SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠CAD=42°, ∠CAF=90°, AD∥EF. Find the measure of ∠EFA.
问题：如图所示，∠CAD=42°，∠CAF=90°，AD平行于EF。求∠EFA的大小。
Construction (CDL):
Shape(CA,AD,DC)
Shape(AF,FG,GD,DA)
Shape(FB,BG,GF)
Shape(EF,FA)
Collinear(CDGB)
Collinear(AFB)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),42)
Equal(MeasureOfAngle(CAF),90)
ParallelBetweenLine(AD,EF)
Goal: Value(MeasureOfAngle(EFA))
Answer: 48
Theorem sequence:
angle_addition(1,CAD,DAF)
parallel_property_alternate_interior_angle(1,AD,EF)
Problem: As shown in the diagram, AO=3, BC=4, BD is the median of △ BCA, CE is the median of △ CAB, F is the midpoint of segment OB, G is the midpoint of segment OC. Find the perimeter of quadrilateral EFGD.
问题：如图所示，AO=3，BC=4，BD是△BCA的中线，CE是△CAB的中线，F平分线段OB，G平分线段OC。求EFGD的周长。
Construction (CDL):
Shape(AE,EO,OA)
Shape(AO,OD,DA)
Shape(OE,EF,FO)
Shape(DO,OG,GD)
Shape(FG,GO,OF)
Shape(EB,BF,FE)
Shape(DG,GC,CD)
Shape(DE,EF,FG,GD)
Shape(GF,FB,BC,CG)
Collinear(AEB)
Collinear(ADC)
Collinear(BFOD)
Collinear(CGOE)
Conditions (CDL):
Equal(LengthOfLine(AO),3)
Equal(LengthOfLine(BC),4)
IsMedianOfTriangle(BD,BCA)
IsMedianOfTriangle(CE,CAB)
IsMidpointOfLine(F,OB)
IsMidpointOfLine(G,OC)
Goal: Value(PerimeterOfQuadrilateral(EFGD))
Answer: 7
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,FG,OBC)
midsegment_of_triangle_judgment_midpoint(1,FE,BOA)
midsegment_of_triangle_judgment_midpoint(1,DG,CAO)
midsegment_of_triangle_judgment_midpoint(1,ED,ABC)
midsegment_of_triangle_property_length(1,FG,OBC)
midsegment_of_triangle_property_length(1,FE,BOA)
midsegment_of_triangle_property_length(1,DG,CAO)
midsegment_of_triangle_property_length(1,ED,ABC)
quadrilateral_perimeter_formula(1,EFGD)
Problem: As shown in the diagram, ∠BEF=55°, AC is parallel to DF. Find the measure of ∠ABG.
问题：如图所示，∠BEF=55°，AC平行于DF。求∠ABG的大小。
Construction (CDL):
Shape(AB,BG)
Shape(GB,BC)
Shape(CB,BE)
Shape(EB,BA)
Shape(BE,EF)
Shape(DE,EB)
Shape(HE,ED)
Shape(FE,EH)
Collinear(ABC)
Collinear(DEF)
Collinear(GBEH)
Conditions (CDL):
Equal(MeasureOfAngle(BEF),55)
ParallelBetweenLine(AC,DF)
Goal: Value(MeasureOfAngle(ABG))
Answer: 125
Theorem sequence:
parallel_property_collinear_extend(3,AC,DF,B)
parallel_property_collinear_extend(3,FD,BA,E)
parallel_property_alternate_interior_angle(2,FE,BA)
adjacent_complementary_angle(1,EBA,ABG)
Problem: As shown in the diagram, ∠FBC=36°, CE is parallel to FH, CB is perpendicular to DB. Find the measure of ∠ADE.
问题：如图所示，∠FBC=36°，CE∥FH，CB垂直于DB。求∠ADE的大小。
Construction (CDL):
Shape(CB,BD,DC)
Shape(FB,BC)
Shape(DB,BH)
Shape(AD,DE)
Collinear(CDE)
Collinear(FBH)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(FBC),36)
ParallelBetweenLine(CE,FH)
PerpendicularBetweenLine(CB,DB)
Goal: Value(MeasureOfAngle(ADE))
Answer: 54
Theorem sequence:
parallel_property_collinear_extend(3,CE,FH,D)
parallel_property_collinear_extend(3,HF,ED,B)
angle_addition(1,FBC,CBD)
adjacent_complementary_angle(1,FBD,DBH)
parallel_property_corresponding_angle(1,DE,BH,A)
Problem: As shown in the diagram, ∠DCE=144°, ∠EBA=70°, AB is parallel to EF, CD is parallel to EF. Find the measure of ∠BEC.
问题：如图所示，∠DCE=144°，∠EBA=70°，AB平行于EF，CD平行于EF。求∠BEC的大小。
Construction (CDL):
Shape(EB,BA)
Shape(BE,EC)
Shape(DC,CE)
Shape(CE,EF)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),144)
Equal(MeasureOfAngle(EBA),70)
ParallelBetweenLine(AB,EF)
ParallelBetweenLine(CD,EF)
Goal: Value(MeasureOfAngle(BEC))
Answer: 34
Theorem sequence:
angle_addition(1,BEC,CEF)
parallel_property_ipsilateral_internal_angle(1,CD,EF)
parallel_property_alternate_interior_angle(2,AB,EF)
Problem: As shown in the diagram, ∠FAB=50°, CD∥AB. Find the measure of ∠CFE.
问题：如图所示，∠FAB=50°，CD∥AB。求∠CFE的大小。
Construction (CDL):
Shape(CF,FE)
Shape(EF,FD)
Shape(DF,FA)
Shape(FA,AB)
Shape(AF,FC)
Collinear(CFD)
Collinear(EFA)
Conditions (CDL):
Equal(MeasureOfAngle(FAB),50)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(CFE))
Answer: 130
Theorem sequence:
parallel_property_collinear_extend(3,CD,AB,F)
parallel_property_corresponding_angle(1,FD,AB,E)
adjacent_complementary_angle(1,CFE,EFD)
Problem: As shown in the diagram, Add(MeasureOfAngle(DAB)=MeasureOfAngle(BCD)), AD and BC are opposite sides of the parallelogram DABC. Find the measure of ∠ABC.
问题：如图所示，Add(MeasureOfAngle(DAB)=MeasureOfAngle(BCD))，DC和AB是平行四边形DABC的一组对边。求∠ABC的大小。
Construction (CDL):
Shape(DA,AB,BC,CD)
Conditions (CDL):
Equal(Add(MeasureOfAngle(DAB),MeasureOfAngle(BCD)),100)
Parallelogram(DABC)
Goal: Value(MeasureOfAngle(ABC))
Answer: 130
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,DABC)
quadrilateral_property_angle_sum(1,DABC)
Problem: As shown in the diagram, ∠BGC=45°, ∠CEB=90°, ∠CGH=22°, ∠GCE=45°, AD is parallel to FH. Find the measure of ∠BCE.
问题：如图所示，∠BGC=45°，∠CEB=90°，∠CGH=22°，∠GCE=45°，AD平行于FH。求∠BCE的大小。
Construction (CDL):
Shape(BA,AF,FG,GB)
Shape(BG,GC,CB)
Shape(EB,BC,CE)
Shape(DC,CG,GH,HD)
Collinear(ABCD)
Collinear(FGH)
Collinear(EBG)
Conditions (CDL):
Equal(MeasureOfAngle(BGC),45)
Equal(MeasureOfAngle(CEB),90)
Equal(MeasureOfAngle(CGH),22)
Equal(MeasureOfAngle(GCE),45)
ParallelBetweenLine(AD,FH)
Goal: Value(MeasureOfAngle(BCE))
Answer: 23
Theorem sequence:
parallel_property_collinear_extend(3,AD,FH,C)
parallel_property_collinear_extend(3,HF,CA,G)
parallel_property_alternate_interior_angle(2,HG,CA)
angle_addition(1,GCB,BCE)
Problem: As shown in the diagram, ∠BFG=∠FBA, ∠GCB=60°. Find the measure of ∠HGL.
问题：如图所示，∠BFG=∠FBA，∠GCB=60°。求∠HGL的大小。
Construction (CDL):
Shape(AB,BI)
Shape(IB,BC)
Shape(BC,CJ)
Shape(JC,CD)
Shape(FB,BA)
Shape(CB,BF,FG,GC)
Shape(DC,CG)
Shape(EF,FB)
Shape(DG,GH)
Shape(KF,FE)
Shape(GF,FK)
Shape(LG,GF)
Shape(HG,GL)
Collinear(ABCD)
Collinear(EFGH)
Collinear(IBFK)
Collinear(JCGL)
Conditions (CDL):
Equal(MeasureOfAngle(BFG),MeasureOfAngle(FBA))
Equal(MeasureOfAngle(GCB),60)
Goal: Value(MeasureOfAngle(HGL))
Answer: 120
Theorem sequence:
parallel_judgment_alternate_interior_angle(2,AB,FG)
parallel_property_collinear_extend(1,GF,BA,H)
parallel_property_collinear_extend(2,AB,GH,C)
parallel_property_alternate_interior_angle(2,AC,GH)
adjacent_complementary_angle(1,JGH,HGL)
Problem: As shown in the diagram, △BEA is congruent to △AGD, triangle BEA is congruent to triangle CHB, △BEA is congruent to △DFC, AE=10, BE=24, ∠EHF=90°, ∠FGE=90°, ∠GEH=90°, ∠HFG=90°, AG and GD are the legs of the right △ AGD, BE and EA are the legs of the right △ BEA, ∠CHB is the right angle of the right triangle CHB, triangle DFC is a right triangle. Find the length of line EF.
问题：如图所示，△BEA全等于△AGD，三角形BEA全等于三角形CHB，三角形BEA全等于三角形DFC，AE=10，BE=24，∠EHF=90°，∠FGE=90°，∠GEH=90°，∠HFG=90°，△AGD为直角△，∠BEA是直角三角形BEA的直角，三角形CHB为直角三角形，∠DFC是直角△DFC的直角。求直线EF的长度。
Construction (CDL):
Shape(EA,AB,BH,HE)
Shape(CF,FH,HB,BC)
Shape(DG,GF,FC,CD)
Shape(DA,AE,EG,GD)
Shape(GE,EF,FG)
Shape(FE,EH,HF)
Collinear(AEG)
Collinear(BHE)
Collinear(DGF)
Collinear(CFH)
Conditions (CDL):
CongruentBetweenTriangle(BEA,AGD)
CongruentBetweenTriangle(BEA,CHB)
CongruentBetweenTriangle(BEA,DFC)
Equal(LengthOfLine(AE),10)
Equal(LengthOfLine(BE),24)
Equal(MeasureOfAngle(EHF),90)
Equal(MeasureOfAngle(FGE),90)
Equal(MeasureOfAngle(GEH),90)
Equal(MeasureOfAngle(HFG),90)
RightTriangle(AGD)
RightTriangle(BEA)
RightTriangle(CHB)
RightTriangle(DFC)
Goal: Value(LengthOfLine(EF))
Answer: 14*sqrt(2)
Theorem sequence:
congruent_triangle_property_line_equal(1,BEA,AGD)
congruent_triangle_property_line_equal(1,ABE,DAG)
congruent_triangle_property_line_equal(1,ABE,CDF)
line_addition(1,AE,EG)
line_addition(1,DG,GF)
right_triangle_judgment_angle(1,FGE)
right_triangle_property_pythagorean(1,FGE)
Problem: As shown in the diagram, ∠ABD=50°, ∠GEF=40°, AC∥EF. Find the measure of ∠BEG.
问题：如图所示，∠ABD=50°，∠GEF=40°，AC∥EF。求∠BEG的大小。
Construction (CDL):
Shape(AB,BD)
Shape(CB,BE)
Shape(BE,EG)
Shape(GE,EF)
Collinear(ABC)
Collinear(DBE)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),50)
Equal(MeasureOfAngle(GEF),40)
ParallelBetweenLine(AC,EF)
Goal: Value(MeasureOfAngle(BEG))
Answer: 90
Theorem sequence:
parallel_property_collinear_extend(3,AC,EF,B)
vertical_angle(1,ABD,CBE)
parallel_property_ipsilateral_internal_angle(1,BC,EF)
angle_addition(1,BEG,GEF)
Problem: As shown in the diagram, ∠BAC=105°, AB∥CD. Find the measure of ∠ACD.
问题：如图所示，∠BAC=105°，AB∥CD。求∠ACD的大小。
Construction (CDL):
Shape(BA,AC)
Shape(AC,CD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),105)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(ACD))
Answer: 75
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,CD)
Problem: As shown in the diagram, ∠BPA=50°, the center of ⊙O is O, the tangent to ⊙O is PA, the tangent to ⊙O is PB. Find the measure of ∠ACB.
问题：如图所示，∠BPA=50°，圆O的圆心为O，PA是⊙O的切线，圆O的切线为PB。求∠ACB的大小。
Construction (CDL):
Shape(OA,AC,CB,BO)
Shape(OBA,AO,OB)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(PA,OBA,BP)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 65
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PB,O,O)
tangent_of_circle_property_perpendicular(2,PA,O,O)
quadrilateral_property_angle_sum(1,AOBP)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, AB=10, BC=8, ED is the perpendicular bisector of CA. Find the perimeter of △EBC.
问题：如图所示，AB=10，BC=8，ED垂直平分AB。求△EBC的周长。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EC,CD)
Shape(EB,BC,CE)
Collinear(ADC)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),8)
IsPerpendicularBisectorOfLine(ED,CA)
Goal: Value(PerimeterOfTriangle(EBC))
Answer: 18
Theorem sequence:
line_addition(1,AE,EB)
perpendicular_bisector_property_distance_equal(1,ED,CA)
triangle_perimeter_formula(1,EBC)
Problem: As shown in the diagram, AB=AC, ∠EAD=30°, ∠FDA=145°, FE is parallel to GC. Find the measure of ∠GCB.
问题：如图所示，AB=AC，∠EAD=30°，∠FDA=145°，FE平行于GC。求∠GCB的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(FD,DA)
Shape(DB,BC,CE,ED)
Shape(BD,DF)
Shape(GC,CB)
Collinear(FDE)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(EAD),30)
Equal(MeasureOfAngle(FDA),145)
ParallelBetweenLine(FE,GC)
Goal: Value(MeasureOfAngle(GCB))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,ADE)
triangle_property_angle_sum(1,ABC)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
adjacent_complementary_angle(1,FDA,ADE)
adjacent_complementary_angle(1,CED,DEA)
parallel_property_ipsilateral_internal_angle(1,CG,EF)
angle_addition(1,GCB,BCE)
Problem: As shown in the diagram, ∠ACE=70°, AE is the angle bisector of ∠BAC, AB∥CD. Find the measure of ∠AED.
问题：如图所示，∠ACE=70°，AE是∠BAC的角平分线，AB∥CD。求∠AED的大小。
Construction (CDL):
Shape(BA,AE)
Shape(AE,ED)
Shape(AC,CE,EA)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),70)
IsBisectorOfAngle(AE,BAC)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(AED))
Answer: 125
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,CD)
angle_addition(1,BAE,EAC)
triangle_property_angle_sum(1,ACE)
adjacent_complementary_angle(1,CEA,AED)
Problem: As shown in the diagram, the perimeter of ABCD is 32, quadrilateral ABCD is a ▱, EO is perpendicular to CO. Find the perimeter of △DEC.
问题：如图所示，ABCD的周长为32，BA和CD是平行四边形ABCD的一组对边，EO⊥CO。求△DEC的周长。
Construction (CDL):
Shape(EA,AO,OE)
Shape(EO,OC,CE)
Shape(DE,EC,CD)
Shape(OA,AB,BO)
Shape(CO,OB,BC)
Shape(AB,BO,OD,DA)
Shape(DO,OB,BC,CD)
Collinear(AED)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),32)
Parallelogram(ABCD)
PerpendicularBetweenLine(EO,CO)
Goal: Value(PerimeterOfTriangle(DEC))
Answer: 16
Theorem sequence:
quadrilateral_perimeter_formula(1,ABCD)
adjacent_complementary_angle(1,AOE,EOC)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
perpendicular_bisector_judgment_per_and_mid(1,EO,AC)
perpendicular_bisector_property_distance_equal(1,EO,AC)
line_addition(1,AE,ED)
triangle_perimeter_formula(1,DEC)
Problem: As shown in the diagram, AB=10, BF=12, AO bisects ∠FAB, BO bisects ∠ABE, quadrilateral ABCD is a parallelogram. Find the length of line AE.
问题：如图所示，AB=10，BF=12，AO是∠FAB的角平分线，BO平分∠ABE，AD和BC是平行四边形ABCD的一组对边。求直线AE的长度。
Construction (CDL):
Shape(FA,AO,OF)
Shape(OA,AB,BO)
Shape(OB,BE,EO)
Shape(FO,OE,EF)
Shape(DF,FE,EC,CD)
Collinear(AFD)
Collinear(BEC)
Collinear(AOE)
Collinear(BOF)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BF),12)
IsBisectorOfAngle(AO,FAB)
IsBisectorOfAngle(BO,ABE)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AE))
Answer: 16
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,F)
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_collinear_extend(3,CB,FA,E)
line_addition(1,BO,OF)
parallel_property_alternate_interior_angle(1,EB,DA)
parallel_property_alternate_interior_angle(2,AF,BC)
isosceles_triangle_judgment_angle_equal(1,ABF)
isosceles_triangle_judgment_angle_equal(1,BEA)
parallelogram_judgment_parallel_and_equal(1,FABE)
parallelogram_property_opposite_line_equal(1,ABEF)
kite_judgment_equal_and_equal(1,FABE)
kite_judgment_equal_and_equal(1,ABEF)
kite_property_diagonal_perpendicular_bisection(1,FABE,O)
kite_property_diagonal_perpendicular_bisection(1,ABEF,O)
right_triangle_judgment_angle(1,AOF)
right_triangle_property_pythagorean(1,AOF)
line_addition(1,AO,OE)
Problem: As shown in the diagram, Add(MeasureOfAngle(GBA)=MeasureOfAngle(FGB)), ∠IHG=50°. Find the measure of ∠DCE.
问题：如图所示，Add(MeasureOfAngle(GBA)=MeasureOfAngle(FGB))，∠IHG=50°。求∠DCE的大小。
Construction (CDL):
Shape(GB,BA)
Shape(CB,BG,GH,HC)
Shape(DC,CE)
Shape(EC,CH)
Shape(FG,GA)
Shape(IG,GF)
Shape(HG,GI,IH)
Collinear(ABCE)
Collinear(FGH)
Collinear(BGI)
Collinear(IHCD)
Conditions (CDL):
Equal(Add(MeasureOfAngle(GBA),MeasureOfAngle(FGB)),180)
Equal(MeasureOfAngle(IHG),50)
Goal: Value(MeasureOfAngle(DCE))
Answer: 50
Theorem sequence:
parallel_judgment_ipsilateral_internal_angle(1,GF,BA)
parallel_property_collinear_extend(1,GF,BA,H)
parallel_property_collinear_extend(2,AB,GH,C)
parallel_property_corresponding_angle(1,HG,CA,I)
vertical_angle(1,ICA,DCE)
Problem: As shown in the diagram, Add(MeasureOfAngle(HAB)=MeasureOfAngle(EDG)), ∠ABC=104°. Find the measure of ∠BEF.
问题：如图所示，Add(MeasureOfAngle(HAB)=MeasureOfAngle(EDG))，∠ABC=104°。求∠BEF的大小。
Construction (CDL):
Shape(HA,AB)
Shape(AB,BC)
Shape(BA,AD,DE,EB)
Shape(ED,DG)
Shape(BE,EF)
Collinear(HADG)
Collinear(CBE)
Collinear(DEF)
Conditions (CDL):
Equal(Add(MeasureOfAngle(HAB),MeasureOfAngle(EDG)),180)
Equal(MeasureOfAngle(ABC),104)
Goal: Value(MeasureOfAngle(BEF))
Answer: 76
Theorem sequence:
adjacent_complementary_angle(1,HDE,EDG)
adjacent_complementary_angle(1,DEB,BEF)
parallel_judgment_corresponding_angle(1,AB,DE,H)
parallel_property_corresponding_angle(2,ED,BA,C)
Problem: As shown in the diagram, AC=10, BD=6, E bisects segment AB, F bisects segment BC, G bisects segment DC, H bisects segment AD, AO is perpendicular to DO, BO⊥AO, CO is perpendicular to BO, DO is perpendicular to CO. Find the area of EFGH.
问题：如图所示，AC=10，BD=6，E平分线段AB，F平分线段BC，G平分线段DC，H是线段AD的中点，AO垂直于DO，BO⊥AO，CO垂直于BO，DO⊥CO。求EFGH的面积。
Construction (CDL):
Shape(DH,HI,IO,OD)
Shape(AI,IH,HA)
Shape(AE,EI,IA)
Shape(EB,BO,OI,IE)
Shape(DO,OC,CG,GD)
Shape(CO,OB,BF,FC)
Shape(HI,IE,EF,FG,GH)
Collinear(HIE)
Collinear(AIOC)
Collinear(BOD)
Collinear(AHD)
Collinear(AEB)
Collinear(CGD)
Collinear(CFB)
Conditions (CDL):
Equal(LengthOfLine(AC),10)
Equal(LengthOfLine(BD),6)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,BC)
IsMidpointOfLine(G,DC)
IsMidpointOfLine(H,AD)
PerpendicularBetweenLine(AO,DO)
PerpendicularBetweenLine(BO,AO)
PerpendicularBetweenLine(CO,BO)
PerpendicularBetweenLine(DO,CO)
Goal: Value(AreaOfQuadrilateral(EFGH))
Answer: 15
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,GF,CDB)
midsegment_of_triangle_judgment_midpoint(1,HG,DAC)
midsegment_of_triangle_judgment_midpoint(1,FE,BCA)
midsegment_of_triangle_judgment_midpoint(1,EH,ABD)
midsegment_of_triangle_property_parallel(1,GF,CDB)
midsegment_of_triangle_property_parallel(1,HG,DAC)
midsegment_of_triangle_property_parallel(1,FE,BCA)
midsegment_of_triangle_property_parallel(1,EH,ABD)
parallel_property_collinear_extend(3,AC,EF,I)
parallel_property_collinear_extend(3,EH,BD,I)
parallel_property_collinear_extend(3,DB,IE,O)
parallel_judgment_par_par(1,GF,DB,HE)
parallel_judgment_par_par(1,HG,AC,EF)
parallel_property_ipsilateral_internal_angle(1,IC,EF)
parallel_property_ipsilateral_internal_angle(1,OB,IE)
midsegment_of_triangle_property_length(1,FE,BCA)
midsegment_of_triangle_property_length(1,EH,ABD)
parallelogram_judgment_parallel_and_parallel(1,HEFG)
parallelogram_area_formula_sine(1,HEFG)
Problem: As shown in the diagram, ∠BDA=65°, ∠DBC=140°, EF∥AC. Find the measure of ∠FED.
问题：如图所示，∠BDA=65°，∠DBC=140°，EF平行于AC。求∠FED的大小。
Construction (CDL):
Shape(DA,AB,BD)
Shape(FE,ED)
Shape(DB,BC)
Collinear(ABC)
Collinear(ADE)
Conditions (CDL):
Equal(MeasureOfAngle(BDA),65)
Equal(MeasureOfAngle(DBC),140)
ParallelBetweenLine(EF,AC)
Goal: Value(MeasureOfAngle(FED))
Answer: 105
Theorem sequence:
adjacent_complementary_angle(1,ABD,DBC)
triangle_property_angle_sum(1,DAB)
parallel_property_ipsilateral_internal_angle(1,EF,AC)
Problem: As shown in the diagram, ∠ACD=50°, OE is the angle bisector of ∠AOB, CD∥OB. Find the measure of ∠ODC.
问题：如图所示，∠ACD=50°，OE是∠AOB的角平分线，CD∥OB。求∠ODC的大小。
Construction (CDL):
Shape(AC,CD)
Shape(DC,CO,OD)
Shape(CD,DE)
Shape(DO,OB)
Collinear(OCA)
Collinear(ODE)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),50)
IsBisectorOfAngle(OE,AOB)
ParallelBetweenLine(CD,OB)
Goal: Value(MeasureOfAngle(ODC))
Answer: 25
Theorem sequence:
angle_addition(1,COD,DOB)
parallel_property_corresponding_angle(1,CD,OB,A)
parallel_property_alternate_interior_angle(2,CD,OB)
Problem: As shown in the diagram, ∠CFB=115°, BF is the angle bisector of ∠ABC, CF bisects ∠BCA. Find the measure of ∠CAB.
问题：如图所示，∠CFB=115°，BF平分∠ABC，CF是∠BCA的角平分线。求∠CAB的大小。
Construction (CDL):
Shape(AB,BF,FC,CA)
Shape(FB,BC,CF)
Conditions (CDL):
Equal(MeasureOfAngle(CFB),115)
IsBisectorOfAngle(BF,ABC)
IsBisectorOfAngle(CF,BCA)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
angle_addition(1,ABF,FBC)
angle_addition(1,BCF,FCA)
triangle_property_angle_sum(1,FBC)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠FEC=90°, ∠FED=30°, BC is parallel to DE. Find the measure of ∠GCE.
问题：如图所示，∠FEC=90°，∠FED=30°，BC平行于DE。求∠GCE的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DE,EC,CB)
Shape(DF,FE,ED)
Shape(GC,CE)
Collinear(ABDF)
Collinear(BCG)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(FEC),90)
Equal(MeasureOfAngle(FED),30)
ParallelBetweenLine(BC,DE)
Goal: Value(MeasureOfAngle(GCE))
Answer: 60
Theorem sequence:
parallel_property_collinear_extend(2,BC,DE,G)
angle_addition(1,FED,DEC)
parallel_property_alternate_interior_angle(1,CG,DE)
Problem: As shown in the diagram, the measure of arc OBC is 60, the measure of ⌒OBD is 180, the center of circle O is O, the diameter of circle O is BD. Find the measure of ∠DBC.
问题：如图所示，弧OBC的角度为60，⌒OBD的角度为180，圆O的圆心为O，圆O的直径为BD。求∠DBC的大小。
Construction (CDL):
Shape(ODA,AB,BO,OD)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DO,OB,BC)
Collinear(BOD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfArc(OBC),60)
Equal(MeasureOfArc(OBD),180)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(MeasureOfAngle(DBC))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OCD,B)
arc_addition_measure(1,OBC,OCD)
Problem: As shown in the diagram, ∠DAE=125°, AD and BC are opposite sides of the ▱ ABCD, AE⊥CE. Find the measure of ∠BCE.
问题：如图所示，∠DAE=125°，四边形ABCD是▱，AE⊥CE。求∠BCE的大小。
Construction (CDL):
Shape(AE,EC,CD,DA)
Shape(EB,BC,CE)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(DAE),125)
Parallelogram(ABCD)
PerpendicularBetweenLine(AE,CE)
Goal: Value(MeasureOfAngle(BCE))
Answer: 35
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,CB,DA)
quadrilateral_property_angle_sum(1,AECD)
angle_addition(1,BCE,ECD)
Problem: As shown in the diagram, ∠BCD=108°, BE is the angle bisector of ∠ABC, quadrilateral ABCD is a parallelogram. Find the measure of ∠BEA.
问题：如图所示，∠BCD=108°，BE平分∠ABC，BA和CD是▱ABCD的一组对边。求∠BEA的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BC,CD,DE)
Collinear(AED)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),108)
IsBisectorOfAngle(BE,ABC)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BEA))
Answer: 36
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,BA,CD)
angle_addition(1,ABE,EBC)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_alternate_interior_angle(2,AE,BC)
Problem: As shown in the diagram, ∠COE=55°, EO is perpendicular to BO. Find the measure of ∠BOD.
问题：如图所示，∠COE=55°，EO垂直于BO。求∠BOD的大小。
Construction (CDL):
Shape(CO,OE)
Shape(EO,OB)
Shape(BO,OD)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(COE),55)
PerpendicularBetweenLine(EO,BO)
Goal: Value(MeasureOfAngle(BOD))
Answer: 35
Theorem sequence:
angle_addition(1,COE,EOB)
adjacent_complementary_angle(1,COB,BOD)
Problem: As shown in the diagram, ∠DAE=35°, ∠ECB=90°, ED perpendicular bisects BA, ∠ACB is the right angle of the right triangle ACB. Find the measure of ∠CBE.
问题：如图所示，∠DAE=35°，∠ECB=90°，ED是BA的垂直平分线，三角形ACB为直角三角形。求∠CBE的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EB,BD)
Shape(EC,CB,BE)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(DAE),35)
Equal(MeasureOfAngle(ECB),90)
IsPerpendicularBisectorOfLine(ED,BA)
RightTriangle(ACB)
Goal: Value(MeasureOfAngle(CBE))
Answer: 20
Theorem sequence:
triangle_property_angle_sum(1,ACB)
perpendicular_bisector_property_distance_equal(1,ED,BA)
isosceles_triangle_judgment_line_equal(1,EBA)
isosceles_triangle_property_angle_equal(1,EBA)
angle_addition(1,CBE,EBD)
Problem: As shown in the diagram, ∠BDF=130°, BC∥EF, AB is perpendicular to CB. Find the measure of ∠GFH.
问题：如图所示，∠BDF=130°，BC平行于EF，AB垂直于CB。求∠GFH的大小。
Construction (CDL):
Shape(AB,BC)
Shape(CB,BD)
Shape(BD,DF)
Shape(FD,DE,EF)
Shape(DF,FG)
Shape(GF,FH)
Collinear(ABDE)
Collinear(DFH)
Collinear(EFG)
Conditions (CDL):
Equal(MeasureOfAngle(BDF),130)
ParallelBetweenLine(BC,EF)
PerpendicularBetweenLine(AB,CB)
Goal: Value(MeasureOfAngle(GFH))
Answer: 40
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBD)
adjacent_complementary_angle(1,ADF,FDE)
adjacent_complementary_angle(1,DFG,GFH)
adjacent_complementary_angle(1,EFD,DFG)
parallel_property_ipsilateral_internal_angle(1,BC,EF)
triangle_property_angle_sum(1,FDE)
Problem: As shown in the diagram, ∠ABC=50°, ∠CAB=90°, AE bisects ∠CAD, AD is perpendicular to CD, BD is perpendicular to AD. Find the measure of ∠EAD.
问题：如图所示，∠ABC=50°，∠CAB=90°，AE是∠CAD的角平分线，AD垂直于CD，BD⊥AD。求∠EAD的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(AE,EC,CA)
Collinear(BDEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
Equal(MeasureOfAngle(CAB),90)
IsBisectorOfAngle(AE,CAD)
PerpendicularBetweenLine(AD,CD)
PerpendicularBetweenLine(BD,AD)
Goal: Value(MeasureOfAngle(EAD))
Answer: 25
Theorem sequence:
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,ABC)
angle_addition(1,CAE,EAD)
Problem: As shown in the diagram, ∠OAB=55°, ABCD is a rectangle. Find the measure of ∠AOD.
问题：如图所示，∠OAB=55°，ABCD是长方形。求∠AOD的大小。
Construction (CDL):
Shape(OA,AB,BO)
Shape(OD,DA,AO)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),55)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(AOD))
Answer: 110
Theorem sequence:
angle_addition(1,DAO,OAB)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
rectangle_property_diagonal_equal(1,ABCD)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,ODA)
Problem: As shown in the diagram, AB=20, BD=30, CE=90, DB⊥AB, EC is perpendicular to AC. Find the length of line BC.
问题：如图所示，AB=20，BD=30，CE=90，DB垂直于AB，EC⊥AC。求直线BC的长度。
Construction (CDL):
Shape(EC,CB,BD,DE)
Shape(DB,BA,AD)
Collinear(ADE)
Collinear(CBA)
Conditions (CDL):
Equal(LengthOfLine(AB),20)
Equal(LengthOfLine(BD),30)
Equal(LengthOfLine(CE),90)
PerpendicularBetweenLine(DB,AB)
PerpendicularBetweenLine(EC,AC)
Goal: Value(LengthOfLine(BC))
Answer: 40
Theorem sequence:
similar_triangle_judgment_aa(1,DBA,ECA)
similar_triangle_property_line_ratio(1,DBA,ECA)
similar_triangle_property_line_ratio(1,ADB,AEC)
line_addition(1,CB,BA)