SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AC=8, AD=5, ∠BOA=90°, ABCD is a parallelogram. Find the length of line BD.
问题：如图所示，AC=8，AD=5，∠BOA=90°，ABCD是▱。求直线BD的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(BOD)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(AD),5)
Equal(MeasureOfAngle(BOA),90)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BD))
Answer: 6
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
adjacent_complementary_angle(1,BOA,AOD)
line_addition(1,AO,OC)
right_triangle_judgment_angle(1,AOD)
right_triangle_property_pythagorean(1,AOD)
line_addition(1,BO,OD)
Problem: As shown in the diagram, EF=3, the perimeter of triangle DOC is 18, E is the midpoint of segment AO, F is the midpoint of segment BO, ABCD is a parallelogram. Find the sum of the length of line AC and the length of line BD.
问题：如图所示，EF=3，三角形DOC的周长为18，E平分线段AO，F平分线段BO，AD和BC是▱ABCD的一组对边。求直线AC的长度与直线BD的长度之和。
Construction (CDL):
Shape(AE,EO,OD,DA)
Shape(AB,BF,FE,EA)
Shape(EF,FO,OE)
Shape(OF,FB,BC,CO)
Shape(DO,OC,CD)
Collinear(AEOC)
Collinear(BFOD)
Conditions (CDL):
Equal(LengthOfLine(EF),3)
Equal(PerimeterOfTriangle(DOC),18)
IsMidpointOfLine(E,AO)
IsMidpointOfLine(F,BO)
Parallelogram(ABCD)
Goal: Value(Add(LengthOfLine(AC),LengthOfLine(BD)))
Answer: 24
Theorem sequence:
line_addition(1,AE,EO)
line_addition(1,BF,FO)
similar_triangle_judgment_sas(1,OEF,OAB)
similar_triangle_property_line_ratio(1,FOE,BOA)
similar_triangle_property_line_ratio(1,OEF,OAB)
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, ∠DAE=115°, ABCD is a parallelogram, CE is perpendicular to BE. Find the measure of ∠BCE.
问题：如图所示，∠DAE=115°，四边形ABCD是▱，CE⊥BE。求∠BCE的大小。
Construction (CDL):
Shape(AE,EC,CD,DA)
Shape(EB,BC,CE)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(DAE),115)
Parallelogram(ABCD)
PerpendicularBetweenLine(CE,BE)
Goal: Value(MeasureOfAngle(BCE))
Answer: 25
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
triangle_property_angle_sum(1,EBC)
Problem: As shown in the diagram, DB=DC, ∠BCD=65°, BA and CD are opposite sides of the ▱ ABCD, AE is perpendicular to DE. Find the measure of ∠DAE.
问题：如图所示，DB=DC，∠BCD=65°，BA和CD是▱ABCD的一组对边，AE垂直于DE。求∠DAE的大小。
Construction (CDL):
Shape(DE,EB,BC,CD)
Shape(EA,AB,BE)
Shape(DA,AE,ED)
Collinear(DEB)
Conditions (CDL):
Equal(LengthOfLine(DB),LengthOfLine(DC))
Equal(MeasureOfAngle(BCD),65)
Parallelogram(ABCD)
PerpendicularBetweenLine(AE,DE)
Goal: Value(MeasureOfAngle(DAE))
Answer: 25
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DBC)
isosceles_triangle_property_angle_equal(1,DBC)
parallel_property_ipsilateral_internal_angle(1,AD,BC)
angle_addition(1,ABE,EBC)
triangle_property_angle_sum(1,DAB)
triangle_property_angle_sum(1,DAE)
Problem: As shown in the diagram, AC=2, ∠ABC=30°, ABCD is a parallelogram, BC is perpendicular to AC. Find the perimeter of ABCD.
问题：如图所示，AC=2，∠ABC=30°，BA和CD是平行四边形ABCD的一组对边，BC⊥AC。求四边形ABCD的周长。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(MeasureOfAngle(ABC),30)
Parallelogram(ABCD)
PerpendicularBetweenLine(BC,AC)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 4*sqrt(3)+8
Theorem sequence:
triangle_property_angle_sum(1,ABC)
sine_theorem(1,ABC)
sine_theorem(1,CAB)
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=6, AD=8, DE is the angle bisector of ∠CDA, BA and CD are opposite sides of the parallelogram ABCD. Find the length of line BE.
问题：如图所示，AB=6，AD=8，DE是∠CDA的角平分线，ABCD是▱。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),6)
Equal(LengthOfLine(AD),8)
IsBisectorOfAngle(DE,CDA)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BE))
Answer: 2
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_alternate_interior_angle(2,AD,EC)
isosceles_triangle_judgment_angle_equal(1,CDE)
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_opposite_line_equal(1,ABCD)
line_addition(1,BE,EC)
Problem: As shown in the diagram, Add(LengthOfLine(AC)=LengthOfLine(BD)), CD=4, AD and BC are opposite sides of the parallelogram ABCD. Find the perimeter of △BOA.
问题：如图所示，Add(LengthOfLine(AC)=LengthOfLine(BD))，CD=4，ABCD是平行四边形。求三角形BOA的周长。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(Add(LengthOfLine(AC),LengthOfLine(BD)),32)
Equal(LengthOfLine(CD),4)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(BOA))
Answer: 20
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
parallelogram_property_opposite_line_equal(1,ABCD)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
triangle_perimeter_formula(1,BOA)
Problem: As shown in the diagram, AE=3, BE=2, AE is the angle bisector of ∠DAB, BE bisects ∠ABC, ABCD is a ▱. Find the area of triangle ABE.
问题：如图所示，AE=3，BE=2，AE是∠DAB的角平分线，BE是∠ABC的角平分线，AD和BC是平行四边形ABCD的一组对边。求△ABE的面积。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EA,AB,BE)
Shape(EB,BC,CE)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(AE),3)
Equal(LengthOfLine(BE),2)
IsBisectorOfAngle(AE,DAB)
IsBisectorOfAngle(BE,ABC)
Parallelogram(ABCD)
Goal: Value(AreaOfTriangle(ABE))
Answer: 3
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
angle_addition(1,DAE,EAB)
angle_addition(1,ABE,EBC)
triangle_property_angle_sum(1,ABE)
triangle_area_formula_sine(1,EAB)
Problem: As shown in the diagram, AB=10, AB=AF, BF=12, AO bisects ∠FAB, AD and BC are opposite sides of the ▱ ABCD. Find the length of line AE.
问题：如图所示，AB=10，AB=AF，BF=12，AO是∠FAB的角平分线，ABCD是▱。求直线AE的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OF,FA)
Shape(OB,BE,EO)
Shape(OE,EF,FO)
Shape(FE,EC,CD,DF)
Collinear(AFD)
Collinear(BEC)
Collinear(AOE)
Collinear(BOF)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AB),LengthOfLine(AF))
Equal(LengthOfLine(BF),12)
IsBisectorOfAngle(AO,FAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AE))
Answer: 16
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABF)
isosceles_triangle_property_line_coincidence(3,ABF,O)
line_addition(1,BO,OF)
parallel_property_collinear_extend(3,AD,BC,F)
parallel_property_collinear_extend(3,CB,FA,E)
parallel_property_alternate_interior_angle(1,AF,BE)
isosceles_triangle_judgment_angle_equal(1,BEA)
adjacent_complementary_angle(1,EOB,BOA)
altitude_of_triangle_judgment(1,BO,BEA)
isosceles_triangle_property_line_coincidence(1,BEA,O)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
line_addition(1,AO,OE)
Problem: As shown in the diagram, AC=10, BD=6, ∠ODA=90°, AD and BC are opposite sides of the parallelogram ABCD. Find the length of line BC.
问题：如图所示，AC=10，BD=6，∠ODA=90°，四边形ABCD是平行四边形。求直线BC的长度。
Construction (CDL):
Shape(DA,AO,OD)
Shape(OA,AB,BO)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AC),10)
Equal(LengthOfLine(BD),6)
Equal(MeasureOfAngle(ODA),90)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BC))
Answer: 4
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,DABC,O)
line_addition(1,DO,OB)
line_addition(1,AO,OC)
right_triangle_judgment_angle(1,ODA)
right_triangle_property_pythagorean(1,ODA)
parallelogram_property_opposite_line_equal(1,DABC)
Problem: As shown in the diagram, BD=12, the perimeter of quadrilateral ABCD is 36, E bisects segment CD, ABCD is a parallelogram. Find the perimeter of △DOE.
问题：如图所示，BD=12，四边形ABCD的周长为36，E是线段CD的中点，四边形ABCD是平行四边形。求△DOE的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(OC,CE,EO)
Shape(OE,ED,DO)
Shape(OD,DA,AO)
Collinear(AOC)
Collinear(BOD)
Collinear(CED)
Conditions (CDL):
Equal(LengthOfLine(BD),12)
Equal(PerimeterOfQuadrilateral(ABCD),36)
IsMidpointOfLine(E,CD)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(DOE))
Answer: 15
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
quadrilateral_perimeter_formula(1,ABCD)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,BO,OD)
line_addition(1,CE,ED)
similar_triangle_judgment_sas(1,DOE,DBC)
similar_triangle_property_line_ratio(1,EDO,CDB)
similar_triangle_property_line_ratio(1,DOE,DBC)
triangle_perimeter_formula(1,DOE)
Problem: As shown in the diagram, BD=12, the perimeter of ABCD is 36, E is the midpoint of segment CD, AD and BC are opposite sides of the ▱ ABCD. Find the perimeter of triangle DOE.
问题：如图所示，BD=12，四边形ABCD的周长为36，E是线段CD的中点，BA和CD是平行四边形ABCD的一组对边。求△DOE的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(OC,CE,EO)
Shape(OE,ED,DO)
Shape(OD,DA,AO)
Collinear(AOC)
Collinear(BOD)
Collinear(CED)
Conditions (CDL):
Equal(LengthOfLine(BD),12)
Equal(PerimeterOfQuadrilateral(ABCD),36)
IsMidpointOfLine(E,CD)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(DOE))
Answer: 15
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
quadrilateral_perimeter_formula(1,ABCD)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,BO,OD)
line_addition(1,CE,ED)
similar_triangle_judgment_sas(1,DOE,DBC)
similar_triangle_property_line_ratio(1,EDO,CDB)
similar_triangle_property_line_ratio(1,DOE,DBC)
triangle_perimeter_formula(1,DOE)
Problem: As shown in the diagram, AE=4, BC=7, CE is the angle bisector of ∠BCD, quadrilateral ABCD is a parallelogram. Find the length of line AB.
问题：如图所示，AE=4，BC=7，CE平分∠BCD，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(AE),4)
Equal(LengthOfLine(BC),7)
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,BCDA)
line_addition(1,AE,ED)
parallelogram_property_opposite_line_equal(1,ABCD)
Problem: As shown in the diagram, ∠ABC=110°, AE is the angle bisector of ∠DAB, ABCD is a parallelogram. Find the measure of ∠AED.
问题：如图所示，∠ABC=110°，AE平分∠DAB，BA和CD是▱ABCD的一组对边。求∠AED的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EA,AB,BC,CE)
Collinear(DEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),110)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(AED))
Answer: 35
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
parallelogram_property_opposite_angle_equal(1,DABC)
angle_addition(1,DAE,EAB)
triangle_property_angle_sum(1,DAE)
Problem: As shown in the diagram, ∠ABD=30°, ∠DAB=110°, ABCD is a parallelogram. Find the measure of ∠DBC.
问题：如图所示，∠ABD=30°，∠DAB=110°，ABCD是▱。求∠DBC的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(BC,CD,DB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),30)
Equal(MeasureOfAngle(DAB),110)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(DBC))
Answer: 40
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
angle_addition(1,ABD,DBC)
Problem: As shown in the diagram, AB=6, AD=9, EF=2, AE bisects ∠DAB, quadrilateral ABCD is a parallelogram. Find the perimeter of △CEF.
问题：如图所示，AB=6，AD=9，EF=2，AE是∠DAB的角平分线，BA和CD是▱ABCD的一组对边。求三角形CEF的周长。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EF,FC,CE)
Shape(AE,EC,CD,DA)
Collinear(AEF)
Collinear(BEC)
Collinear(FCD)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AD),9)
Equal(LengthOfLine(EF),2)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(CEF))
Answer: 8
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)
parallel_property_collinear_extend(2,DC,AB,F)
parallel_property_alternate_interior_angle(2,BA,FD)
vertical_angle(1,BEA,CEF)
isosceles_triangle_judgment_angle_equal(1,CEF)
triangle_perimeter_formula(1,CEF)
Problem: As shown in the diagram, AD=6, BE=2, DE bisects ∠CDA, AD and BC are opposite sides of the ▱ ABCD. Find the perimeter of ABCD.
问题：如图所示，AD=6，BE=2，DE是∠CDA的角平分线，ABCD是平行四边形。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),6)
Equal(LengthOfLine(BE),2)
IsBisectorOfAngle(DE,CDA)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 20
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_alternate_interior_angle(2,AD,EC)
isosceles_triangle_judgment_angle_equal(1,CDE)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,BE,EC)
parallelogram_property_opposite_line_equal(1,ABCD)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, the perimeter of quadrilateral ABCD is 22, the perimeter of △ABC is 17, BA and CD are opposite sides of the ▱ ABCD. Find the length of line AC.
问题：如图所示，四边形ABCD的周长为22，△ABC的周长为17，ABCD是▱。求直线AC的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),22)
Equal(PerimeterOfTriangle(ABC),17)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AC))
Answer: 6
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
quadrilateral_perimeter_formula(1,ABCD)
triangle_perimeter_formula(1,ABC)
Problem: As shown in the diagram, ∠DAC=42°, ∠DBC=23°, ABCD is a ▱. Find the measure of ∠DOC.
问题：如图所示，∠DAC=42°，∠DBC=23°，AD和BC是平行四边形ABCD的一组对边。求∠DOC的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Shape(OD,DA,AO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(MeasureOfAngle(DAC),42)
Equal(MeasureOfAngle(DBC),23)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(DOC))
Answer: 65
Theorem sequence:
parallel_property_alternate_interior_angle(2,AD,BC)
triangle_property_angle_sum(1,AOD)
adjacent_complementary_angle(1,AOD,DOC)
Problem: As shown in the diagram, the perimeter of △ABD is 16, E bisects segment CD, AD and BC are opposite sides of the ▱ ABCD. Find the perimeter of triangle DOE.
问题：如图所示，△ABD的周长为16，E是线段CD的中点，AD和BC是平行四边形ABCD的一组对边。求△DOE的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(OD,DA,AO)
Shape(OC,CE,EO)
Shape(OE,ED,DO)
Collinear(AOC)
Collinear(BOD)
Collinear(CED)
Conditions (CDL):
Equal(PerimeterOfTriangle(ABD),16)
IsMidpointOfLine(E,CD)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(DOE))
Answer: 8
Theorem sequence:
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,BO,OD)
line_addition(1,DE,EC)
similar_triangle_judgment_sas(1,DOE,DBC)
similar_triangle_property_line_ratio(1,EDO,CDB)
similar_triangle_property_line_ratio(1,DOE,DBC)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
triangle_perimeter_formula(1,ABD)
triangle_perimeter_formula(1,OED)
Problem: As shown in the diagram, MC=2, the perimeter of ABCD is 14, BM bisects ∠ABC, quadrilateral ABCD is a ▱. Find the length of line DM.
问题：如图所示，MC=2，四边形ABCD的周长为14，BM是∠ABC的角平分线，ABCD是▱。求直线DM的长度。
Construction (CDL):
Shape(DA,AB,BM,MD)
Shape(MB,BC,CM)
Collinear(DMC)
Conditions (CDL):
Equal(LengthOfLine(MC),2)
Equal(PerimeterOfQuadrilateral(ABCD),14)
IsBisectorOfAngle(BM,ABC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(DM))
Answer: 3
Theorem sequence:
parallel_property_collinear_extend(3,DC,AB,M)
parallel_property_alternate_interior_angle(1,MC,AB)
isosceles_triangle_judgment_angle_equal(1,CMB)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
quadrilateral_perimeter_formula(1,ABCD)
line_addition(1,DM,MC)
Problem: As shown in the diagram, DE=2, DF=3, E is the midpoint of segment CD, BA and CD are opposite sides of the ▱ ABCD. Find the perimeter of ABCD.
问题：如图所示，DE=2，DF=3，E是线段CD的中点，AD和BC是平行四边形ABCD的一组对边。求ABCD的周长。
Construction (CDL):
Shape(FD,DE,EF)
Shape(DA,AB,BE,ED)
Shape(EB,BC,CE)
Collinear(FEB)
Collinear(DEC)
Collinear(FDA)
Conditions (CDL):
Equal(LengthOfLine(DE),2)
Equal(LengthOfLine(DF),3)
IsMidpointOfLine(E,CD)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 14
Theorem sequence:
parallel_property_collinear_extend(2,AD,BC,F)
parallel_property_alternate_interior_angle(2,AF,BC)
vertical_angle(1,DEF,CEB)
congruent_triangle_judgment_aas(1,DEF,CEB)
congruent_triangle_property_line_equal(1,EFD,EBC)
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, AB=4, BC=6, quadrilateral ABCD is a ▱, AO⊥EO. Find the perimeter of △CDE.
问题：如图所示，AB=4，BC=6，BA和CD是平行四边形ABCD的一组对边，AO垂直于EO。求三角形CDE的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(OC,CG,GO)
Shape(OG,GE,EO)
Shape(OE,EA,AO)
Shape(GC,CD,DG)
Shape(GD,DE,EG)
Collinear(BOGD)
Collinear(AOC)
Collinear(EGC)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BC),6)
Parallelogram(ABCD)
PerpendicularBetweenLine(AO,EO)
Goal: Value(PerimeterOfTriangle(CDE))
Answer: 10
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_diagonal_bisection(1,ABCD,O)
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,ECD)
Problem: As shown in the diagram, the perimeter of triangle ABD is 36, E is the midpoint of segment CD, quadrilateral ABCD is a ▱. Find the perimeter of triangle DOE.
问题：如图所示，三角形ABD的周长为36，E是线段CD的中点，四边形ABCD是▱。求三角形DOE的周长。
Construction (CDL):
Shape(AO,OD,DA)
Shape(OA,AB,BO)
Shape(OB,BC,CO)
Shape(OC,CE,EO)
Shape(OE,ED,DO)
Collinear(AOC)
Collinear(BOD)
Collinear(CED)
Conditions (CDL):
Equal(PerimeterOfTriangle(ABD),36)
IsMidpointOfLine(E,CD)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(DOE))
Answer: 18
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,BO,OD)
line_addition(1,CE,ED)
similar_triangle_judgment_sas(1,DOE,DBC)
similar_triangle_property_line_ratio(1,EDO,CDB)
similar_triangle_property_line_ratio(1,DOE,DBC)
triangle_perimeter_formula(1,ABD)
triangle_perimeter_formula(1,DOE)
Problem: As shown in the diagram, BD=14, the perimeter of ABCD is 36, E is the midpoint of segment CD, ABCD is a ▱. Find the perimeter of triangle DOE.
问题：如图所示，BD=14，ABCD的周长为36，E平分线段CD，四边形ABCD是▱。求△DOE的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(OC,CE,EO)
Shape(OE,ED,DO)
Shape(OD,DA,AO)
Collinear(AOC)
Collinear(BOD)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(BD),14)
Equal(PerimeterOfQuadrilateral(ABCD),36)
IsMidpointOfLine(E,CD)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(DOE))
Answer: 16
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,BO,OD)
line_addition(1,CE,ED)
similar_triangle_judgment_sas(1,DOE,DBC)
similar_triangle_property_line_ratio(1,EDO,CDB)
similar_triangle_property_line_ratio(1,DOE,DBC)
quadrilateral_perimeter_formula(1,ABCD)
triangle_perimeter_formula(1,DOE)
Problem: As shown in the diagram, AB=18, PC=6, AP bisects ∠DAB, AD and BC are opposite sides of the parallelogram ABCD. Find the perimeter of ABCD.
问题：如图所示，AB=18，PC=6，AP是∠DAB的角平分线，AD和BC是平行四边形ABCD的一组对边。求四边形ABCD的周长。
Construction (CDL):
Shape(DA,AP,PD)
Shape(AB,BC,CP,PA)
Collinear(DPC)
Conditions (CDL):
Equal(LengthOfLine(AB),18)
Equal(LengthOfLine(PC),6)
IsBisectorOfAngle(AP,DAB)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 60
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,DABC)
parallel_property_collinear_extend(3,DC,AB,P)
line_addition(1,DP,PC)
parallel_property_alternate_interior_angle(2,DP,AB)
isosceles_triangle_judgment_angle_equal(1,DAP)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, AM=MD, AM=MF, ∠MAF=70°, BA and CD are opposite sides of the parallelogram ABCD, quadrilateral MABN is a ▱, quadrilateral MFEN is a ▱. Find the measure of ∠FMA.
问题：如图所示，AM=MD，AM=MF，∠MAF=70°，ABCD是平行四边形，MN和AB是▱MABN的一组对边，MFEN是▱。求∠FMA的大小。
Construction (CDL):
Shape(DM,MN,NC)
Shape(MF,FB,BN,NM)
Shape(MA,AF,FM)
Shape(NB,BE,EN)
Collinear(DMA)
Collinear(CNB)
Collinear(AFBE)
Conditions (CDL):
Equal(LengthOfLine(AM),LengthOfLine(MD))
Equal(LengthOfLine(AM),LengthOfLine(MF))
Equal(MeasureOfAngle(MAF),70)
Parallelogram(ABCD)
Parallelogram(MABN)
Parallelogram(MFEN)
Goal: Value(MeasureOfAngle(FMA))
Answer: 40
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,MAF)
isosceles_triangle_property_angle_equal(1,MAF)
triangle_property_angle_sum(1,MAF)
Problem: As shown in the diagram, AB=8, BC=12, EF perpendicular bisects AC, quadrilateral ABCD is a ▱. Find the perimeter of △CDE.
问题：如图所示，AB=8，BC=12，EF垂直平分AB，AD和BC是▱ABCD的一组对边。求△CDE的周长。
Construction (CDL):
Shape(AB,BC,CF,FA)
Shape(AF,FE,EA)
Shape(EC,CD,DE)
Shape(EF,FC,CE)
Collinear(AFC)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BC),12)
IsPerpendicularBisectorOfLine(EF,AC)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(CDE))
Answer: 20
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,AE,ED)
perpendicular_bisector_property_distance_equal(1,EF,AC)
triangle_perimeter_formula(1,CDE)
Problem: As shown in the diagram, the perimeter of quadrilateral ABCD is 20, AD and BC are opposite sides of the parallelogram ABCD, AO is perpendicular to EO. Find the perimeter of △CDE.
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，AO⊥EO。求三角形CDE的周长。
Construction (CDL):
Shape(AO,OE,EA)
Shape(EO,OF,FE)
Shape(EF,FD,DE)
Shape(OC,CF,FO)
Shape(FC,CD,DF)
Shape(BO,OA,AB)
Shape(OB,BC,CO)
Collinear(AOC)
Collinear(BOFD)
Collinear(EFC)
Collinear(AED)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),20)
Parallelogram(ABCD)
PerpendicularBetweenLine(AO,EO)
Goal: Value(PerimeterOfTriangle(CDE))
Answer: 10
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
quadrilateral_perimeter_formula(1,ABCD)
line_addition(1,AE,ED)
parallelogram_property_diagonal_bisection(1,ABCD,O)
perpendicular_bisector_judgment_per_and_mid(1,EO,AC)
perpendicular_bisector_property_distance_equal(1,EO,AC)
triangle_perimeter_formula(1,CDE)
Problem: As shown in the diagram, DE=Mul(LengthOfLine(AB),2), ∠ABD=50°, O bisects segment ED, AD and BC are opposite sides of the ▱ ABCD, BF is perpendicular to AF, DA is perpendicular to FA. Find the measure of ∠BDA.
问题：如图所示，DE=Mul(LengthOfLine(AB),2)，∠ABD=50°，O平分线段ED，四边形ABCD是▱，BF垂直于AF，DA⊥FA。求∠BDA的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EO,OA)
Shape(AO,OD,DA)
Shape(BF,FE,EB)
Shape(EF,FC,CO,OE)
Shape(OC,CD,DO)
Collinear(AEF)
Collinear(BEOD)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(DE),Mul(LengthOfLine(AB),2))
Equal(MeasureOfAngle(ABD),50)
IsMidpointOfLine(O,ED)
Parallelogram(ABCD)
PerpendicularBetweenLine(BF,AF)
PerpendicularBetweenLine(DA,FA)
Goal: Value(MeasureOfAngle(BDA))
Answer: 25
Theorem sequence:
median_of_triangle_judgment(1,AO,AED)
right_triangle_judgment_angle(1,DAE)
right_triangle_property_length_of_median(1,DAE,O)
isosceles_triangle_judgment_line_equal(1,ABO)
isosceles_triangle_property_angle_equal(1,ABO)
line_addition(1,EO,OD)
adjacent_complementary_angle(1,EOA,AOD)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,ODA)