SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BAE=30°, ∠DCF=∠BAE, ∠GBC=50°, CF is parallel to GB, GB∥AE. Find the measure of ∠DCB.
问题：如图所示，∠BAE=30°，∠DCF=∠BAE，∠GBC=50°，CF平行于GB，GB平行于AE。求∠DCB的大小。
Construction (CDL):
Shape(DC,CF)
Shape(FC,CB)
Shape(GB,BC)
Shape(AB,BG)
Shape(BA,AE)
Conditions (CDL):
Equal(MeasureOfAngle(BAE),30)
Equal(MeasureOfAngle(DCF),MeasureOfAngle(BAE))
Equal(MeasureOfAngle(GBC),50)
ParallelBetweenLine(CF,GB)
ParallelBetweenLine(GB,AE)
Goal: Value(MeasureOfAngle(DCB))
Answer: 80
Theorem sequence:
parallel_property_alternate_interior_angle(1,CF,GB)
angle_addition(1,DCF,FCB)
Problem: As shown in the diagram, ∠COB=40°, AO⊥BO. Find the measure of ∠AOC.
问题：如图所示，∠COB=40°，AO垂直于BO。求∠AOC的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OB)
Conditions (CDL):
Equal(MeasureOfAngle(COB),40)
PerpendicularBetweenLine(AO,BO)
Goal: Value(MeasureOfAngle(AOC))
Answer: 50
Theorem sequence:
angle_addition(1,AOC,COB)
Problem: As shown in the diagram, CD=CF, ∠ABC=72°, BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠CFA.
问题：如图所示，CD=CF，∠ABC=72°，AD和BC是▱ABCD的一组对边。求∠CFA的大小。
Construction (CDL):
Shape(AB,BC,CF,FA)
Shape(FC,CD,DF)
Collinear(AFD)
Conditions (CDL):
Equal(LengthOfLine(CD),LengthOfLine(CF))
Equal(MeasureOfAngle(ABC),72)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(CFA))
Answer: 108
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
isosceles_triangle_judgment_line_equal(1,CDF)
isosceles_triangle_property_angle_equal(1,CDF)
adjacent_complementary_angle(1,DFC,CFA)
Problem: As shown in the diagram, ∠EOC=70°, OA is the angle bisector of ∠EOC. Find the measure of ∠BOE.
问题：如图所示，∠EOC=70°，OA平分∠EOC。求∠BOE的大小。
Construction (CDL):
Shape(BO,OD)
Shape(DO,OE)
Shape(EO,OA)
Shape(AO,OC)
Shape(CO,OB)
Collinear(BOA)
Collinear(DOC)
Conditions (CDL):
Equal(MeasureOfAngle(EOC),70)
IsBisectorOfAngle(OA,EOC)
Goal: Value(MeasureOfAngle(BOE))
Answer: 145
Theorem sequence:
angle_addition(1,EOA,AOC)
adjacent_complementary_angle(1,BOE,EOA)
Problem: As shown in the diagram, ∠AOE=80°, OC bisects ∠EOB. Find the measure of ∠DOA.
问题：如图所示，∠AOE=80°，OC平分∠EOB。求∠DOA的大小。
Construction (CDL):
Shape(DO,OA)
Shape(AO,OE)
Shape(EO,OC)
Shape(CO,OB)
Shape(BO,OD)
Collinear(AOB)
Collinear(DOC)
Conditions (CDL):
Equal(MeasureOfAngle(AOE),80)
IsBisectorOfAngle(OC,EOB)
Goal: Value(MeasureOfAngle(DOA))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,AOE,EOB)
angle_addition(1,EOC,COB)
angle_addition(1,AOE,EOC)
adjacent_complementary_angle(1,DOA,AOC)
Problem: As shown in the diagram, ∠BCD=50°, O is the center of circle O, the tangent to ⊙O is DC. Find the measure of ∠COA.
问题：如图所示，∠BCD=50°，O是圆O的圆心，圆O的切线为DC。求∠COA的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(CO,OA,OAC)
Shape(CB,BO,OC)
Shape(BC,OCB)
Shape(OCB,CD)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 80
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DC,O,O)
angle_addition(1,OCB,BCD)
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)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, ∠CAB=40°, P is the center of the inscribed circle of triangle ABC. Find the measure of ∠CPB.
问题：如图所示，∠CAB=40°，P是三角形ABC的内心。求∠CPB的大小。
Construction (CDL):
Shape(AB,BP,PC,CA)
Shape(PB,BC,CP)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsIncenterOfTriangle(P,ABC)
Goal: Value(MeasureOfAngle(CPB))
Answer: 110
Theorem sequence:
angle_addition(1,ABP,PBC)
angle_addition(1,BCP,PCA)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,PBC)
Problem: As shown in the diagram, ∠EAF=50°, the center of circle I is I, the tangent to circle I is AE, the tangent to ⊙I is AF, BD is the tangent to circle I, BF is the tangent to ⊙I, CD is the tangent to circle I, the tangent to circle I is CE. Find the measure of ∠FDE.
问题：如图所示，∠EAF=50°，I是⊙I的圆心，AE是圆O的切线，圆O的切线为AF，BD是圆O的切线，BF是圆O的切线，圆O的切线为CD，⊙O的切线为CE。求∠FDE的大小。
Construction (CDL):
Shape(EA,AF,IEF)
Shape(FD,DE,EI,IF)
Shape(FI,IE,IEF)
Shape(ED,IDE)
Shape(DF,IFD)
Shape(FB,BD,IFD)
Shape(DC,CE,IDE)
Collinear(AEC)
Collinear(AFB)
Collinear(BDC)
Cocircular(I,FDE)
Conditions (CDL):
Equal(MeasureOfAngle(EAF),50)
IsCentreOfCircle(I,I)
IsTangentOfCircle(AE,I)
IsTangentOfCircle(AF,I)
IsTangentOfCircle(BD,I)
IsTangentOfCircle(BF,I)
IsTangentOfCircle(CD,I)
IsTangentOfCircle(CE,I)
Goal: Value(MeasureOfAngle(FDE))
Answer: 65
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AF,I,I)
tangent_of_circle_property_perpendicular(1,AE,I,I)
quadrilateral_property_angle_sum(1,AFIE)
arc_property_center_angle(1,IEF,I)
arc_property_circumference_angle_external(1,IEF,D)
Problem: As shown in the diagram, AB=3, AC=2, AC∥BD, BA is parallel to DC. Find the perimeter of quadrilateral ABDC.
问题：如图所示，AB=3，AC=2，AC平行于BD，BA∥DC。求ABDC的周长。
Construction (CDL):
Shape(AB,BD,DC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AC),2)
ParallelBetweenLine(AC,BD)
ParallelBetweenLine(BA,DC)
Goal: Value(PerimeterOfQuadrilateral(ABDC))
Answer: 10
Theorem sequence:
parallelogram_judgment_parallel_and_parallel(1,ABDC)
parallelogram_property_opposite_line_equal(1,ABDC)
parallelogram_property_opposite_line_equal(1,BDCA)
quadrilateral_perimeter_formula(1,ABDC)
Problem: As shown in the diagram, ∠CBO=30°, O is the center of ⊙O. Find the measure of ∠OAC.
问题：如图所示，∠CBO=30°，⊙O的圆心为O。求∠OAC的大小。
Construction (CDL):
Shape(BO,OA,AC,CB)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Shape(CA,OAC)
Collinear(BOA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 60
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,BAC)
Problem: As shown in the diagram, ∠DAC=25°, BA is perpendicular to DA. Find the measure of ∠BAC.
问题：如图所示，∠DAC=25°，BA垂直于DA。求∠BAC的大小。
Construction (CDL):
Shape(BA,AD)
Shape(DA,AC)
Conditions (CDL):
Equal(MeasureOfAngle(DAC),25)
PerpendicularBetweenLine(BA,DA)
Goal: Value(MeasureOfAngle(BAC))
Answer: 115
Theorem sequence:
angle_addition(1,BAD,DAC)
Problem: As shown in the diagram, ∠FEA=90°, ∠FED=35°, BC∥DE. Find the measure of ∠BCA.
问题：如图所示，∠FEA=90°，∠FED=35°，BC∥DE。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DE,EC,CB)
Shape(DF,FE,ED)
Collinear(ABDF)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(FEA),90)
Equal(MeasureOfAngle(FED),35)
ParallelBetweenLine(BC,DE)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
Theorem sequence:
parallel_property_corresponding_angle(2,ED,CB,A)
angle_addition(1,FED,DEC)
Problem: As shown in the diagram, ∠EAF=50°, BE is the altitude of triangle BCA, CF is the altitude of triangle CAB. Find the measure of ∠CHB.
问题：如图所示，∠EAF=50°，BE是△BCA的高，CF是三角形CAB的高。求∠CHB的大小。
Construction (CDL):
Shape(AF,FH,HE,EA)
Shape(FB,BH,HF)
Shape(HB,BC,CH)
Shape(HC,CE,EH)
Collinear(AFB)
Collinear(AEC)
Collinear(FHC)
Collinear(BHE)
Conditions (CDL):
Equal(MeasureOfAngle(EAF),50)
IsAltitudeOfTriangle(BE,BCA)
IsAltitudeOfTriangle(CF,CAB)
Goal: Value(MeasureOfAngle(CHB))
Answer: 130
Theorem sequence:
triangle_property_angle_sum(1,ABE)
triangle_property_angle_sum(1,BHF)
adjacent_complementary_angle(1,CHB,BHF)
Problem: As shown in the diagram, ⌒OAC is congruent to ⌒OCB, ∠OAD=50°, O is the center of ⊙O. Find the measure of ∠BOD.
问题：如图所示，弧OAC与弧OCB全等，∠OAD=50°，⊙O的圆心为O。求∠BOD的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(OD,DB,BO)
Shape(AO,OB,OBA)
Shape(CD,DA,OAC)
Shape(BD,DC,OCB)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
CongruentBetweenArc(OAC,OCB)
Equal(MeasureOfAngle(OAD),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOD))
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, ∠BOA=15°, ∠DOC=105°. Find the measure of ∠COA.
问题：如图所示，∠BOA=15°，∠DOC=105°。求∠COA的大小。
Construction (CDL):
Shape(DO,OC)
Shape(CO,OB)
Shape(BO,OA)
Shape(AO,OD)
Collinear(DOB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),15)
Equal(MeasureOfAngle(DOC),105)
Goal: Value(MeasureOfAngle(COA))
Answer: 90
Theorem sequence:
adjacent_complementary_angle(1,DOC,COB)
angle_addition(1,COB,BOA)
Problem: As shown in the diagram, BC=12, ∠CAB=30°, the center of ⊙O is O. Find the diameter of circle O.
问题：如图所示，BC=12，∠CAB=30°，圆O的圆心为O。求圆O的直径。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EO,OC)
Shape(DE,EA,OAD)
Shape(ED,DB,BE)
Shape(CO,OE,EB,BC)
Shape(BD,ODB)
Shape(CB,OBC)
Collinear(AEB)
Collinear(COED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(BC),12)
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
Goal: Value(DiameterOfCircle(O))
Answer: 24
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,DOC,O)
diameter_of_circle_property_right_angle(1,DBC,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
sine_theorem(1,CDB)
diameter_of_circle_property_length_equal(1,DC,O)
Problem: As shown in the diagram, AC=BC, BD=5, DE=3, ∠ACD=90°, AD bisects ∠EAC, DE is perpendicular to AE. Find the length of line AC.
问题：如图所示，AC=BC，BD=5，DE=3，∠ACD=90°，AD是∠EAC的角平分线，DE垂直于AE。求直线AC的长度。
Construction (CDL):
Shape(AC,CD,DA)
Shape(AD,DE,EA)
Shape(ED,DB,BE)
Collinear(CDB)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AC),LengthOfLine(BC))
Equal(LengthOfLine(BD),5)
Equal(LengthOfLine(DE),3)
Equal(MeasureOfAngle(ACD),90)
IsBisectorOfAngle(AD,EAC)
PerpendicularBetweenLine(DE,AE)
Goal: Value(LengthOfLine(AC))
Answer: 8
Theorem sequence:
mirror_congruent_triangle_judgment_aas(1,DAC,DEA)
mirror_congruent_triangle_property_line_equal(1,ACD,ADE)
line_addition(1,CD,DB)
Problem: As shown in the diagram, ∠ABE=145°, ∠BCD=100°. Find the measure of ∠BAC.
问题：如图所示，∠ABE=145°，∠BCD=100°。求∠BAC的大小。
Construction (CDL):
Shape(AB,BE)
Shape(AC,CB,BA)
Shape(BC,CD)
Collinear(EBC)
Collinear(ACD)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),145)
Equal(MeasureOfAngle(BCD),100)
Goal: Value(MeasureOfAngle(BAC))
Answer: 65
Theorem sequence:
adjacent_complementary_angle(1,CBA,ABE)
adjacent_complementary_angle(1,ACB,BCD)
triangle_property_angle_sum(1,ACB)
Problem: As shown in the diagram, ∠AOC=Add(MeasureOfAngle(DOB),30)°, ∠COD=90°. Find the measure of ∠DOB.
问题：如图所示，∠AOC=Add(MeasureOfAngle(DOB),30)°，∠COD=90°。求∠DOB的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OD)
Shape(DO,OB)
Collinear(AOB)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),Add(MeasureOfAngle(DOB),30))
Equal(MeasureOfAngle(COD),90)
Goal: Value(MeasureOfAngle(DOB))
Answer: 30
Theorem sequence:
angle_addition(1,AOC,COD)
adjacent_complementary_angle(1,AOD,DOB)
Problem: As shown in the diagram, ∠AOB=40°, ∠AOC=90°, ∠BOD=90°. Find the measure of ∠COD.
问题：如图所示，∠AOB=40°，∠AOC=90°，∠BOD=90°。求∠COD的大小。
Construction (CDL):
Shape(AO,OB)
Shape(BO,OC)
Shape(CO,OD)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),40)
Equal(MeasureOfAngle(AOC),90)
Equal(MeasureOfAngle(BOD),90)
Goal: Value(MeasureOfAngle(COD))
Answer: 40
Theorem sequence:
angle_addition(1,AOB,BOC)
angle_addition(1,BOC,COD)
Problem: As shown in the diagram, △ABC is congruent to △DEF, ∠CAG=100°, ∠GFD=47°. Find the measure of ∠DEF.
问题：如图所示，三角形ABC与三角形DEF是全等三角形，∠CAG=100°，∠GFD=47°。求∠DEF的大小。
Construction (CDL):
Shape(FD,DH,HG,GF)
Shape(HB,BE,EH)
Shape(GH,HE,EG)
Shape(AG,GE,EC,CA)
Collinear(DHE)
Collinear(FGE)
Collinear(BHGA)
Collinear(BEC)
Conditions (CDL):
CongruentBetweenTriangle(ABC,DEF)
Equal(MeasureOfAngle(CAG),100)
Equal(MeasureOfAngle(GFD),47)
Goal: Value(MeasureOfAngle(DEF))
Answer: 33
Theorem sequence:
congruent_triangle_property_angle_equal(1,ABC,DEF)
triangle_property_angle_sum(1,FDE)
Problem: As shown in the diagram, ∠DCP=25°, ∠PBA=32°, AB∥DC, AB∥PE, PE∥DC. Find the measure of ∠BPC.
问题：如图所示，∠DCP=25°，∠PBA=32°，AB平行于DC，AB∥PE，PE平行于DC。求∠BPC的大小。
Construction (CDL):
Shape(PB,BA)
Shape(BP,PE)
Shape(EP,PC)
Shape(DC,CP)
Conditions (CDL):
Equal(MeasureOfAngle(DCP),25)
Equal(MeasureOfAngle(PBA),32)
ParallelBetweenLine(AB,DC)
ParallelBetweenLine(AB,PE)
ParallelBetweenLine(PE,DC)
Goal: Value(MeasureOfAngle(BPC))
Answer: 57
Theorem sequence:
parallel_property_alternate_interior_angle(1,PE,DC)
parallel_property_alternate_interior_angle(2,AB,PE)
angle_addition(1,BPE,EPC)
Problem: As shown in the diagram, ∠ABC=142°, CD∥AB. Find the measure of ∠DCB.
问题：如图所示，∠ABC=142°，CD∥AB。求∠DCB的大小。
Construction (CDL):
Shape(AB,BC)
Shape(CB,BE)
Shape(DC,CB)
Collinear(ABE)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),142)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(DCB))
Answer: 142
Theorem sequence:
parallel_property_collinear_extend(1,BA,DC,E)
adjacent_complementary_angle(1,ABC,CBE)
parallel_property_ipsilateral_internal_angle(1,CD,BE)
Problem: As shown in the diagram, AB=10, AC=CB, ∠ACD=90°, AD bisects ∠EAC, DE is perpendicular to AE. Find the perimeter of triangle EDB.
问题：如图所示，AB=10，AC=CB，∠ACD=90°，AD平分∠EAC，DE⊥AE。求三角形EDB的周长。
Construction (CDL):
Shape(AC,CD,DA)
Shape(AD,DE,EA)
Shape(ED,DB,BE)
Collinear(CDB)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),LengthOfLine(CB))
Equal(MeasureOfAngle(ACD),90)
IsBisectorOfAngle(AD,EAC)
PerpendicularBetweenLine(DE,AE)
Goal: Value(PerimeterOfTriangle(EDB))
Answer: 10
Theorem sequence:
mirror_congruent_triangle_judgment_aas(1,DAC,DEA)
mirror_congruent_triangle_property_line_equal(1,DAC,DEA)
mirror_congruent_triangle_property_line_equal(1,ACD,ADE)
line_addition(1,CD,DB)
line_addition(1,AE,EB)
triangle_perimeter_formula(1,EDB)
Problem: As shown in the diagram, BD=3, DE=3/2, ∠DCA=90°, AD is the angle bisector of ∠CAE, DE⊥BE. Find the length of line BC.
问题：如图所示，BD=3，DE=3/2，∠DCA=90°，AD是∠CAE的角平分线，DE⊥BE。求直线BC的长度。
Construction (CDL):
Shape(BD,DE,EB)
Shape(ED,DA,AE)
Shape(AD,DC,CA)
Collinear(AEB)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(BD),3)
Equal(LengthOfLine(DE),3/2)
Equal(MeasureOfAngle(DCA),90)
IsBisectorOfAngle(AD,CAE)
PerpendicularBetweenLine(DE,BE)
Goal: Value(LengthOfLine(BC))
Answer: 9/2
Theorem sequence:
adjacent_complementary_angle(1,AED,DEB)
mirror_congruent_triangle_judgment_aas(1,DAE,DCA)
mirror_congruent_triangle_property_line_equal(1,AED,ADC)
line_addition(1,BD,DC)
Problem: As shown in the diagram, ∠CAB=80°, AD bisects ∠CAB, AE bisects ∠DAB. Find the measure of ∠DAE.
问题：如图所示，∠CAB=80°，AD是∠CAB的角平分线，AE是∠DAB的角平分线。求∠DAE的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AD,DC,CA)
Shape(AE,ED,DA)
Collinear(BEDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),80)
IsBisectorOfAngle(AD,CAB)
IsBisectorOfAngle(AE,DAB)
Goal: Value(MeasureOfAngle(DAE))
Answer: 20
Theorem sequence:
angle_addition(1,CAD,DAB)
angle_addition(1,DAe,EAB)
Problem: As shown in the diagram, AC=AB, AD=AE, ∠CAB=∠EAD, ∠DAB=25°, ∠OCE=30°. Find the measure of ∠ADO.
问题：如图所示，AC=AB，AD=AE，∠CAB=∠EAD，∠DAB=25°，∠OCE=30°。求∠ADO的大小。
Construction (CDL):
Shape(BD,DA,AB)
Shape(DO,OA,AD)
Shape(OE,EA,AO)
Shape(OD,DB,BC,CO)
Shape(OC,CE,EO)
Collinear(BDOE)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AC),LengthOfLine(AB))
Equal(LengthOfLine(AD),LengthOfLine(AE))
Equal(MeasureOfAngle(CAB),MeasureOfAngle(EAD))
Equal(MeasureOfAngle(DAB),25)
Equal(MeasureOfAngle(OCE),30)
Goal: Value(MeasureOfAngle(ADO))
Answer: 55
Theorem sequence:
angle_addition(1,OAD,DAB)
angle_addition(1,EAO,OAD)
congruent_triangle_judgment_sas(1,ABD,ACE)
congruent_triangle_property_angle_equal(1,BDA,CEA)
triangle_property_angle_sum(1,ABD)
adjacent_complementary_angle(1,BDA,ADO)
Problem: As shown in the diagram, AB=8, CF=5, E bisects segment DF, BA is parallel to CF. Find the length of line DB.
问题：如图所示，AB=8，CF=5，E是线段DF的中点，BA∥CF。求直线DB的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BC,CE)
Shape(FE,EC,CF)
Collinear(ADB)
Collinear(AEC)
Collinear(DEF)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(CF),5)
IsMidpointOfLine(E,DF)
ParallelBetweenLine(BA,CF)
Goal: Value(LengthOfLine(DB))
Answer: 3
Theorem sequence:
vertical_angle(1,DEA,FEC)
parallel_property_alternate_interior_angle(2,BA,CF)
congruent_triangle_judgment_aas(1,DEA,FEC)
congruent_triangle_property_line_equal(1,EAD,ECF)
line_addition(1,AD,DB)
Problem: As shown in the diagram, ∠ABD=46°, ∠DCA=54°, AD is the angle bisector of ∠CAB. Find the measure of ∠DAB.
问题：如图所示，∠ABD=46°，∠DCA=54°，AD平分∠CAB。求∠DAB的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),46)
Equal(MeasureOfAngle(DCA),54)
IsBisectorOfAngle(AD,CAB)
Goal: Value(MeasureOfAngle(DAB))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,ABC)
angle_addition(1,CAD,DAB)
Problem: As shown in the diagram, ∠ABD=∠CAD, ∠CAB=78°. Find the measure of ∠ADC.
问题：如图所示，∠ABD=∠CAD，∠CAB=78°。求∠ADC的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),MeasureOfAngle(CAD))
Equal(MeasureOfAngle(CAB),78)
Goal: Value(MeasureOfAngle(ADC))
Answer: 78
Theorem sequence:
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠BOF=120°. Find the sum of the measure of ∠LAG, the measure of ∠GBH, the measure of ∠HCI, the measure of ∠IDJ, the measure of ∠JEK and the measure of ∠KFL.
问题：如图所示，∠BOF=120°。求∠LAG的大小、∠GBH的大小、∠HCI的大小、∠IDJ的大小、∠JEK的大小与∠KFL的大小之和。
Construction (CDL):
Shape(BH,HG,GB)
Shape(HC,CI,IH)
Shape(GH,HI,IO,OG)
Shape(AG,GO,OL,LA)
Shape(FL,LK,KF)
Shape(LO,OJ,JK,KL)
Shape(KJ,JE,EK)
Shape(ID,DJ,JO,OI)
Collinear(DIHB)
Collinear(BGOJE)
Collinear(EKLA)
Collinear(AGHC)
Collinear(CIOLF)
Collinear(FKJD)
Conditions (CDL):
Equal(MeasureOfAngle(BOF),120)
Goal: Value(Add(MeasureOfAngle(LAG),MeasureOfAngle(GBH),MeasureOfAngle(HCI),MeasureOfAngle(IDJ),MeasureOfAngle(JEK),MeasureOfAngle(KFL)))
Answer: 240
Theorem sequence:
triangle_property_angle_sum(1,BIO)
triangle_property_angle_sum(1,COG)
triangle_property_angle_sum(1,LOE)
triangle_property_angle_sum(1,CLA)
triangle_property_angle_sum(1,AGE)
triangle_property_angle_sum(1,FID)
adjacent_complementary_angle(1,HIO,OID)
adjacent_complementary_angle(1,BOL,LOJ)
adjacent_complementary_angle(1,ELO,OLA)
adjacent_complementary_angle(1,AGO,OGH)
Problem: As shown in the diagram, BC=2, BD=1/2, DE=1, triangle ADE is similar to triangle ABC.. Find the length of line AB.
问题：如图所示，BC=2，BD=1/2，DE=1，三角形ADE与三角形ABC是相似三角形。求直线AB的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(BC),2)
Equal(LengthOfLine(BD),1/2)
Equal(LengthOfLine(DE),1)
SimilarBetweenTriangle(ADE,ABC)
Goal: Value(LengthOfLine(AB))
Answer: 1
Theorem sequence:
line_addition(1,AD,DB)
similar_triangle_property_line_ratio(1,ADE,ABC)
similar_triangle_property_line_ratio(1,EAD,CAB)
Problem: As shown in the diagram, AB=10, AC=6, AD is the angle bisector of ∠BAC, DE is perpendicular to AE, ∠ACB is the right angle of the right triangle ACB. Find the length of line BE.
问题：如图所示，AB=10，AC=6，AD平分∠BAC，DE⊥AE，三角形ACB为直角三角形。求直线BE的长度。
Construction (CDL):
Shape(AC,CD,DA)
Shape(AD,DE,EA)
Shape(DB,BE,ED)
Collinear(AEB)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),6)
IsBisectorOfAngle(AD,BAC)
PerpendicularBetweenLine(DE,AE)
RightTriangle(ACB)
Goal: Value(LengthOfLine(BE))
Answer: 4
Theorem sequence:
mirror_congruent_triangle_judgment_aas(1,DAC,DEA)
mirror_congruent_triangle_property_line_equal(1,DAC,DEA)
line_addition(1,AE,EB)
Problem: As shown in the diagram, ∠CAD=35°, DC is perpendicular to OC, OB⊥AB. Find the measure of ∠ADC.
问题：如图所示，∠CAD=35°，DC垂直于OC，OB垂直于AB。求∠ADC的大小。
Construction (CDL):
Shape(AO,OB,BA)
Shape(OD,DC,CO)
Shape(OA,AD,DO)
Collinear(BOD)
Collinear(AOC)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),35)
PerpendicularBetweenLine(DC,OC)
PerpendicularBetweenLine(OB,AB)
Goal: Value(MeasureOfAngle(ADC))
Answer: 55
Theorem sequence:
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, ∠ABC=70°, ∠BCA=60°, AF⊥CF, BE⊥AE. Find the measure of ∠FHE.
问题：如图所示，∠ABC=70°，∠BCA=60°，AF垂直于CF，BE垂直于AE。求∠FHE的大小。
Construction (CDL):
Shape(AF,FH,HE,EA)
Shape(FB,BH,HF)
Shape(EH,HC,CE)
Shape(HB,BC,CH)
Collinear(AFB)
Collinear(AEC)
Collinear(FHC)
Collinear(BHE)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),70)
Equal(MeasureOfAngle(BCA),60)
PerpendicularBetweenLine(AF,CF)
PerpendicularBetweenLine(BE,AE)
Goal: Value(MeasureOfAngle(FHE))
Answer: 130
Theorem sequence:
triangle_property_angle_sum(1,ABC)
quadrilateral_property_angle_sum(1,AFHE)
Problem: As shown in the diagram, ∠ACB=90°, ∠DAE=24°, triangle CBD is mirror congruent to triangle CDE. Find the measure of ∠BDC.
问题：如图所示，∠ACB=90°，∠DAE=24°，△CBD与△CDE是镜像全等三角形。求∠BDC的大小。
Construction (CDL):
Shape(CB,BD,DC)
Shape(CD,DE,EC)
Shape(ED,DA,AE)
Collinear(CEA)
Collinear(BDA)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(DAE),24)
MirrorCongruentBetweenTriangle(CBD,CDE)
Goal: Value(MeasureOfAngle(BDC))
Answer: 69
Theorem sequence:
triangle_property_angle_sum(1,ACB)
mirror_congruent_triangle_property_angle_equal(1,CBD,CDE)
angle_addition(1,ECD,DCB)
triangle_property_angle_sum(1,CBD)
Problem: As shown in the diagram, ∠DBC=60°, ∠DEA=45°, DE is parallel to BC. Find the measure of ∠EAD.
问题：如图所示，∠DBC=60°，∠DEA=45°，DE平行于BC。求∠EAD的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(DBC),60)
Equal(MeasureOfAngle(DEA),45)
ParallelBetweenLine(DE,BC)
Goal: Value(MeasureOfAngle(EAD))
Answer: 75
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
triangle_property_angle_sum(1,ADE)
Problem: As shown in the diagram, ∠BAO=66°, ∠DOC=66°, AB is parallel to CD. Find the measure of ∠OCD.
问题：如图所示，∠BAO=66°，∠DOC=66°，AB平行于CD。求∠OCD的大小。
Construction (CDL):
Shape(AO,OB,BA)
Shape(OC,CD,DO)
Collinear(AOD)
Collinear(COB)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),66)
Equal(MeasureOfAngle(DOC),66)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(OCD))
Answer: 48
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,CD)
triangle_property_angle_sum(1,OCD)
Problem: As shown in the diagram, ∠ODE=50°, the center of circle O is O, CD is the diameter of circle O, AO∥DE. Find the measure of ∠OCA.
问题：如图所示，∠ODE=50°，⊙O的圆心为O，⊙O的直径为CD，AO平行于DE。求∠OCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OC,CA)
Shape(DO,OA,OAD)
Shape(CO,OD,DE,OEC)
Shape(ED,ODE)
Collinear(COD)
Cocircular(O,ADEC)
Conditions (CDL):
Equal(MeasureOfAngle(ODE),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(AO,DE)
Goal: Value(MeasureOfAngle(OCA))
Answer: 25
Theorem sequence:
parallel_property_alternate_interior_angle(2,AO,DE)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, ∠BCA=30°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=30°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(OA,AD,DO)
Shape(DB,BC,CD)
Shape(BD,DA,OAB)
Shape(CB,OBC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠DAB=120°, BD is the angle bisector of ∠ABE, DE is the perpendicular bisector of BC. Find the measure of ∠ECD.
问题：如图所示，∠DAB=120°，BD是∠ABE的角平分线，DE是BC的垂直平分线。求∠ECD的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BE,ED)
Shape(DE,EC,CD)
Collinear(ADC)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),120)
IsBisectorOfAngle(BD,ABE)
IsPerpendicularBisectorOfLine(DE,BC)
Goal: Value(MeasureOfAngle(ECD))
Answer: 20
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,ABD,DBE)