SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=2, BC=3, BG is the angle bisector of ∠ABC, CG bisects ∠BCD, BA and CD are opposite sides of the ▱ ABCD. Find the length of line FE.
问题：如图所示，AB=2，BC=3，BG平分∠ABC，CG是∠BCD的角平分线，AD和BC是平行四边形ABCD的一组对边。求直线FE的长度。
Construction (CDL):
Shape(AB,BG,GF,FA)
Shape(EF,FG,GE)
Shape(DE,EG,GC,CD)
Shape(GB,BC,CG)
Collinear(AFED)
Collinear(FGC)
Collinear(BGE)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(BC),3)
IsBisectorOfAngle(BG,ABC)
IsBisectorOfAngle(CG,BCD)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(FE))
Answer: 1
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,F)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_alternate_interior_angle(1,FD,BC)
parallel_property_alternate_interior_angle(2,AE,BC)
isosceles_triangle_judgment_angle_equal(1,ABE)
isosceles_triangle_judgment_angle_equal(1,DFC)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,AE,ED)
line_addition(1,FE,ED)
Problem: As shown in the diagram, AB=4, BC=5, OE=3/2, quadrilateral ABCD is a ▱. Find the perimeter of quadrilateral EFCD.
问题：如图所示，AB=4，BC=5，OE=3/2，AD和BC是▱ABCD的一组对边。求四边形EFCD的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OE,EA)
Shape(EO,OD,DE)
Shape(OB,BF,FO)
Shape(OF,FC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(EOF)
Collinear(BOD)
Collinear(AED)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(OE),3/2)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(EFCD))
Answer: 12
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallel_property_alternate_interior_angle(1,AD,BC)
vertical_angle(1,AOE,COF)
parallelogram_property_diagonal_bisection(1,ABCD,O)
congruent_triangle_judgment_aas(2,EAO,FCO)
congruent_triangle_property_line_equal(1,AOE,COF)
congruent_triangle_property_line_equal(1,OEA,OFC)
line_addition(1,EO,OF)
line_addition(1,AE,ED)
quadrilateral_perimeter_formula(1,EFCD)
Problem: As shown in the diagram, AB=4, AD=7, BE is the angle bisector of ∠ABC, BA and CD are opposite sides of the ▱ ABCD. Find the length of line DF.
问题：如图所示，AB=4，AD=7，BE平分∠ABC，AD和BC是▱ABCD的一组对边。求直线DF的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(ED,DF,FE)
Shape(EB,BC,CD,DE)
Collinear(AED)
Collinear(BEF)
Collinear(CDF)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(AD),7)
IsBisectorOfAngle(BE,ABC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(DF))
Answer: 3
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallel_property_collinear_extend(1,DC,AB,F)
parallel_property_alternate_interior_angle(1,FD,AB)
isosceles_triangle_judgment_angle_equal(1,CFB)
line_addition(1,CD,DF)
Problem: As shown in the diagram, AB=6, AD=9, BE is the angle bisector of ∠ABC, BA and CD are opposite sides of the parallelogram ABCD. Find the length of line DF.
问题：如图所示，AB=6，AD=9，BE是∠ABC的角平分线，AD和BC是平行四边形ABCD的一组对边。求直线DF的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(ED,DF,FE)
Shape(EB,BC,CD,DE)
Collinear(AED)
Collinear(BEF)
Collinear(CDF)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AD),9)
IsBisectorOfAngle(BE,ABC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(DF))
Answer: 3
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallel_property_collinear_extend(1,DC,AB,F)
parallel_property_alternate_interior_angle(1,FD,AB)
isosceles_triangle_judgment_angle_equal(1,CFB)
line_addition(1,CD,DF)
Problem: As shown in the diagram, the perimeter of quadrilateral AFDE is 30, triangleABC is an isosceles triangle, FA and DE are opposite sides of the ▱ AFDE. Find the length of line AB.
问题：如图所示，四边形AFDE的周长为30，△ABC为等腰△，AE和FD是平行四边形AFDE的一组对边。求直线AB的长度。
Construction (CDL):
Shape(AF,FD,DE,EA)
Shape(FB,BD,DF)
Shape(ED,DC,CE)
Collinear(AFB)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(AFDE),30)
IsoscelesTriangle(ABC)
Parallelogram(AFDE)
Goal: Value(LengthOfLine(AB))
Answer: 15
Theorem sequence:
isosceles_triangle_property_angle_equal(1,ABC)
parallel_property_collinear_extend(1,FA,DE,B)
parallel_property_corresponding_angle(2,BF,DE,C)
isosceles_triangle_judgment_angle_equal(1,EDC)
parallelogram_property_opposite_line_equal(1,AFDE)
parallelogram_property_opposite_line_equal(1,FDEA)
quadrilateral_perimeter_formula(1,AFDE)
line_addition(1,AE,EC)
Problem: As shown in the diagram, AB=6, the perimeter of triangle OCD is 23, quadrilateral ABCD is a ▱. Find the sum of the length of line AC and the length of line BD.
问题：如图所示，AB=6，三角形OCD的周长为23，ABCD是▱。求直线AC的长度与直线BD的长度之和。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(PerimeterOfTriangle(OCD),23)
Parallelogram(ABCD)
Goal: Value(Add(LengthOfLine(AC),LengthOfLine(BD)))
Answer: 34
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
triangle_perimeter_formula(1,DOC)
Problem: As shown in the diagram, the perimeter of ABCD is 10, AD and BC are opposite sides of the parallelogram ABCD, AO is perpendicular to EO. Find the perimeter of triangle DEC.
问题：如图所示，四边形ABCD的周长为10，AD和BC是▱ABCD的一组对边，AO垂直于EO。求△DEC的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OE,EA)
Shape(EO,OF,FE)
Shape(EF,FD,DE)
Shape(OB,BC,CO)
Shape(OC,CF,FO)
Shape(FC,CD,DF)
Collinear(AED)
Collinear(AOC)
Collinear(BOFD)
Collinear(EFC)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),10)
Parallelogram(ABCD)
PerpendicularBetweenLine(AO,EO)
Goal: Value(PerimeterOfTriangle(DEC))
Answer: 5
Theorem sequence:
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)
quadrilateral_perimeter_formula(1,ABCD)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
triangle_perimeter_formula(1,DEC)
Problem: As shown in the diagram, DE=7, EC=2, AE bisects ∠DAB, quadrilateral ABCD is a ▱. Find the perimeter of quadrilateral ABCD.
问题：如图所示，DE=7，EC=2，AE是∠DAB的角平分线，BA和CD是平行四边形ABCD的一组对边。求四边形ABCD的周长。
Construction (CDL):
Shape(DA,AE,ED)
Shape(AB,BC,CE,EA)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(DE),7)
Equal(LengthOfLine(EC),2)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 32
Theorem sequence:
parallel_property_collinear_extend(3,DC,AB,E)
parallel_property_alternate_interior_angle(2,DE,AB)
isosceles_triangle_judgment_angle_equal(1,DAE)
line_addition(1,DE,EC)
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, DC=Mul(2,LengthOfLine(AD)), M bisects segment CD, quadrilateral ABCD is a parallelogram. Find the measure of ∠BMA.
问题：如图所示，DC=Mul(2,LengthOfLine(AD))，M是线段CD的中点，BA和CD是▱ABCD的一组对边。求∠BMA的大小。
Construction (CDL):
Shape(DA,AM,MD)
Shape(MA,AB,BM)
Shape(MB,BC,CM)
Collinear(DMC)
Conditions (CDL):
Equal(LengthOfLine(DC),Mul(2,LengthOfLine(AD)))
IsMidpointOfLine(M,CD)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BMA))
Answer: 90
Theorem sequence:
parallel_property_collinear_extend(3,DC,AB,M)
parallel_property_alternate_interior_angle(1,MC,AB)
parallel_property_alternate_interior_angle(2,DM,AB)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,DM,MC)
isosceles_triangle_judgment_line_equal(1,DAM)
isosceles_triangle_judgment_line_equal(1,CMB)
isosceles_triangle_property_angle_equal(1,DAM)
isosceles_triangle_property_angle_equal(1,CMB)
parallel_property_ipsilateral_internal_angle(1,AD,BC)
angle_addition(1,DAM,MAB)
angle_addition(1,ABM,MBC)
triangle_property_angle_sum(1,MAB)
Problem: As shown in the diagram, ∠DAE=125°, AD and BC are opposite sides of the ▱ ABCD, PE is perpendicular to BE. Find the measure of ∠EPC.
问题：如图所示，∠DAE=125°，ABCD是▱，PE⊥BE。求∠EPC的大小。
Construction (CDL):
Shape(AE,EP,PC,CD,DA)
Shape(EB,BP,PE)
Collinear(AEB)
Collinear(BPC)
Conditions (CDL):
Equal(MeasureOfAngle(DAE),125)
Parallelogram(ABCD)
PerpendicularBetweenLine(PE,BE)
Goal: Value(MeasureOfAngle(EPC))
Answer: 145
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
triangle_property_angle_sum(1,EBP)
adjacent_complementary_angle(1,BPE,EPC)
Problem: As shown in the diagram, AB=DC, AD=BC, BF=ED, ∠BEA=120°, ∠FDA=30°. Find the measure of ∠BCF.
问题：如图所示，AB=DC，AD=BC，BF=ED，∠BEA=120°，∠FDA=30°。求∠BCF的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EF,FD,DA)
Shape(BC,CF,FE,EB)
Shape(FC,CD,DF)
Collinear(BEFD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(DC))
Equal(LengthOfLine(AD),LengthOfLine(BC))
Equal(LengthOfLine(BF),LengthOfLine(ED))
Equal(MeasureOfAngle(BEA),120)
Equal(MeasureOfAngle(FDA),30)
Goal: Value(MeasureOfAngle(BCF))
Answer: 90
Theorem sequence:
parallelogram_judgment_equal_and_equal(1,ABCD)
parallel_property_alternate_interior_angle(2,AD,BC)
congruent_triangle_judgment_sas(1,DAE,BCF)
congruent_triangle_property_angle_equal(1,AED,CFB)
adjacent_complementary_angle(1,BEA,AEF)
triangle_property_angle_sum(1,AED)
Problem: As shown in the diagram, AB=6, BC=9, ∠ABC=30°, quadrilateral ABCD is a parallelogram. Find the area of ABCD.
问题：如图所示，AB=6，BC=9，∠ABC=30°，ABCD是平行四边形。求四边形ABCD的面积。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EC,CD,DA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(BC),9)
Equal(MeasureOfAngle(ABC),30)
Parallelogram(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 27
Theorem sequence:
parallelogram_area_formula_sine(1,ABCD)
Problem: As shown in the diagram, BD=CD, ∠BCE=15°, ABCD is a parallelogram, DE is perpendicular to CE. Find the measure of ∠DAB.
问题：如图所示，BD=CD，∠BCE=15°，四边形ABCD是平行四边形，DE垂直于CE。求∠DAB的大小。
Construction (CDL):
Shape(DA,AB,BE,ED)
Shape(DE,EC,CD)
Shape(CE,EB,BC)
Collinear(DEB)
Conditions (CDL):
Equal(LengthOfLine(BD),LengthOfLine(CD))
Equal(MeasureOfAngle(BCE),15)
Parallelogram(ABCD)
PerpendicularBetweenLine(DE,CE)
Goal: Value(MeasureOfAngle(DAB))
Answer: 75
Theorem sequence:
adjacent_complementary_angle(1,DEC,CEB)
triangle_property_angle_sum(1,CEB)
isosceles_triangle_judgment_line_equal(1,DBC)
isosceles_triangle_property_angle_equal(1,DBC)
parallelogram_property_opposite_angle_equal(1,ABCD)
Problem: As shown in the diagram, the perimeter of quadrilateral ABCD is 28, the perimeter of triangle ABC is 22, quadrilateral ABCD is a ▱. Find the length of line AC.
问题：如图所示，ABCD的周长为28，△ABC的周长为22，BA和CD是▱ABCD的一组对边。求直线AC的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),28)
Equal(PerimeterOfTriangle(ABC),22)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AC))
Answer: 8
Theorem sequence:
quadrilateral_perimeter_formula(1,ABCD)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
triangle_perimeter_formula(1,ABC)
Problem: As shown in the diagram, AB=4, BC=5, OE=3/2, AD and BC are opposite sides of the ▱ ABCD. Find the perimeter of EFCD.
问题：如图所示，AB=4，BC=5，OE=3/2，四边形ABCD是平行四边形。求EFCD的周长。
Construction (CDL):
Shape(AO,OE,EA)
Shape(AB,BO,OA)
Shape(EO,OD,DE)
Shape(OB,BF,FO)
Shape(OF,FC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(EOF)
Collinear(BOD)
Collinear(AED)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(OE),3/2)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(EFCD))
Answer: 12
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,BC)
vertical_angle(1,AOE,COF)
parallelogram_property_diagonal_bisection(1,ABCD,O)
congruent_triangle_judgment_aas(2,EAO,FCO)
congruent_triangle_property_line_equal(1,AOE,COF)
congruent_triangle_property_line_equal(1,OEA,OFC)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,EO,OF)
line_addition(1,AE,ED)
quadrilateral_perimeter_formula(1,DEFC)
Problem: As shown in the diagram, AB=4, AD=3, OF=3/2, ABCD is a parallelogram. Find the perimeter of quadrilateral BCEF.
问题：如图所示，AB=4，AD=3，OF=3/2，ABCD是▱。求四边形BCEF的周长。
Construction (CDL):
Shape(DO,OE,ED)
Shape(DA,AO,OD)
Shape(EO,OC,CE)
Shape(OA,AF,FO)
Shape(OF,FB,BO)
Shape(OB,BC,CO)
Collinear(AOC)
Collinear(EOF)
Collinear(BOD)
Collinear(DEC)
Collinear(AFB)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(OF),3/2)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(BCEF))
Answer: 10
Theorem sequence:
parallel_property_alternate_interior_angle(1,BA,CD)
vertical_angle(1,DOE,BOF)
parallelogram_property_diagonal_bisection(1,BCDA,O)
congruent_triangle_judgment_aas(2,EDO,FBO)
congruent_triangle_property_line_equal(1,DOE,BOF)
congruent_triangle_property_line_equal(1,OED,OFB)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,EO,OF)
line_addition(1,DE,EC)
quadrilateral_perimeter_formula(1,BCEF)
Problem: As shown in the diagram, ∠BCA=36°, ∠CAB=68°, AD and BC are opposite sides of the ▱ ABCD. Find the measure of ∠CDA.
问题：如图所示，∠BCA=36°，∠CAB=68°，BA和CD是平行四边形ABCD的一组对边。求∠CDA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),36)
Equal(MeasureOfAngle(CAB),68)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(CDA))
Answer: 76
Theorem sequence:
triangle_property_angle_sum(1,ABC)
parallelogram_property_opposite_angle_equal(1,BCDA)
Problem: As shown in the diagram, AB=5, AD=8, DE is the angle bisector of ∠CDA, AD and BC are opposite sides of the ▱ ABCD. Find the length of line BE.
问题：如图所示，AB=5，AD=8，DE平分∠CDA，BA和CD是平行四边形ABCD的一组对边。求直线BE的长度。
Construction (CDL):
Shape(AB,BE,ED,DA)
Shape(DE,EC,CD)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(AD),8)
IsBisectorOfAngle(DE,CDA)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BE))
Answer: 3
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)
Problem: As shown in the diagram, AD=Mul(2,LengthOfLine(AB)), AE=3, CE bisects ∠BCD, quadrilateral ABCD is a ▱. Find the length of line AB.
问题：如图所示，AD=Mul(2,LengthOfLine(AB))，AE=3，CE是∠BCD的角平分线，四边形ABCD是平行四边形。求直线AB的长度。
Construction (CDL):
Shape(AB,BC,CE,EA)
Shape(EC,CD,DE)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AD),Mul(2,LengthOfLine(AB)))
Equal(LengthOfLine(AE),3)
IsBisectorOfAngle(CE,BCD)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 3
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_alternate_interior_angle(1,ED,BC)
isosceles_triangle_judgment_angle_equal(1,DEC)
parallelogram_property_opposite_line_equal(1,ABCD)
line_addition(1,AE,ED)
Problem: As shown in the diagram, Add(LengthOfLine(AC)=LengthOfLine(BD)), the perimeter of triangle OCD is 23, ABCD is a parallelogram. Find the length of line AB.
问题：如图所示，Add(LengthOfLine(AC)=LengthOfLine(BD))，三角形OCD的周长为23，ABCD是▱。求直线AB的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(Add(LengthOfLine(AC),LengthOfLine(BD)),36)
Equal(PerimeterOfTriangle(OCD),23)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 5
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
triangle_perimeter_formula(1,DOC)
parallelogram_property_opposite_line_equal(1,ABCD)
Problem: As shown in the diagram, the area of triangle PCD is 3, Div(LengthOfLine(AP)=LengthOfLine(PD)), ABCD is a ▱. Find the area of quadrilateral ABCD.
问题：如图所示，三角形PCD的面积为3，Div(LengthOfLine(AP)=LengthOfLine(PD))，四边形ABCD是▱。求ABCD的面积。
Construction (CDL):
Shape(AB,BC,CP,PA)
Shape(PC,CM,MP)
Shape(DM,MC,CD)
Collinear(APMD)
Conditions (CDL):
Equal(AreaOfTriangle(PCD),3)
Equal(Div(LengthOfLine(AP),LengthOfLine(PD)),1/2)
Parallelogram(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 9
Theorem sequence:
triangle_area_formula_sine(1,DPC)
line_addition(1,AP,PD)
parallelogram_area_formula_sine(1,CDAB)
Problem: As shown in the diagram, ∠BEA=36°, BE is the angle bisector of ∠ABC, BA and CD are opposite sides of the parallelogram ABCD. Find the measure of ∠BCD.
问题：如图所示，∠BEA=36°，BE是∠ABC的角平分线，四边形ABCD是平行四边形。求∠BCD的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BC,CD,DE)
Collinear(AED)
Conditions (CDL):
Equal(MeasureOfAngle(BEA),36)
IsBisectorOfAngle(BE,ABC)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BCD))
Answer: 108
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_alternate_interior_angle(2,AE,BC)
triangle_property_angle_sum(1,ABE)
parallelogram_property_opposite_angle_equal(1,ABCD)
Problem: As shown in the diagram, AB=4, AD=5, OF=2, BA and CD are opposite sides of the parallelogram ABCD. Find the perimeter of ABFE.
问题：如图所示，AB=4，AD=5，OF=2，AD和BC是平行四边形ABCD的一组对边。求四边形ABFE的周长。
Construction (CDL):
Shape(AO,OE,EA)
Shape(AB,BO,OA)
Shape(EO,OD,DE)
Shape(OB,BF,FO)
Shape(OF,FC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(EOF)
Collinear(BOD)
Collinear(AED)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(AD),5)
Equal(LengthOfLine(OF),2)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABFE))
Answer: 13
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,BC)
vertical_angle(1,AOE,COF)
parallelogram_property_diagonal_bisection(1,ABCD,O)
congruent_triangle_judgment_aas(2,EAO,FCO)
congruent_triangle_property_line_equal(1,AOE,COF)
congruent_triangle_property_line_equal(1,OEA,OFC)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,EO,OF)
line_addition(1,BF,FC)
quadrilateral_perimeter_formula(1,ABFE)
Problem: As shown in the diagram, BC=CD, BE=6, ED=10, AD and BC are opposite sides of the ▱ ABCD. Find the perimeter of △AEB.
问题：如图所示，BC=CD，BE=6，ED=10，四边形ABCD是▱。求三角形AEB的周长。
Construction (CDL):
Shape(AE,EB,BA)
Shape(AB,BC,CD,DA)
Collinear(EAD)
Conditions (CDL):
Equal(LengthOfLine(BC),LengthOfLine(CD))
Equal(LengthOfLine(BE),6)
Equal(LengthOfLine(ED),10)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(AEB))
Answer: 16
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,EA,AD)
triangle_perimeter_formula(1,AEB)
Problem: As shown in the diagram, ∠DAB=154°, ABCD is a parallelogram. Find the measure of ∠ABC.
问题：如图所示，∠DAB=154°，BA和CD是平行四边形ABCD的一组对边。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),154)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(ABC))
Answer: 26
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
Problem: As shown in the diagram, ∠DCE=70°, BA and CD are opposite sides of the parallelogram ABCD. Find the measure of ∠DAB.
问题：如图所示，∠DCE=70°，四边形ABCD是▱。求∠DAB的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Shape(DC,CE)
Collinear(BCE)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),70)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(DAB))
Answer: 110
Theorem sequence:
adjacent_complementary_angle(1,BCD,DCE)
parallelogram_property_opposite_angle_equal(1,ABCD)
Problem: As shown in the diagram, AB=3, BC=4, BE is the angle bisector of ∠ABC, ABCD is a ▱. Find the length of line ED.
问题：如图所示，AB=3，BC=4，BE平分∠ABC，AD和BC是平行四边形ABCD的一组对边。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),3)
Equal(LengthOfLine(BC),4)
IsBisectorOfAngle(BE,ABC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(ED))
Answer: 1
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)
line_addition(1,AE,ED)
parallelogram_property_opposite_line_equal(1,BCDA)
Problem: As shown in the diagram, AB=5, the perimeter of △OCD is 23, BA and CD are opposite sides of the parallelogram ABCD. Find the sum of the length of line AC and the length of line BD.
问题：如图所示，AB=5，三角形OCD的周长为23，ABCD是平行四边形。求直线AC的长度与直线BD的长度之和。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(PerimeterOfTriangle(OCD),23)
Parallelogram(ABCD)
Goal: Value(Add(LengthOfLine(AC),LengthOfLine(BD)))
Answer: 36
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
triangle_perimeter_formula(1,DOC)
Problem: As shown in the diagram, AB=4, AD=7, BE bisects ∠ABC, AD and BC are opposite sides of the ▱ ABCD. Find the length of line ED.
问题：如图所示，AB=4，AD=7，BE平分∠ABC，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(AD),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)
line_addition(1,AE,ED)
Problem: As shown in the diagram, AC=3, the perimeter of △ABC is 10, quadrilateral ABCD is a parallelogram. Find the perimeter of ABCD.
问题：如图所示，AC=3，三角形ABC的周长为10，ABCD是平行四边形。求四边形ABCD的周长。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(PerimeterOfTriangle(ABC),10)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 14
Theorem sequence:
triangle_perimeter_formula(1,ABC)
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, AB=7, AC=12, BD=8, BA and CD are opposite sides of the ▱ ABCD. Find the perimeter of △OAB.
问题：如图所示，AB=7，AC=12，BD=8，ABCD是平行四边形。求△OAB的周长。
Construction (CDL):
Shape(DA,AO,OD)
Shape(DO,OC,CD)
Shape(OA,AB,BO)
Shape(CO,OB,BC)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AB),7)
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BD),8)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(OAB))
Answer: 17
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)
triangle_perimeter_formula(1,OAB)
Problem: As shown in the diagram, DB=DC, ∠CDE=40°, quadrilateral ABCD is a ▱, AE⊥DE. Find the measure of ∠DAE.
问题：如图所示，DB=DC，∠CDE=40°，AD和BC是▱ABCD的一组对边，AE垂直于DE。求∠DAE的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(AB,BE,EA)
Shape(CD,DE,EB,BC)
Collinear(DEB)
Conditions (CDL):
Equal(LengthOfLine(DB),LengthOfLine(DC))
Equal(MeasureOfAngle(CDE),40)
Parallelogram(ABCD)
PerpendicularBetweenLine(AE,DE)
Goal: Value(MeasureOfAngle(DAE))
Answer: 20
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DBC)
isosceles_triangle_property_angle_equal(1,DBC)
triangle_property_angle_sum(1,DBC)
parallel_property_alternate_interior_angle(2,AD,BC)
triangle_property_angle_sum(1,DAE)
Problem: As shown in the diagram, AB=9, AD=6, BE bisects ∠CBA, quadrilateral ADCB is a parallelogram. Find the length of line DE.
问题：如图所示，AB=9，AD=6，BE是∠CBA的角平分线，四边形ADCB是▱。求直线DE的长度。
Construction (CDL):
Shape(AD,DE,EB,BA)
Shape(BE,EC,CB)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(AB),9)
Equal(LengthOfLine(AD),6)
IsBisectorOfAngle(BE,CBA)
Parallelogram(ADCB)
Goal: Value(LengthOfLine(DE))
Answer: 3
Theorem sequence:
parallel_property_collinear_extend(3,CD,BA,E)
parallel_property_alternate_interior_angle(2,CE,BA)
isosceles_triangle_judgment_angle_equal(1,CBE)
line_addition(1,DE,EC)
parallelogram_property_opposite_line_equal(1,ADCB)
parallelogram_property_opposite_line_equal(1,DCBA)