SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠CAD=60°, ∠DBE=40°, DE perpendicular bisects BC. Find the measure of ∠DCA.
问题：如图所示，∠CAD=60°，∠DBE=40°，DE是BC的垂直平分线。求∠DCA的大小。
Construction (CDL):
Shape(BE,ED,DB)
Shape(DE,EC,CD)
Shape(DC,CA,AD)
Collinear(ADB)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),60)
Equal(MeasureOfAngle(DBE),40)
IsPerpendicularBisectorOfLine(DE,BC)
Goal: Value(MeasureOfAngle(DCA))
Answer: 40
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,BC)
isosceles_triangle_judgment_line_equal(1,DBC)
isosceles_triangle_property_angle_equal(1,DBC)
triangle_property_angle_sum(1,ABC)
angle_addition(1,ECD,DCA)
Problem: As shown in the diagram, ∠DBF=50°, D is the midpoint of segment AB, E bisects segment AC, triangle DEA is mirror congruent to triangle DFE. Find the measure of ∠FDB.
问题：如图所示，∠DBF=50°，D平分线段AB，E平分线段AC，三角形DEA与三角形DFE是镜像全等三角形。求∠FDB的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BF,FD)
Shape(DF,FE,ED)
Shape(EF,FC,CE)
Collinear(ADB)
Collinear(BFC)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(DBF),50)
IsMidpointOfLine(D,AB)
IsMidpointOfLine(E,AC)
MirrorCongruentBetweenTriangle(DEA,DFE)
Goal: Value(MeasureOfAngle(FDB))
Answer: 80
Theorem sequence:
mirror_congruent_triangle_property_line_equal(1,EAD,EDF)
isosceles_triangle_judgment_line_equal(1,DBF)
isosceles_triangle_property_angle_equal(1,DBF)
triangle_property_angle_sum(1,DBF)
Problem: As shown in the diagram, ∠COA=100°, the center of circle O is O. Find the measure of ∠ABC.
问题：如图所示，∠COA=100°，O是圆O的圆心。求∠ABC的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(AB,BC,CO,OA)
Shape(BA,OAB)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 130
Theorem sequence:
round_angle(1,AOC,COA)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, BD=CD, ∠CAB=80°, AB is perpendicular to DB, DC is perpendicular to AC. Find the measure of ∠DAB.
问题：如图所示，BD=CD，∠CAB=80°，AB垂直于DB，DC垂直于AC。求∠DAB的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Conditions (CDL):
Equal(LengthOfLine(BD),LengthOfLine(CD))
Equal(MeasureOfAngle(CAB),80)
PerpendicularBetweenLine(AB,DB)
PerpendicularBetweenLine(DC,AC)
Goal: Value(MeasureOfAngle(DAB))
Answer: 40
Theorem sequence:
mirror_congruent_triangle_judgment_hl(1,ABD,ADC)
mirror_congruent_triangle_property_angle_equal(1,ABD,ADC)
angle_addition(1,CAD,DAB)
Problem: As shown in the diagram, BD=CD, ∠ADG=130°, ∠CAB=40°, AB⊥DB, DC⊥AC. Find the measure of ∠FGD.
问题：如图所示，BD=CD，∠ADG=130°，∠CAB=40°，AB⊥DB，DC垂直于AC。求∠FGD的大小。
Construction (CDL):
Shape(AD,DC,CA)
Shape(CD,DG,GC)
Shape(AB,BD,DA)
Shape(GD,DB,BE,EF,FG)
Collinear(ACGF)
Collinear(GDB)
Collinear(ABE)
Conditions (CDL):
Equal(LengthOfLine(BD),LengthOfLine(CD))
Equal(MeasureOfAngle(ADG),130)
Equal(MeasureOfAngle(CAB),40)
PerpendicularBetweenLine(AB,DB)
PerpendicularBetweenLine(DC,AC)
Goal: Value(MeasureOfAngle(FGD))
Answer: 150
Theorem sequence:
mirror_congruent_triangle_judgment_hl(1,ABD,ADC)
mirror_congruent_triangle_property_angle_equal(1,ABD,ADC)
angle_addition(1,CAD,DAB)
triangle_property_angle_sum(1,GAD)
adjacent_complementary_angle(1,FGB,BGA)
Problem: As shown in the diagram, ∠COA=75°, OC bisects ∠BOA. Find the measure of ∠BOA.
问题：如图所示，∠COA=75°，OC是∠BOA的角平分线。求∠BOA的大小。
Construction (CDL):
Shape(BO,OC)
Shape(CO,OA)
Conditions (CDL):
Equal(MeasureOfAngle(COA),75)
IsBisectorOfAngle(OC,BOA)
Goal: Value(MeasureOfAngle(BOA))
Answer: 150
Theorem sequence:
angle_addition(1,BOC,COA)
Problem: As shown in the diagram, ∠EFB=70°, AB is parallel to CD. Find the measure of ∠IHC.
问题：如图所示，∠EFB=70°，AB∥CD。求∠IHC的大小。
Construction (CDL):
Shape(AF,FE)
Shape(EF,FB)
Shape(HF,FA)
Shape(BF,FH)
Shape(CH,HF)
Shape(FH,HD)
Shape(IH,HC)
Shape(DH,HI)
Collinear(AFB)
Collinear(CHD)
Collinear(EFHI)
Conditions (CDL):
Equal(MeasureOfAngle(EFB),70)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(IHC))
Answer: 70
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_collinear_extend(3,DC,BF,H)
parallel_property_corresponding_angle(1,FB,HD,E)
vertical_angle(1,EHD,IHC)
Problem: As shown in the diagram, ∠BCO=40°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠BCO=40°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(BO,OC,CA,AB)
Shape(OB,BC,CO)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(BCO),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠EDC=150°, BD is the angle bisector of ∠ABC, DC is parallel to AB. Find the measure of ∠BCD.
问题：如图所示，∠EDC=150°，BD平分∠ABC，DC∥AB。求∠BCD的大小。
Construction (CDL):
Shape(ED,DC)
Shape(DB,BC,CD)
Shape(AB,BD)
Collinear(EDB)
Conditions (CDL):
Equal(MeasureOfAngle(EDC),150)
IsBisectorOfAngle(BD,ABC)
ParallelBetweenLine(DC,AB)
Goal: Value(MeasureOfAngle(BCD))
Answer: 120
Theorem sequence:
adjacent_complementary_angle(1,EDC,CDB)
parallel_property_alternate_interior_angle(1,DC,AB)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, ⌒OAC is congruent to ⌒OCB, ∠OAB=50°, O is the center of ⊙O. Find the measure of ∠BOC.
问题：如图所示，⌒OAC全等于⌒OCB，∠OAB=50°，圆O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(OA,AD,DO)
Shape(OD,DB,BO)
Shape(CD,DA,OAC)
Shape(BD,DC,OCB)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
CongruentBetweenArc(OAC,OCB)
Equal(MeasureOfAngle(OAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 40
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,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_property_center_angle(1,OAC,O)
arc_property_center_angle(1,OCB,O)
angle_addition(1,BOD,DOA)
Problem: As shown in the diagram, ∠BOC=70°, O is the center of circle O. Find the measure of ∠BAC.
问题：如图所示，∠BOC=70°，O是⊙O的圆心。求∠BAC的大小。
Construction (CDL):
Shape(CO,OD,DA,OAC)
Shape(CD,DO,OC)
Shape(DB,BA,AD)
Shape(BO,OC,OCB)
Shape(AB,OBA)
Collinear(ODB)
Collinear(CDA)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, triangle ABC is congruent to triangle ADE, ∠DAB=120°, ∠EAC=120°, AE∥BD. Find the measure of ∠DAC.
问题：如图所示，三角形ABC与三角形ADE是全等三角形，∠DAB=120°，∠EAC=120°，AE∥BD。求∠DAC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CB,BD,DA)
Shape(AD,DE,EA)
Conditions (CDL):
CongruentBetweenTriangle(ABC,ADE)
Equal(MeasureOfAngle(DAB),120)
Equal(MeasureOfAngle(EAC),120)
ParallelBetweenLine(AE,BD)
Goal: Value(MeasureOfAngle(DAC))
Answer: 90
Theorem sequence:
congruent_triangle_property_line_equal(1,CAB,EAD)
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
triangle_property_angle_sum(1,ABD)
parallel_property_alternate_interior_angle(1,AE,BD)
angle_addition(1,EAD,DAC)
Problem: As shown in the diagram, ∠CAB=65°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠ADC.
问题：如图所示，∠CAB=65°，O是⊙O的圆心，AB是圆O的直径。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(AD,DE,EA)
Shape(DA,OAD)
Shape(CE,EO,OB,BC)
Shape(CB,OBC)
Shape(BO,OE,ED,ODB)
Collinear(CED)
Collinear(AEOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),65)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 25
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, . Find the sum of the measure of ∠OAH, the measure of ∠ICO, the measure of ∠HBI, the measure of ∠HDE and the measure of ∠DEI.
问题：如图所示，。求∠OAH的大小、∠ICO的大小、∠HBI的大小、∠HDE的大小与∠DEI的大小之和。
Construction (CDL):
Shape(AH,HO,OA)
Shape(OH,HB,BI,IO)
Shape(CO,OI,IC)
Shape(HD,DE,EI,IB,BH)
Collinear(AHB)
Collinear(AOIE)
Collinear(DHOC)
Collinear(BIC)
Conditions (CDL):
Goal: Value(Add(MeasureOfAngle(OAH),MeasureOfAngle(ICO),MeasureOfAngle(HBI),MeasureOfAngle(HDE),MeasureOfAngle(DEI)))
Answer: 180
Theorem sequence:
adjacent_complementary_angle(1,AHO,OHB)
adjacent_complementary_angle(1,IOH,HOA)
triangle_property_angle_sum(1,AHO)
triangle_property_angle_sum(1,ODE)
triangle_property_angle_sum(1,HBC)
Problem: As shown in the diagram, OC=3, the radius of ⊙O is 5, the center of circle O is O, OC⊥BC. Find the length of line AB.
问题：如图所示，OC=3，⊙O的半径为5，⊙O的圆心为O，OC⊥BC。求直线AB的长度。
Construction (CDL):
Shape(AC,CO,OA)
Shape(OC,CB,BO)
Shape(AO,OB,OBA)
Shape(BC,CA,OAB)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OC),3)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OC,BC)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
adjacent_complementary_angle(1,ACO,OCB)
right_triangle_judgment_angle(1,ACO)
right_triangle_judgment_angle(1,OCB)
right_triangle_property_pythagorean(1,ACO)
right_triangle_property_pythagorean(1,OCB)
line_addition(1,AC,CB)
Problem: As shown in the diagram, ∠CDA=54°, O is the center of ⊙O, the diameter of circle O is BE, quadrilateral ABCD is a ▱. Find the measure of ∠BEA.
问题：如图所示，∠CDA=54°，⊙O的圆心为O，圆O的直径为BE，AD和BC是▱ABCD的一组对边。求∠BEA的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(BA,OAB)
Shape(EC,CO,OB,OBE)
Shape(DF,FE,OED)
Shape(AB,BO,OC,CF,FA)
Shape(AF,FD,DA)
Shape(FC,CE,EF)
Collinear(AFE)
Collinear(BOCE)
Collinear(DFC)
Cocircular(O,ABED)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),54)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BEA))
Answer: 36
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
diameter_of_circle_property_right_angle(1,EAB,O)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, ∠AOC=70°, ∠FOD=40°. Find the measure of ∠BOF.
问题：如图所示，∠AOC=70°，∠FOD=40°。求∠BOF的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OB)
Shape(BO,OF)
Shape(FO,OD)
Shape(DO,OA)
Collinear(COD)
Collinear(AOB)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),70)
Equal(MeasureOfAngle(FOD),40)
Goal: Value(MeasureOfAngle(BOF))
Answer: 30
Theorem sequence:
vertical_angle(1,AOC,BOD)
angle_addition(1,BOF,FOD)
Problem: As shown in the diagram, AE=Mul(LengthOfLine(ED),2), CD=3, quadrilateral ABCD is a parallelogram. Find the length of line AF.
问题：如图所示，AE=Mul(LengthOfLine(ED),2)，CD=3，AD和BC是▱ABCD的一组对边。求直线AF的长度。
Construction (CDL):
Shape(FA,AE,EF)
Shape(EC,CD,DE)
Shape(AB,BC,CE,EA)
Collinear(AED)
Collinear(FEC)
Collinear(FAB)
Conditions (CDL):
Equal(LengthOfLine(AE),Mul(LengthOfLine(ED),2))
Equal(LengthOfLine(CD),3)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AF))
Answer: 6
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
vertical_angle(1,AEF,DEC)
parallel_property_corresponding_angle(1,AD,BC,F)
similar_triangle_judgment_aa(1,FAE,CDE)
similar_triangle_property_line_ratio(1,FAE,CDE)
similar_triangle_property_line_ratio(1,EFA,ECD)
Problem: As shown in the diagram, ∠CAB=40°, ∠FDE=90°. Find the sum of the measure of ∠ABD and the measure of ∠DCA.
问题：如图所示，∠CAB=40°，∠FDE=90°。求∠ABD的大小与∠DCA的大小之和。
Construction (CDL):
Shape(AB,BD,DC,CA)
Shape(DB,BC,CD)
Shape(BE,EF,FC,CB)
Collinear(EBD)
Collinear(DCF)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
Equal(MeasureOfAngle(FDE),90)
Goal: Value(Add(MeasureOfAngle(ABD),MeasureOfAngle(DCA)))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,DBC)
triangle_property_angle_sum(1,ABC)
angle_addition(1,ABD,DBC)
angle_addition(1,BCD,DCA)
Problem: As shown in the diagram, BD=CF, BE=CD, ∠DCF=50°, ∠EBD=∠DCF. Find the measure of ∠EDF.
问题：如图所示，BD=CF，BE=CD，∠DCF=50°，∠EBD=∠DCF。求∠EDF的大小。
Construction (CDL):
Shape(AE,ED,DF,FA)
Shape(EB,BD,DE)
Shape(FD,DC,CF)
Collinear(AEB)
Collinear(AFC)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(BD),LengthOfLine(CF))
Equal(LengthOfLine(BE),LengthOfLine(CD))
Equal(MeasureOfAngle(DCF),50)
Equal(MeasureOfAngle(EBD),MeasureOfAngle(DCF))
Goal: Value(MeasureOfAngle(EDF))
Answer: 50
Theorem sequence:
congruent_triangle_judgment_sas(1,BDE,CFD)
congruent_triangle_property_angle_equal(1,DEB,FDC)
triangle_property_angle_sum(1,DCF)
angle_addition(1,BDE,EDF)
adjacent_complementary_angle(1,BDF,FDC)
Problem: As shown in the diagram, ∠ACB=70°, the center of circle O is O, the tangent to circle O is CA, the tangent to circle O is CB. Find the measure of ∠BOA.
问题：如图所示，∠ACB=70°，O是圆O的圆心，⊙O的切线为CA，⊙O的切线为CB。求∠BOA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AC,CB,OAB)
Shape(AB,BO,OA)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),70)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CA,O)
IsTangentOfCircle(CB,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 110
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CA,O,O)
tangent_of_circle_property_perpendicular(2,CB,O,O)
quadrilateral_property_angle_sum(1,ACBO)
Problem: As shown in the diagram, ∠CAB=55°, the center of ⊙O is O. Find the measure of ∠OBC.
问题：如图所示，∠CAB=55°，⊙O的圆心为O。求∠OBC的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),55)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 35
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠BPA=40°, the center of ⊙O is O, the diameter of ⊙O is AB, PA is the tangent to ⊙O, the tangent to circle O is PB. Find the measure of ∠BAC.
问题：如图所示，∠BPA=40°，O是圆O的圆心，AB是⊙O的直径，圆O的切线为PA，圆O的切线为PB。求∠BAC的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(BA,AO,OB)
Shape(OC,CB,BO)
Shape(AB,OBA)
Shape(BC,OCB)
Shape(BP,PA,OBA)
Collinear(AOC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 20
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)
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,OBA)
Problem: As shown in the diagram, AB=10, AC=6, BE=EC, ∠CAD=90°, ∠DEC=90°, ∠ECD=∠DCA. Find the perimeter of △BED.
问题：如图所示，AB=10，AC=6，BE=EC，∠CAD=90°，∠DEC=90°，∠ECD=∠DCA。求△BED的周长。
Construction (CDL):
Shape(AD,DC,CA)
Shape(DE,EC,CD)
Shape(DB,BE,ED)
Collinear(BDA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(BE),LengthOfLine(EC))
Equal(MeasureOfAngle(CAD),90)
Equal(MeasureOfAngle(DEC),90)
Equal(MeasureOfAngle(ECD),MeasureOfAngle(DCA))
Goal: Value(PerimeterOfTriangle(BED))
Answer: 16
Theorem sequence:
mirror_congruent_triangle_judgment_aas(3,DEC,DCA)
mirror_congruent_triangle_property_line_equal(1,CDE,CAD)
mirror_congruent_triangle_property_line_equal(1,DEC,DCA)
line_addition(1,BD,DA)
triangle_perimeter_formula(1,DBE)
Problem: As shown in the diagram, AD=DC, ∠DAB=120°, BM bisects ∠ABC. Find the measure of ∠BMA.
问题：如图所示，AD=DC，∠DAB=120°，BM平分∠ABC。求∠BMA的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OM,MA)
Shape(OB,BD,DO)
Shape(OD,DC,CM,MO)
Collinear(AOD)
Collinear(AMC)
Collinear(BDC)
Collinear(BOM)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(DC))
Equal(MeasureOfAngle(DAB),120)
IsBisectorOfAngle(BM,ABC)
Goal: Value(MeasureOfAngle(BMA))
Answer: 30
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,DCA)
triangle_property_angle_sum(1,DCA)
adjacent_complementary_angle(1,BDA,ADC)
angle_addition(1,ABO,OBD)
triangle_property_angle_sum(1,ABD)
triangle_property_angle_sum(1,MBC)
adjacent_complementary_angle(1,CMB,BMA)
Problem: As shown in the diagram, ∠AOB=35°, ∠GOA=∠AOQ, ∠QOB=∠BOH. Find the measure of ∠GOH.
问题：如图所示，∠AOB=35°，∠GOA=∠AOQ，∠QOB=∠BOH。求∠GOH的大小。
Construction (CDL):
Shape(GO,OA,AG)
Shape(AO,OQ,QA)
Shape(AQ,QP,PA)
Shape(OB,BQ,QO)
Shape(PQ,QB,BP)
Shape(OH,HB,BO)
Shape(AB,BP,PA)
Shape(GA,AM)
Shape(MA,AP)
Shape(PB,BN)
Shape(NB,BH)
Collinear(OAM)
Collinear(GAQBH)
Collinear(OBN)
Collinear(OQP)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),35)
Equal(MeasureOfAngle(GOA),MeasureOfAngle(AOQ))
Equal(MeasureOfAngle(QOB),MeasureOfAngle(BOH))
Goal: Value(MeasureOfAngle(GOH))
Answer: 70
Theorem sequence:
angle_addition(1,AOQ,QOB)
angle_addition(1,GOA,AOQ)
angle_addition(1,GOQ,QOB)
angle_addition(1,GOB,BOH)
Problem: As shown in the diagram, ∠COD=25°, ∠COE=90°, OD bisects ∠AOE. Find the measure of ∠DOB.
问题：如图所示，∠COD=25°，∠COE=90°，OD平分∠AOE。求∠DOB的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OD)
Shape(DO,OE)
Shape(EO,OB)
Collinear(AOB)
Conditions (CDL):
Equal(MeasureOfAngle(COD),25)
Equal(MeasureOfAngle(COE),90)
IsBisectorOfAngle(OD,AOE)
Goal: Value(MeasureOfAngle(DOB))
Answer: 115
Theorem sequence:
angle_addition(1,COD,DOE)
angle_addition(1,AOD,DOE)
adjacent_complementary_angle(1,AOE,EOB)
angle_addition(1,DOE,EOB)
Problem: As shown in the diagram, ∠BEF=105°, ∠CDA=∠GHD. Find the measure of ∠HIE.
问题：如图所示，∠BEF=105°，∠CDA=∠GHD。求∠HIE的大小。
Construction (CDL):
Shape(CD,DA)
Shape(AD,DE)
Shape(DE,EB)
Shape(BE,EF)
Shape(HD,DC)
Shape(GH,HD)
Shape(FE,EI)
Shape(EI,IJ)
Shape(KH,HG)
Shape(IH,HK)
Shape(LI,IH)
Shape(JI,IL)
Shape(DH,HI,IE,ED)
Collinear(CDEF)
Collinear(GHIJ)
Collinear(ADHK)
Collinear(BEIL)
Conditions (CDL):
Equal(MeasureOfAngle(BEF),105)
Equal(MeasureOfAngle(CDA),MeasureOfAngle(GHD))
Goal: Value(MeasureOfAngle(HIE))
Answer: 75
Theorem sequence:
parallel_judgment_corresponding_angle(2,HG,DC,A)
parallel_property_collinear_extend(2,CD,GH,F)
parallel_property_collinear_extend(1,HG,FC,J)
parallel_property_collinear_extend(3,CF,GJ,E)
parallel_property_collinear_extend(3,JG,FE,I)
parallel_property_corresponding_angle(1,EF,IJ,B)
adjacent_complementary_angle(1,HIE,EIJ)
Problem: As shown in the diagram, ∠ABC=45°, ∠CAB=90°, BC∥DE. Find the measure of ∠GED.
问题：如图所示，∠ABC=45°，∠CAB=90°，BC平行于DE。求∠GED的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DE,EC,CB)
Shape(DF,FG,GE,ED)
Collinear(ABDF)
Collinear(ACEG)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),45)
Equal(MeasureOfAngle(CAB),90)
ParallelBetweenLine(BC,DE)
Goal: Value(MeasureOfAngle(GED))
Answer: 135
Theorem sequence:
triangle_property_angle_sum(1,ABC)
parallel_property_corresponding_angle(2,ED,CB,A)
adjacent_complementary_angle(1,GED,DEC)
Problem: As shown in the diagram, △CAB is congruent to △DAF, ∠CAB=80°, DC∥AB. Find the measure of ∠HAB.
问题：如图所示，△CAB与△DAF是全等三角形，∠CAB=80°，DC∥AB。求∠HAB的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(DE,EC,CD)
Shape(CE,EG,GC)
Shape(EA,AH,HG,GE)
Shape(GH,HF,FG)
Shape(HA,AB,BH)
Shape(FH,HB)
Collinear(DEGF)
Collinear(AEC)
Collinear(CGHB)
Collinear(AHF)
Conditions (CDL):
CongruentBetweenTriangle(CAB,DAF)
Equal(MeasureOfAngle(CAB),80)
ParallelBetweenLine(DC,AB)
Goal: Value(MeasureOfAngle(HAB))
Answer: 20
Theorem sequence:
parallel_property_alternate_interior_angle(2,DC,AB)
congruent_triangle_property_line_equal(1,BCA,FDA)
isosceles_triangle_judgment_line_equal(1,ACD)
isosceles_triangle_property_angle_equal(1,ACD)
triangle_property_angle_sum(1,ACD)
congruent_triangle_property_angle_equal(1,ABC,AFD)
angle_addition(1,DAC,CAF)
angle_addition(1,CAF,FAB)
Problem: As shown in the diagram, ∠BOD=90°, ∠DAC=35°, the center of ⊙O is O. Find the measure of arc ODC.
问题：如图所示，∠BOD=90°，∠DAC=35°，圆O的圆心为O。求⌒ODC的角度。
Construction (CDL):
Shape(BO,OC,CB)
Shape(CO,OD,ODC)
Shape(BC,OCB)
Shape(DA,AC,ODC)
Shape(DO,OB,OBD)
Collinear(BCA)
Collinear(ODA)
Cocircular(O,DCB)
Conditions (CDL):
Equal(MeasureOfAngle(BOD),90)
Equal(MeasureOfAngle(DAC),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfArc(ODC))
Answer: 20
Theorem sequence:
triangle_property_angle_sum(1,OAB)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,OCB)
angle_addition(1,BOC,COD)
arc_property_center_angle(1,ODC,O)
Problem: As shown in the diagram, ∠CBA=25°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠BDC.
问题：如图所示，∠CBA=25°，O是⊙O的圆心，圆O的直径为AB。求∠BDC的大小。
Construction (CDL):
Shape(AC,CO,OA)
Shape(OC,CB,BO)
Shape(OB,BD,DO)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(DB,OBD)
Shape(AO,OD,ODA)
Collinear(AOB)
Collinear(COD)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 65
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, AB=AC, BC=10, the perimeter of triangle DBC is 22, DE is the perpendicular bisector of AB. Find the length of line AB.
问题：如图所示，AB=AC，BC=10，△DBC的周长为22，DE是AB的垂直平分线。求直线AB的长度。
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(LengthOfLine(BC),10)
Equal(PerimeterOfTriangle(DBC),22)
IsPerpendicularBisectorOfLine(DE,AB)
Goal: Value(LengthOfLine(AB))
Answer: 12
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,AB)
line_addition(1,AD,DC)
triangle_perimeter_formula(1,DBC)
Problem: As shown in the diagram, triangle BCA is congruent to triangle EDA, ∠BCA=90°, ∠CAB=70°. Find the measure of ∠EBD.
问题：如图所示，△BCA与△EDA是全等三角形，∠BCA=90°，∠CAB=70°。求∠EBD的大小。
Construction (CDL):
Shape(BC,CA,AB)
Shape(BD,DE,EB)
Shape(ED,DA,AE)
Collinear(BDA)
Conditions (CDL):
CongruentBetweenTriangle(BCA,EDA)
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CAB),70)
Goal: Value(MeasureOfAngle(EBD))
Answer: 55
Theorem sequence:
congruent_triangle_property_line_equal(1,CAB,DAE)
isosceles_triangle_judgment_line_equal(1,AEB)
isosceles_triangle_property_angle_equal(1,AEB)
congruent_triangle_property_angle_equal(1,ABC,AED)
triangle_property_angle_sum(1,AEB)
Problem: As shown in the diagram, AB=5, the ratio of similar triangles ABC and DEF is 1/2, triangle ABC is similar to triangle DEF.. Find the length of line DE.
问题：如图所示，AB=5，三角形ABC和三角形DEF的相似比为1/2，△ABC与△DEF是相似三角形。求直线DE的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(RatioOfSimilarTriangle(ABC,DEF),1/2)
SimilarBetweenTriangle(ABC,DEF)
Goal: Value(LengthOfLine(DE))
Answer: 10
Theorem sequence:
similar_triangle_property_line_ratio(1,CAB,FDE)
Problem: As shown in the diagram, ∠APB=66°, the center of circle O is O, the tangent to ⊙O is PA, PB is the tangent to ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠APB=66°，O是⊙O的圆心，PA是⊙O的切线，PB是⊙O的切线。求∠BCA的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(BC,CA,AO,OB)
Shape(BO,OA,OAB)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(APB),66)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 57
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
tangent_of_circle_property_perpendicular(2,PB,O,O)
quadrilateral_property_angle_sum(1,APBO)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)