SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=8, the perimeter of △OCD is 23, BA and CD are opposite sides of the parallelogram ABCD. Find the sum of the length of line BD and the length of line AC.
问题：如图所示，AB=8，△OCD的周长为23，ABCD是▱。求直线BD的长度与直线AC的长度之和。
Construction (CDL):
Shape(AO,OD,DA)
Shape(OA,AB,BO)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(PerimeterOfTriangle(OCD),23)
Parallelogram(ABCD)
Goal: Value(Add(LengthOfLine(BD),LengthOfLine(AC)))
Answer: 30
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
triangle_perimeter_formula(1,OCD)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
Problem: As shown in the diagram, AD=8, EC=2, AE is the angle bisector of ∠DAB, quadrilateral ABCD is a ▱. Find the length of line AB.
问题：如图所示，AD=8，EC=2，AE平分∠DAB，四边形ABCD是平行四边形。求直线AB的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EC,CD,DA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AD),8)
Equal(LengthOfLine(EC),2)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 6
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_alternate_interior_angle(1,AD,BE)
isosceles_triangle_judgment_angle_equal(1,BEA)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,BE,EC)
Problem: As shown in the diagram, AC=24, AD=28, BD=38, quadrilateral ABCD is a ▱. Find the perimeter of triangle COB.
问题：如图所示，AC=24，AD=28，BD=38，AD和BC是平行四边形ABCD的一组对边。求△COB的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AC),24)
Equal(LengthOfLine(AD),28)
Equal(LengthOfLine(BD),38)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(COB))
Answer: 59
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
triangle_perimeter_formula(1,COB)
Problem: As shown in the diagram, ∠EAD=54°, AD and BC are opposite sides of the ▱ ABCD, FE is perpendicular to AE. Find the measure of ∠BCE.
问题：如图所示，∠EAD=54°，四边形ABCD是平行四边形，FE垂直于AE。求∠BCE的大小。
Construction (CDL):
Shape(EA,AF,FE)
Shape(AB,BC,CF,FA)
Shape(FC,CD,DF)
Collinear(EAB)
Collinear(AFD)
Collinear(EFC)
Conditions (CDL):
Equal(MeasureOfAngle(EAD),54)
Parallelogram(ABCD)
PerpendicularBetweenLine(FE,AE)
Goal: Value(MeasureOfAngle(BCE))
Answer: 36
Theorem sequence:
parallel_property_corresponding_angle(1,AD,BC,E)
triangle_property_angle_sum(1,EBC)
Problem: As shown in the diagram, BD=CD, ∠BCD=70°, quadrilateral ABCD is a parallelogram, BE is perpendicular to AE. Find the measure of ∠EAB.
问题：如图所示，BD=CD，∠BCD=70°，BA和CD是平行四边形ABCD的一组对边，BE⊥AE。求∠EAB的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(AB,BE,EA)
Shape(EB,BC,CD,DE)
Collinear(DEB)
Conditions (CDL):
Equal(LengthOfLine(BD),LengthOfLine(CD))
Equal(MeasureOfAngle(BCD),70)
Parallelogram(ABCD)
PerpendicularBetweenLine(BE,AE)
Goal: Value(MeasureOfAngle(EAB))
Answer: 50
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DBC)
isosceles_triangle_property_angle_equal(1,DBC)
parallel_property_ipsilateral_internal_angle(1,BA,CD)
angle_addition(1,ABD,DBC)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, AB=10, AD=6, OF=3, quadrilateral ABCD is a parallelogram. Find the perimeter of BCEF.
问题：如图所示，AB=10，AD=6，OF=3，BA和CD是▱ABCD的一组对边。求四边形BCEF的周长。
Construction (CDL):
Shape(OA,AF,FO)
Shape(OF,FB,BO)
Shape(OB,BC,CO)
Shape(OC,CE,EO)
Shape(OE,ED,DO)
Shape(OD,DA,AO)
Collinear(DEC)
Collinear(AFB)
Collinear(EOF)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AD),6)
Equal(LengthOfLine(OF),3)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(BCEF))
Answer: 22
Theorem sequence:
parallel_property_alternate_interior_angle(1,DC,AB)
parallel_property_alternate_interior_angle(2,DC,AB)
vertical_angle(1,DOE,BOF)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
vertical_angle(1,EOC,FOA)
congruent_triangle_judgment_aas(2,EDO,FBO)
congruent_triangle_judgment_aas(2,EOC,FOA)
congruent_triangle_property_line_equal(1,DOE,BOF)
congruent_triangle_property_line_equal(1,OCE,OAF)
line_addition(1,AF,FB)
line_addition(1,EO,OF)
parallelogram_property_opposite_line_equal(1,DABC)
quadrilateral_perimeter_formula(1,EFBC)
Problem: As shown in the diagram, the perimeter of quadrilateral ABCD is 20, AD and BC are opposite sides of the parallelogram ABCD, BO is perpendicular to EO. Find the perimeter of triangle ABE.
问题：如图所示，四边形ABCD的周长为20，BA和CD是▱ABCD的一组对边，BO垂直于EO。求△ABE的周长。
Construction (CDL):
Shape(AB,BF,FA)
Shape(AF,FE,EA)
Shape(BO,OF,FB)
Shape(FO,OE,EF)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Shape(OD,DE,EO)
Collinear(AED)
Collinear(AFOC)
Collinear(BOD)
Collinear(BFE)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),20)
Parallelogram(ABCD)
PerpendicularBetweenLine(BO,EO)
Goal: Value(PerimeterOfTriangle(ABE))
Answer: 10
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_diagonal_bisection(1,BCDA,O)
perpendicular_bisector_judgment_per_and_mid(1,EO,BD)
perpendicular_bisector_property_distance_equal(1,EO,BD)
line_addition(1,AE,ED)
quadrilateral_perimeter_formula(1,ABCD)
triangle_perimeter_formula(1,BEA)
Problem: As shown in the diagram, AC=38, AD=14, BD=24, quadrilateral ABCD is a ▱. Find the perimeter of triangle OBC.
问题：如图所示，AC=38，AD=14，BD=24，BA和CD是平行四边形ABCD的一组对边。求三角形OBC的周长。
Construction (CDL):
Shape(DO,OC,CD)
Shape(DA,AO,OD)
Shape(AB,BO,OA)
Shape(BC,CO,OB)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AC),38)
Equal(LengthOfLine(AD),14)
Equal(LengthOfLine(BD),24)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(OBC))
Answer: 45
Theorem sequence:
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,DO,OB)
line_addition(1,AO,OC)
triangle_perimeter_formula(1,OBC)
Problem: As shown in the diagram, AC=10, AD=5, BD=6, BA and CD are opposite sides of the ▱ ABCD. Find the perimeter of △BCO.
问题：如图所示，AC=10，AD=5，BD=6，AD和BC是平行四边形ABCD的一组对边。求三角形BCO的周长。
Construction (CDL):
Shape(DA,AO,OD)
Shape(OA,AB,BO)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(DOB)
Conditions (CDL):
Equal(LengthOfLine(AC),10)
Equal(LengthOfLine(AD),5)
Equal(LengthOfLine(BD),6)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(BCO))
Answer: 13
Theorem sequence:
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,AO,OC)
line_addition(1,DO,OB)
triangle_perimeter_formula(1,OBC)
Problem: As shown in the diagram, ∠DAB=120°, ABCD is a parallelogram. Find the measure of ∠DCE.
问题：如图所示，∠DAB=120°，BA和CD是▱ABCD的一组对边。求∠DCE的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Shape(DC,CE)
Collinear(BCE)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),120)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(DCE))
Answer: 60
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,ABCD)
adjacent_complementary_angle(1,BCD,DCE)
Problem: As shown in the diagram, ∠CDA=55°, BA and CD are opposite sides of the parallelogram ABCD, CE is perpendicular to BE. Find the measure of ∠BCE.
问题：如图所示，∠CDA=55°，ABCD是▱，CE⊥BE。求∠BCE的大小。
Construction (CDL):
Shape(AE,EC,CD,DA)
Shape(EB,BC,CE)
Collinear(BEA)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),55)
Parallelogram(ABCD)
PerpendicularBetweenLine(CE,BE)
Goal: Value(MeasureOfAngle(BCE))
Answer: 35
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
triangle_property_angle_sum(1,EBC)
Problem: As shown in the diagram, AB=6, AD=9, AF bisects ∠DAB, BA and CD are opposite sides of the ▱ ABCD. Find the length of line CE.
问题：如图所示，AB=6，AD=9，AF是∠DAB的角平分线，AD和BC是▱ABCD的一组对边。求直线CE的长度。
Construction (CDL):
Shape(AB,BF,FA)
Shape(AF,FC,CD,DA)
Shape(FE,EC,CF)
Collinear(BFC)
Collinear(AFE)
Collinear(ECD)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AD),9)
IsBisectorOfAngle(AF,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(CE))
Answer: 3
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,F)
parallel_property_alternate_interior_angle(1,AD,BF)
isosceles_triangle_judgment_angle_equal(1,BFA)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,BF,FC)
parallel_property_collinear_extend(2,DC,AB,E)
parallel_property_alternate_interior_angle(2,DE,AB)
vertical_angle(1,BFA,CFE)
isosceles_triangle_judgment_angle_equal(1,CFE)
Problem: As shown in the diagram, DE=8, ∠DAE=30°, the height of DABC is DE, quadrilateral ABCD is a ▱. Find the length of line BC.
问题：如图所示，DE=8，∠DAE=30°，DABC的高为DE，ABCD是平行四边形。求直线BC的长度。
Construction (CDL):
Shape(DA,AE,ED)
Shape(DE,EB,BC,CD)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(DE),8)
Equal(MeasureOfAngle(DAE),30)
IsAltitudeOfQuadrilateral(DE,DABC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BC))
Answer: 16
Theorem sequence:
sine_theorem(1,DAE)
parallelogram_property_opposite_line_equal(1,DABC)
Problem: As shown in the diagram, BD=CD, ∠BCD=70°, quadrilateral ABCD is a parallelogram, BE is perpendicular to AE. Find the measure of ∠DAE.
问题：如图所示，BD=CD，∠BCD=70°，BA和CD是平行四边形ABCD的一组对边，BE⊥AE。求∠DAE的大小。
Construction (CDL):
Shape(DE,EB,BC,CD)
Shape(DA,AE,ED)
Shape(EA,AB,BE)
Collinear(DEB)
Conditions (CDL):
Equal(LengthOfLine(BD),LengthOfLine(CD))
Equal(MeasureOfAngle(BCD),70)
Parallelogram(ABCD)
PerpendicularBetweenLine(BE,AE)
Goal: Value(MeasureOfAngle(DAE))
Answer: 20
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_angle_equal(1,ABCD)
isosceles_triangle_judgment_line_equal(1,BDA)
isosceles_triangle_property_angle_equal(1,BDA)
adjacent_complementary_angle(1,BEA,AED)
triangle_property_angle_sum(1,AED)
Problem: As shown in the diagram, AB=4, AD=7, CE bisects ∠BCD, quadrilateral ABCD is a parallelogram. Find the length of line AF.
问题：如图所示，AB=4，AD=7，CE平分∠BCD，AD和BC是▱ABCD的一组对边。求直线AF的长度。
Construction (CDL):
Shape(FA,AE,EF)
Shape(AB,BC,CE,EA)
Shape(EC,CD,DE)
Collinear(AED)
Collinear(FEC)
Collinear(BAF)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(AD),7)
IsBisectorOfAngle(CE,BCD)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AF))
Answer: 3
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_alternate_interior_angle(1,ED,BC)
isosceles_triangle_judgment_angle_equal(1,DEC)
line_addition(1,AE,ED)
parallel_property_collinear_extend(2,BA,CD,F)
parallel_property_alternate_interior_angle(2,BF,CD)
vertical_angle(1,AEF,DEC)
isosceles_triangle_judgment_angle_equal(1,AEF)
Problem: As shown in the diagram, AB=4, BC=7, BE is the angle bisector of ∠ABC, BA and CD are opposite sides of the ▱ ABCD. Find the length of line ED.
问题：如图所示，AB=4，BC=7，BE是∠ABC的角平分线，ABCD是▱。求直线ED的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BC,CD,DE)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BC),7)
IsBisectorOfAngle(BE,ABC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(ED))
Answer: 3
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_alternate_interior_angle(2,AE,BC)
isosceles_triangle_judgment_angle_equal(1,ABE)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,AE,ED)
Problem: As shown in the diagram, BF=BE, ∠FBE=70°, E bisects segment BC, AD and BC are opposite sides of the ▱ ABCD, FG⊥DG. Find the measure of ∠CGE.
问题：如图所示，BF=BE，∠FBE=70°，E是线段BC的中点，AD和BC是▱ABCD的一组对边，FG⊥DG。求∠CGE的大小。
Construction (CDL):
Shape(AF,FG,GD,DA)
Shape(FE,EG,GF)
Shape(FB,BE,EF)
Shape(GE,EC,CG)
Shape(BH,HE,EB)
Collinear(AFBH)
Collinear(DGC)
Collinear(BEC)
Collinear(HEG)
Conditions (CDL):
Equal(LengthOfLine(BF),LengthOfLine(BE))
Equal(MeasureOfAngle(FBE),70)
IsMidpointOfLine(E,BC)
Parallelogram(ABCD)
PerpendicularBetweenLine(FG,DG)
Goal: Value(MeasureOfAngle(CGE))
Answer: 55
Theorem sequence:
parallel_property_collinear_extend(1,BA,CD,H)
parallel_property_collinear_extend(2,HB,CD,F)
parallel_property_collinear_extend(3,DC,BH,G)
parallel_property_collinear_extend(3,DC,FH,G)
parallel_property_alternate_interior_angle(1,GC,BH)
parallel_property_alternate_interior_angle(2,GC,BH)
parallel_property_alternate_interior_angle(2,DG,FH)
congruent_triangle_judgment_aas(1,ECG,EBH)
congruent_triangle_property_line_equal(1,CGE,BHE)
right_triangle_judgment_angle(1,GFH)
median_of_triangle_judgment(1,FE,FHG)
right_triangle_property_length_of_median(1,GFH,E)
line_addition(1,HE,EG)
isosceles_triangle_judgment_line_equal(1,BEF)
isosceles_triangle_property_angle_equal(1,BEF)
triangle_property_angle_sum(1,BEF)
isosceles_triangle_judgment_line_equal(1,EFH)
isosceles_triangle_property_angle_equal(1,EFH)
Problem: As shown in the diagram, BC=BD, ∠BCD=74°, ABCD is a ▱. Find the measure of ∠BDA.
问题：如图所示，BC=BD，∠BCD=74°，AD和BC是平行四边形ABCD的一组对边。求∠BDA的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Conditions (CDL):
Equal(LengthOfLine(BC),LengthOfLine(BD))
Equal(MeasureOfAngle(BCD),74)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BDA))
Answer: 32
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BCD)
isosceles_triangle_property_angle_equal(1,BCD)
parallel_property_ipsilateral_internal_angle(1,CB,DA)
angle_addition(1,CDB,BDA)
Problem: As shown in the diagram, AD=12, BE=4, DE is the angle bisector of ∠CDA, ABCD is a parallelogram. Find the perimeter of quadrilateral ABCD.
问题：如图所示，AD=12，BE=4，DE是∠CDA的角平分线，BA和CD是▱ABCD的一组对边。求四边形ABCD的周长。
Construction (CDL):
Shape(AB,BE,ED,DA)
Shape(DE,EC,CD)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AD),12)
Equal(LengthOfLine(BE),4)
IsBisectorOfAngle(DE,CDA)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 40
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_alternate_interior_angle(2,CE,DA)
isosceles_triangle_judgment_angle_equal(1,CDE)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,BE,EC)
quadrilateral_perimeter_formula(1,DABC)
Problem: As shown in the diagram, AB=4, BC=6, EF is the perpendicular bisector of AC, AD and BC are opposite sides of the parallelogram ABCD. Find the perimeter of triangle CDE.
问题：如图所示，AB=4，BC=6，EF垂直平分AB，ABCD是▱。求△CDE的周长。
Construction (CDL):
Shape(AB,BC,CF,FA)
Shape(AF,FE,EA)
Shape(EF,FC,CE)
Shape(EC,CD,DE)
Collinear(AED)
Collinear(AFC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BC),6)
IsPerpendicularBisectorOfLine(EF,AC)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(CDE))
Answer: 10
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,EF,AC)
line_addition(1,AE,ED)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
triangle_perimeter_formula(1,ECD)
Problem: As shown in the diagram, AB=10, BC=6, AO is the angle bisector of ∠DAB, ABCD is a parallelogram. Find the length of line EC.
问题：如图所示，AB=10，BC=6，AO平分∠DAB，四边形ABCD是平行四边形。求直线EC的长度。
Construction (CDL):
Shape(AB,BC,CO,OA)
Shape(OC,CE,EO)
Shape(AO,OD,DA)
Collinear(AOE)
Collinear(COD)
Collinear(BCE)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),6)
IsBisectorOfAngle(AO,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(EC))
Answer: 4
Theorem sequence:
parallel_property_collinear_extend(1,CB,DA,E)
parallel_property_alternate_interior_angle(1,EC,DA)
isosceles_triangle_judgment_angle_equal(1,BEA)
line_addition(1,BC,CE)
Problem: As shown in the diagram, AD=8, FE=2, AG is the angle bisector of ∠DAB, DG bisects ∠CDA, BA and CD are opposite sides of the ▱ ABCD. Find the length of line AB.
问题：如图所示，AD=8，FE=2，AG平分∠DAB，DG平分∠CDA，ABCD是▱。求直线AB的长度。
Construction (CDL):
Shape(AB,BF,FG,GA)
Shape(GF,FE,EG)
Shape(AG,GD,DA)
Shape(DG,GE,EC,CD)
Collinear(AGE)
Collinear(FGD)
Collinear(BFEC)
Conditions (CDL):
Equal(LengthOfLine(AD),8)
Equal(LengthOfLine(FE),2)
IsBisectorOfAngle(AG,DAB)
IsBisectorOfAngle(DG,CDA)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 5
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,F)
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_alternate_interior_angle(1,EB,DA)
parallel_property_alternate_interior_angle(2,CF,DA)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
isosceles_triangle_judgment_angle_equal(1,BEA)
isosceles_triangle_judgment_angle_equal(1,CDF)
line_addition(1,BF,FE)
line_addition(1,BF,FC)
Problem: As shown in the diagram, the perimeter of triangle CDE is 10, quadrilateral ABCD is a parallelogram, EO⊥BO. Find the perimeter of quadrilateral ABCD.
问题：如图所示，三角形CDE的周长为10，AD和BC是平行四边形ABCD的一组对边，EO垂直于BO。求ABCD的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BE,EO)
Shape(OE,EF,FO)
Shape(OF,FD,DO)
Shape(FE,EC,CF)
Shape(DF,FC,CD)
Collinear(AOFC)
Collinear(BEC)
Collinear(BOD)
Collinear(EFD)
Conditions (CDL):
Equal(PerimeterOfTriangle(CDE),10)
Parallelogram(ABCD)
PerpendicularBetweenLine(EO,BO)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 20
Theorem sequence:
parallelogram_property_diagonal_bisection(1,BCDA,O)
adjacent_complementary_angle(1,DOE,EOB)
mirror_congruent_triangle_judgment_sas(1,OBE,OED)
mirror_congruent_triangle_property_line_equal(1,OBE,OED)
line_addition(1,BE,EC)
triangle_perimeter_formula(1,ECD)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, DE=Mul(2,LengthOfLine(AB)), ∠EBA=50°, O is the midpoint of segment DE, quadrilateral ADCB is a ▱, EF⊥BF. Find the measure of ∠ADO.
问题：如图所示，DE=Mul(2,LengthOfLine(AB))，∠EBA=50°，O是线段DE的中点，ADCB是平行四边形，EF⊥BF。求∠ADO的大小。
Construction (CDL):
Shape(AD,DO,OA)
Shape(AO,OE,EA)
Shape(AE,EB,BA)
Shape(BE,EF,FB)
Shape(EO,OC,CF,FE)
Shape(OD,DC,CO)
Collinear(AEF)
Collinear(DOEB)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(DE),Mul(2,LengthOfLine(AB)))
Equal(MeasureOfAngle(EBA),50)
IsMidpointOfLine(O,DE)
Parallelogram(ADCB)
PerpendicularBetweenLine(EF,BF)
Goal: Value(MeasureOfAngle(ADO))
Answer: 25
Theorem sequence:
parallel_property_collinear_extend(3,BC,AD,F)
parallel_property_alternate_interior_angle(2,BF,AD)
right_triangle_judgment_angle(1,EAD)
median_of_triangle_judgment(1,AO,ADE)
right_triangle_property_length_of_median(1,EAD,O)
line_addition(1,DO,OE)
isosceles_triangle_judgment_line_equal(1,OAD)
isosceles_triangle_judgment_line_equal(1,AOB)
isosceles_triangle_property_angle_equal(1,OAD)
isosceles_triangle_property_angle_equal(1,AOB)
adjacent_complementary_angle(1,DOA,AOE)
triangle_property_angle_sum(1,ADO)
Problem: As shown in the diagram, ∠DAE=125°, BA and CD are opposite sides of the parallelogram ABCD, CE is perpendicular to BE. Find the measure of ∠BCE.
问题：如图所示，∠DAE=125°，ABCD是▱，CE垂直于BE。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(CE,BE)
Goal: Value(MeasureOfAngle(BCE))
Answer: 35
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
triangle_property_angle_sum(1,EBC)
Problem: As shown in the diagram, the perimeter of triangle CDM is 5, AD and BC are opposite sides of the ▱ ABCD, AO⊥MO. Find the perimeter of quadrilateral ABCD.
问题：如图所示，三角形CDM的周长为5，AD和BC是▱ABCD的一组对边，AO垂直于MO。求四边形ABCD的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OM,MA)
Shape(MO,ON,NM)
Shape(MN,ND,DM)
Shape(OB,BC,CO)
Shape(OC,CN,NO)
Shape(NC,CD,DN)
Collinear(BOND)
Collinear(AMD)
Collinear(AOC)
Collinear(MNC)
Conditions (CDL):
Equal(PerimeterOfTriangle(CDM),5)
Parallelogram(ABCD)
PerpendicularBetweenLine(AO,MO)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 10
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
adjacent_complementary_angle(1,AOM,MOC)
mirror_congruent_triangle_judgment_sas(1,OMA,OCM)
mirror_congruent_triangle_property_line_equal(1,OMA,OCM)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
triangle_perimeter_formula(1,CDM)
line_addition(1,AM,MD)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, Add(LengthOfLine(AC)=LengthOfLine(BD)), BC=6, ABCD is a ▱. Find the perimeter of △AOD.
问题：如图所示，Add(LengthOfLine(AC)=LengthOfLine(BD))，BC=6，四边形ABCD是平行四边形。求△AOD的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(Add(LengthOfLine(AC),LengthOfLine(BD)),18)
Equal(LengthOfLine(BC),6)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(AOD))
Answer: 15
Theorem sequence:
line_addition(1,AO,OC)
line_addition(1,BO,OD)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
parallelogram_property_opposite_line_equal(1,BCDA)
triangle_perimeter_formula(1,AOD)
Problem: As shown in the diagram, AD=3, DC=5, EF perpendicular bisects DB, quadrilateral ABCD is a parallelogram. Find the perimeter of triangle BCE.
问题：如图所示，AD=3，DC=5，EF垂直平分AB，ABCD是平行四边形。求三角形BCE的周长。
Construction (CDL):
Shape(AB,BF,FD,DA)
Shape(DF,FE,ED)
Shape(EF,FB,BE)
Shape(CE,EB,BC)
Collinear(DEC)
Collinear(DFB)
Conditions (CDL):
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(DC),5)
IsPerpendicularBisectorOfLine(EF,DB)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(BCE))
Answer: 8
Theorem sequence:
mirror_congruent_triangle_judgment_sas(1,FED,FBE)
mirror_congruent_triangle_property_line_equal(1,FED,FBE)
line_addition(1,DE,EC)
parallelogram_property_opposite_line_equal(1,BCDA)
triangle_perimeter_formula(1,CEB)
Problem: As shown in the diagram, ∠ABC=110°, ABCD is a parallelogram. Find the sum of the measure of ∠FED and the measure of ∠DFE.
问题：如图所示，∠ABC=110°，AD和BC是▱ABCD的一组对边。求∠FED的大小与∠DFE的大小之和。
Construction (CDL):
Shape(ED,DF,FE)
Shape(AB,BC,CD,DA)
Collinear(EDC)
Collinear(FDA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),110)
Parallelogram(ABCD)
Goal: Value(Add(MeasureOfAngle(FED),MeasureOfAngle(DFE)))
Answer: 70
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
vertical_angle(1,EDF,CDA)
triangle_property_angle_sum(1,EDF)
Problem: As shown in the diagram, AB=8, E is the midpoint of segment AD, quadrilateral ABCD is a parallelogram. Find the length of line OE.
问题：如图所示，AB=8，E平分线段AD，AD和BC是平行四边形ABCD的一组对边。求直线OE的长度。
Construction (CDL):
Shape(DE,EO,OD)
Shape(EA,AO,OE)
Shape(DO,OC,CD)
Shape(OA,AB,BO)
Shape(OB,BC,CO)
Collinear(DOB)
Collinear(AOC)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
IsMidpointOfLine(E,AD)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(OE))
Answer: 4
Theorem sequence:
parallelogram_property_diagonal_bisection(1,BCDA,O)
midsegment_of_triangle_judgment_midpoint(1,EO,DAB)
midsegment_of_triangle_property_length(1,EO,DAB)
Problem: As shown in the diagram, ∠CDA=53°, ABCD is a parallelogram, CE is perpendicular to BE. Find the measure of ∠BCE.
问题：如图所示，∠CDA=53°，四边形ABCD是▱，CE垂直于BE。求∠BCE的大小。
Construction (CDL):
Shape(AE,EC,CD,DA)
Shape(EB,BC,CE)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),53)
Parallelogram(ABCD)
PerpendicularBetweenLine(CE,BE)
Goal: Value(MeasureOfAngle(BCE))
Answer: 37
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
triangle_property_angle_sum(1,EBC)
Problem: As shown in the diagram, the perimeter of ABCD is 16, quadrilateral ABCD is a parallelogram, AO is perpendicular to EO. Find the perimeter of △DEC.
问题：如图所示，四边形ABCD的周长为16，ABCD是平行四边形，AO⊥EO。求△DEC的周长。
Construction (CDL):
Shape(AO,OE,EA)
Shape(EO,OF,FE)
Shape(EF,FD,DE)
Shape(FO,OC,CF)
Shape(FC,CD,DF)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Collinear(AOC)
Collinear(BOFD)
Collinear(AED)
Collinear(EFC)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),16)
Parallelogram(ABCD)
PerpendicularBetweenLine(AO,EO)
Goal: Value(PerimeterOfTriangle(DEC))
Answer: 8
Theorem sequence:
quadrilateral_perimeter_formula(1,ABCD)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_diagonal_bisection(1,ABCD,O)
adjacent_complementary_angle(1,AOE,EOC)
mirror_congruent_triangle_judgment_sas(1,OEA,OCE)
mirror_congruent_triangle_property_line_equal(1,OEA,OCE)
line_addition(1,AE,ED)
triangle_perimeter_formula(1,DEC)