(MeasureOfAngle(CDB))
Answer: 115
Theorem sequence:
triangle_property_angle_sum(1,ABC)
angle_addition(1,FBD,DBC)
angle_addition(1,BCD,DCE)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, ∠CBF=25°, ∠EAC=45°, AE∥DC, DC∥BF. Find the measure of ∠BCA.
问题：如图所示，∠CBF=25°，∠EAC=45°，AE∥DC，DC∥BF。求∠BCA的大小。
Construction (CDL):
Shape(EA,AC)
Shape(BC,CD)
Shape(DC,CA)
Shape(CB,BF)
Conditions (CDL):
Equal(MeasureOfAngle(CBF),25)
Equal(MeasureOfAngle(EAC),45)
ParallelBetweenLine(AE,DC)
ParallelBetweenLine(DC,BF)
Goal: Value(MeasureOfAngle(BCA))
Answer: 70
Theorem sequence:
parallel_property_alternate_interior_angle(1,AE,DC)
parallel_property_alternate_interior_angle(2,DC,BF)
angle_addition(1,BCD,DCA)
Problem: As shown in the diagram, AB=AC, ∠DAE=36°, DE perpendicular bisects AB. Find the measure of ∠CDB.
问题：如图所示，AB=AC，∠DAE=36°，DE垂直平分AB。求∠CDB的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EB,BD,DE)
Shape(DB,BC,CD)
Collinear(AEB)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(DAE),36)
IsPerpendicularBisectorOfLine(DE,AB)
Goal: Value(MeasureOfAngle(CDB))
Answer: 72
Theorem sequence:
triangle_property_angle_sum(1,ABC)
perpendicular_bisector_property_distance_equal(1,DE,AB)
isosceles_triangle_judgment_line_equal(1,DAB)
isosceles_triangle_property_angle_equal(1,DAB)
angle_addition(1,EBD,DBC)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, ∠BCA=90°, ∠CDE=150°, DE is parallel to AB. Find the measure of ∠CAB.
问题：如图所示，∠BCA=90°，∠CDE=150°，DE∥AB。求∠CAB的大小。
Construction (CDL):
Shape(AB,BD,DC,CA)
Shape(CD,DE)
Shape(ED,DB)
Collinear(CDB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CDE),150)
ParallelBetweenLine(DE,AB)
Goal: Value(MeasureOfAngle(CAB))
Answer: 60
Theorem sequence:
adjacent_complementary_angle(1,CDE,EDB)
parallel_property_alternate_interior_angle(1,DE,AB)
triangle_property_angle_sum(1,BCA)
Problem: As shown in the diagram, ∠DCF=25°, ∠EAD=∠ABC, AD is parallel to BC. Find the measure of ∠CDA.
问题：如图所示，∠DCF=25°，∠EAD=∠ABC，AD平行于BC。求∠CDA的大小。
Construction (CDL):
Shape(EA,AD)
Shape(DA,AB,BC,CD)
Shape(DC,CF)
Collinear(BAE)
Collinear(BCF)
Conditions (CDL):
Equal(MeasureOfAngle(DCF),25)
Equal(MeasureOfAngle(EAD),MeasureOfAngle(ABC))
ParallelBetweenLine(AD,BC)
Goal: Value(MeasureOfAngle(CDA))
Answer: 25
Theorem sequence:
adjacent_complementary_angle(1,EAD,DAB)
adjacent_complementary_angle(1,BCD,DCF)
quadrilateral_property_angle_sum(1,ABCD)
Problem: As shown in the diagram, ∠CPQ=50°, CD is parallel to AB, FE⊥BE. Find the measure of ∠PEF.
问题：如图所示，∠CPQ=50°，CD∥AB，FE⊥BE。求∠PEF的大小。
Construction (CDL):
Shape(CP,PQ)
Shape(QP,PF)
Shape(PF,FM)
Shape(MF,FD)
Shape(EP,PC)
Shape(AE,EP)
Shape(PE,EF,FP)
Shape(DF,FE)
Shape(FE,EB)
Shape(NE,EA)
Shape(BE,EN)
Collinear(CPFD)
Collinear(QPE)
Collinear(AEB)
Collinear(MFEN)
Conditions (CDL):
Equal(MeasureOfAngle(CPQ),50)
ParallelBetweenLine(CD,AB)
PerpendicularBetweenLine(FE,BE)
Goal: Value(MeasureOfAngle(PEF))
Answer: 40
Theorem sequence:
parallel_property_collinear_extend(3,CD,AB,F)
parallel_property_collinear_extend(3,BA,FC,E)
parallel_property_alternate_interior_angle(2,CF,EB)
vertical_angle(1,CPQ,FPE)
triangle_property_angle_sum(1,PEF)
Problem: As shown in the diagram, ∠CAO=35°, the center of ⊙O is O, PC is the tangent to ⊙O. Find the measure of ∠BPC.
问题：如图所示，∠CAO=35°，O是⊙O的圆心，PC是圆O的切线。求∠BPC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OB,OBC)
Shape(BO,OA,OAB)
Shape(BP,PC,OBC)
Collinear(AOBP)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),35)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PC,O)
Goal: Value(MeasureOfAngle(BPC))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
adjacent_complementary_angle(1,AOC,COB)
arc_property_center_angle(1,OCA,O)
arc_property_center_angle(1,OBC,O)
circle_property_circular_power_tangent_and_segment_angle(2,PC,PBA,O)
Problem: As shown in the diagram, ∠EBF=130°, EM∥DN, AE is perpendicular to ME, BO⊥FO. Find the measure of ∠NFC.
问题：如图所示，∠EBF=130°，EM∥DN，AE垂直于ME，BO垂直于FO。求∠NFC的大小。
Construction (CDL):
Shape(AE,EM)
Shape(ME,EB)
Shape(EB,BF)
Shape(BF,FN)
Shape(DO,OB)
Shape(BO,OF,FB)
Shape(NF,FC)
Shape(CF,FD)
Collinear(AEBO)
Collinear(BFC)
Collinear(DOFN)
Conditions (CDL):
Equal(MeasureOfAngle(EBF),130)
ParallelBetweenLine(EM,DN)
PerpendicularBetweenLine(AE,ME)
PerpendicularBetweenLine(BO,FO)
Goal: Value(MeasureOfAngle(NFC))
Answer: 40
Theorem sequence:
adjacent_complementary_angle(1,EBF,FBO)
triangle_property_angle_sum(1,BOF)
vertical_angle(1,OFB,NFC)
Problem: As shown in the diagram, ∠BCA=50°, ∠EBD=60°, AC∥BE. Find the measure of ∠CBD.
问题：如图所示，∠BCA=50°，∠EBD=60°，AC平行于BE。求∠CBD的大小。
Construction (CDL):
Shape(CA,AB,BC)
Shape(CB,BE)
Shape(EB,BD)
Collinear(ABD)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
Equal(MeasureOfAngle(EBD),60)
ParallelBetweenLine(AC,BE)
Goal: Value(MeasureOfAngle(CBD))
Answer: 110
Theorem sequence:
parallel_property_alternate_interior_angle(2,AC,BE)
angle_addition(1,CBE,EBD)
Problem: As shown in the diagram, Add(MeasureOfAngle(HCD)=MeasureOfAngle(ABF)), ∠DCE=90°, DE is parallel to HC, FA∥DE, FA is parallel to HC, DC⊥EC. Find the measure of ∠BFA.
问题：如图所示，Add(MeasureOfAngle(HCD)=MeasureOfAngle(ABF))，∠DCE=90°，DE∥HC，FA∥DE，FA∥HC，DC⊥EC。求∠BFA的大小。
Construction (CDL):
Shape(BF,FA,AB)
Shape(FD,DE,EA,AF)
Shape(DC,CE,ED)
Shape(HC,CD)
Collinear(BFD)
Collinear(BAEC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(HCD),MeasureOfAngle(ABF)),70)
Equal(MeasureOfAngle(DCE),90)
ParallelBetweenLine(DE,HC)
ParallelBetweenLine(FA,DE)
ParallelBetweenLine(FA,HC)
PerpendicularBetweenLine(DC,EC)
Goal: Value(MeasureOfAngle(BFA))
Answer: 20
Theorem sequence:
parallel_property_corresponding_angle(1,FA,DE,B)
parallel_property_alternate_interior_angle(1,DE,HC)
angle_addition(1,FDE,EDC)
triangle_property_angle_sum(1,BDC)
Problem: As shown in the diagram, ∠CAO=25°, O is the center of circle O, OE⊥CE. Find the measure of ∠DOA.
问题：如图所示，∠CAO=25°，⊙O的圆心为O，OE垂直于CE。求∠DOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OC)
Shape(CO,OE,EC)
Shape(OBC,CE,EB)
Shape(OAD,DA)
Shape(DO,OA,AD)
Shape(OD,DE,EO)
Shape(ODB,BE,ED)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),25)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(MeasureOfAngle(DOA))
Answer: 130
Theorem sequence:
adjacent_complementary_angle(1,DEO,OEC)
adjacent_complementary_angle(1,EOD,DOA)
circle_property_chord_perpendicular_bisect_arc(1,ODC,OEB)
congruent_arc_judgment_length_equal(1,OBC,ODB)
congruent_arc_property_measure_equal(1,OBC,ODB)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, triangle DEA is congruent to triangle CBA, ∠ACF=36°, ∠EAB=60°, ∠FAC=60°. Find the measure of ∠CFA.
问题：如图所示，△DEA与△CBA是全等三角形，∠ACF=36°，∠EAB=60°，∠FAC=60°。求∠CFA的大小。
Construction (CDL):
Shape(CF,FA,AC)
Shape(DP,PF,FD)
Shape(PB,BQ,QP)
Shape(FP,PQ,QA,AF)
Shape(QE,EA,AQ)
Collinear(DFA)
Collinear(BPFC)
Collinear(BQA)
Collinear(DPQE)
Conditions (CDL):
CongruentBetweenTriangle(DEA,CBA)
Equal(MeasureOfAngle(ACF),36)
Equal(MeasureOfAngle(EAB),60)
Equal(MeasureOfAngle(FAC),60)
Goal: Value(MeasureOfAngle(CFA))
Answer: 84
Theorem sequence:
triangle_property_angle_sum(1,CFA)
Problem: As shown in the diagram, ∠BAO=40°, O is the center of circle O, O is the circumcenter of △ACB. Find the measure of ∠DCB.
问题：如图所示，∠BAO=40°，O是⊙O的圆心，O是△ABC外接圆的圆心。求∠DCB的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(BA,AD,DB)
Shape(AO,OD,DA)
Shape(DC,CB,BD)
Shape(OAC,CD,DO,OA)
Shape(OCB,BC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),40)
IsCentreOfCircle(O,O)
IsCircumcenterOfTriangle(O,ACB)
Goal: Value(MeasureOfAngle(DCB))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,AOB)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠ACF=25°, AC is parallel to ME, AC is parallel to PD, PD is parallel to ME, ME⊥DE. Find the measure of ∠EDC.
问题：如图所示，∠ACF=25°，AC∥ME，AC∥PD，PD∥ME，ME垂直于DE。求∠EDC的大小。
Construction (CDL):
Shape(AC,CF)
Shape(FC,CB)
Shape(DC,CA)
Shape(BC,CD)
Shape(PD,DC)
Shape(CD,DQ)
Shape(ED,DP)
Shape(QD,DE)
Shape(ME,ED)
Shape(DE,EN)
Shape(ED,DC)
Collinear(ACB)
Collinear(PDQ)
Collinear(MEN)
Collinear(FCD)
Conditions (CDL):
Equal(MeasureOfAngle(ACF),25)
ParallelBetweenLine(AC,ME)
ParallelBetweenLine(AC,PD)
ParallelBetweenLine(PD,ME)
PerpendicularBetweenLine(ME,DE)
Goal: Value(MeasureOfAngle(EDC))
Answer: 115
Theorem sequence:
parallel_property_corresponding_angle(2,DP,CA,F)
parallel_property_ipsilateral_internal_angle(1,EM,DP)
angle_addition(1,EDP,PDC)
Problem: As shown in the diagram, ∠CNF=75°, ∠FME=35°, AF∥CN, FM∥ND. Find the measure of ∠MEF.
问题：如图所示，∠CNF=75°，∠FME=35°，AF∥CN，FM∥ND。求∠MEF的大小。
Construction (CDL):
Shape(ME,EF,FM)
Shape(AF,FE)
Shape(NF,FA)
Shape(MF,FN)
Shape(CN,NF)
Shape(FN,ND)
Collinear(AFMB)
Collinear(EFN)
Collinear(CND)
Conditions (CDL):
Equal(MeasureOfAngle(CNF),75)
Equal(MeasureOfAngle(FME),35)
ParallelBetweenLine(AF,CN)
ParallelBetweenLine(FM,ND)
Goal: Value(MeasureOfAngle(MEF))
Answer: 40
Theorem sequence:
parallel_property_corresponding_angle(2,NC,FA,E)
flat_angle(1,AFM)
angle_addition(1,AFE,EFM)
triangle_property_angle_sum(1,EFM)
Problem: As shown in the diagram, AB=AC, ∠DAB=35°, D is the midpoint of segment BC. Find the measure of ∠DCA.
问题：如图所示，AB=AC，∠DAB=35°，D平分线段BC。求∠DCA的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(DAB),35)
IsMidpointOfLine(D,BC)
Goal: Value(MeasureOfAngle(DCA))
Answer: 55
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
median_of_triangle_judgment(1,AD,ABC)
isosceles_triangle_property_line_coincidence(2,ABC,D)
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, Add(MeasureOfAngle(FQB)=MeasureOfAngle(BFQ)), ∠EDP=90°, EP∥CD, QB∥CD, QB∥EP. Find the measure of ∠CDE.
问题：如图所示，Add(MeasureOfAngle(FQB)=MeasureOfAngle(BFQ))，∠EDP=90°，EP∥CD，QB∥CD，QB平行于EP。求∠CDE的大小。
Construction (CDL):
Shape(QE,EP,PB,BQ)
Shape(FQ,QB,BF)
Shape(ED,DP,PE)
Shape(CD,DE)
Collinear(AQB)
Collinear(EQF)
Collinear(FBPD)
Conditions (CDL):
Equal(Add(MeasureOfAngle(FQB),MeasureOfAngle(BFQ)),70)
Equal(MeasureOfAngle(EDP),90)
ParallelBetweenLine(EP,CD)
ParallelBetweenLine(QB,CD)
ParallelBetweenLine(QB,EP)
Goal: Value(MeasureOfAngle(CDE))
Answer: 20
Theorem sequence:
parallel_property_corresponding_angle(1,QB,EP,F)
parallel_property_alternate_interior_angle(1,EP,CD)
triangle_property_angle_sum(1,FED)
angle_addition(1,QEP,PED)
Problem: As shown in the diagram, ∠BPA=50°, the center of circle O is O, PA is the tangent to circle O, PB is the tangent to circle O. Find the measure of ∠AOB.
问题：如图所示，∠BPA=50°，圆O的圆心为O，⊙O的切线为PA，⊙O的切线为PB。求∠AOB的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OAB,BO,OA)
Shape(OBA,BP,PA)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 130
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
quadrilateral_property_angle_sum(1,AOBP)