SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AC=2, ∠DAC=45°, AD and BC are opposite sides of the ▱ ABCD, CA is perpendicular to BA. Find the length of line BD.
问题：如图所示，AC=2，∠DAC=45°，AD和BC是▱ABCD的一组对边，CA垂直于BA。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(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(MeasureOfAngle(DAC),45)
Parallelogram(ABCD)
PerpendicularBetweenLine(CA,BA)
Goal: Value(LengthOfLine(BD))
Answer: 2*sqrt(5)
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
line_addition(1,AO,OC)
parallel_property_alternate_interior_angle(1,AD,BC)
triangle_property_angle_sum(1,ABC)
isosceles_triangle_judgment_angle_equal(1,ABC)
right_triangle_judgment_angle(1,OAB)
right_triangle_property_pythagorean(1,OAB)
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,BO,OD)
Problem: As shown in the diagram, AB=6, EF=2, BF bisects ∠ABC, CE bisects ∠BCD, BA and CD are opposite sides of the parallelogram ABCD. Find the length of line BC.
问题：如图所示，AB=6，EF=2，BF平分∠ABC，CE平分∠BCD，ABCD是▱。求直线BC的长度。
Construction (CDL):
Shape(AB,BO,OE,EA)
Shape(EO,OF,FE)
Shape(FO,OC,CD,DF)
Shape(OB,BC,CO)
Collinear(AEFD)
Collinear(EOC)
Collinear(BOF)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(EF),2)
IsBisectorOfAngle(BF,ABC)
IsBisectorOfAngle(CE,BCD)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BC))
Answer: 10
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_collinear_extend(3,AD,BC,F)
parallel_property_alternate_interior_angle(2,AF,BC)
isosceles_triangle_judgment_angle_equal(1,ABF)
parallel_property_alternate_interior_angle(1,ED,BC)
isosceles_triangle_judgment_angle_equal(1,DEC)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,EF,FD)
line_addition(1,AF,FD)
Problem: As shown in the diagram, AC=24, AD=14, BD=38, quadrilateral ABCD is a parallelogram. Find the perimeter of triangle OBC.
问题：如图所示，AC=24，AD=14，BD=38，ABCD是▱。求三角形OBC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(LengthOfLine(AC),24)
Equal(LengthOfLine(AD),14)
Equal(LengthOfLine(BD),38)
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,AO,OC)
line_addition(1,BO,OD)
triangle_perimeter_formula(1,OBC)
Problem: As shown in the diagram, AB=3, BC=5, CF=2, BE is the angle bisector of ∠ABC, CF bisects ∠BCD, FC is parallel to AM, ABCD is a ▱. Find the length of line BE.
问题：如图所示，AB=3，BC=5，CF=2，BE平分∠ABC，CF是∠BCD的角平分线，FC平行于AM，四边形ABCD是平行四边形。求直线BE的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OG,GF,FA)
Shape(FG,GE,EF)
Shape(GO,OM,MC,CG)
Shape(BM,MO,OB)
Shape(EG,GC,CD,DE)
Collinear(AFED)
Collinear(BOGE)
Collinear(FGC)
Collinear(AOM)
Collinear(BMC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(CF),2)
IsBisectorOfAngle(BE,ABC)
IsBisectorOfAngle(CF,BCD)
ParallelBetweenLine(FC,AM)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BE))
Answer: 4*sqrt(2)
Theorem sequence:
line_addition(1,AO,OM)
line_addition(1,BO,OE)
parallel_property_collinear_extend(3,AD,BC,F)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_collinear_extend(3,CB,DA,M)
parallel_property_alternate_interior_angle(1,MB,DA)
parallel_property_alternate_interior_angle(2,AE,BC)
isosceles_triangle_judgment_angle_equal(1,ABE)
parallel_property_corresponding_angle(1,MA,CF,B)
parallelogram_property_opposite_angle_equal(1,ABCD)
angle_addition(1,DAM,MAB)
angle_addition(1,BCF,FCD)
bisector_of_angle_judgment_angle_equal(1,AO,EAB)
isosceles_triangle_property_line_coincidence(3,ABE,O)
vertical_angle(1,AOE,MOB)
congruent_triangle_judgment_aas(3,EAO,BMO)
parallel_property_collinear_extend(3,CB,FA,M)
parallelogram_judgment_parallel_and_parallel(1,FAMC)
parallelogram_property_opposite_line_equal(1,AMCF)
congruent_triangle_property_line_equal(1,EAO,BMO)
congruent_triangle_property_line_equal(1,OEA,OBM)
right_triangle_judgment_angle(1,MOB)
right_triangle_property_pythagorean(1,MOB)
Problem: As shown in the diagram, ∠AFD=44°, ∠BCA=44°, △BCA is mirror congruent to △EAC, quadrilateral ABCD is a parallelogram. Find the measure of ∠ABC.
问题：如图所示，∠AFD=44°，∠BCA=44°，△BCA与△EAC是镜像全等三角形，ABCD是平行四边形。求∠ABC的大小。
Construction (CDL):
Shape(AF,FD,DA)
Shape(EF,FC,CE)
Shape(FA,AC,CF)
Shape(CA,AB,BC)
Collinear(DFC)
Collinear(EFA)
Conditions (CDL):
Equal(MeasureOfAngle(AFD),44)
Equal(MeasureOfAngle(BCA),44)
MirrorCongruentBetweenTriangle(BCA,EAC)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(ABC))
Answer: 114
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,BCA,EAC)
mirror_congruent_triangle_property_angle_equal(1,CAB,CEA)
mirror_congruent_triangle_property_angle_equal(1,ABC,ACE)
angle_addition(1,ACD,DCE)
adjacent_complementary_angle(1,CFA,AFD)
adjacent_complementary_angle(1,EFC,CFA)
parallel_property_alternate_interior_angle(2,BA,CD)
triangle_property_angle_sum(1,EFC)
triangle_property_angle_sum(1,FAC)
Problem: As shown in the diagram, Add(LengthOfLine(AC)=LengthOfLine(BD)), CD=6, ABCD is a parallelogram. Find the perimeter of triangle ABO.
问题：如图所示，Add(LengthOfLine(AC)=LengthOfLine(BD))，CD=6，AD和BC是▱ABCD的一组对边。求△ABO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)),16)
Equal(LengthOfLine(CD),6)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(ABO))
Answer: 14
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,ABO)
Problem: As shown in the diagram, EF=6, E bisects segment AB, EF is parallel to BC, quadrilateral ABCD is a rhombus. Find the length of line CD.
问题：如图所示，EF=6，E是线段AB的中点，EF∥BC，ABCD是菱形。求直线CD的长度。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BC,CF,FE)
Shape(AF,FC,CD,DA)
Collinear(AEB)
Collinear(AFC)
Conditions (CDL):
Equal(LengthOfLine(EF),6)
IsMidpointOfLine(E,AB)
ParallelBetweenLine(EF,BC)
Rhombus(ABCD)
Goal: Value(LengthOfLine(CD))
Answer: 12
Theorem sequence:
midsegment_of_triangle_judgment_parallel(1,EF,ABC)
midsegment_of_triangle_property_length(1,EF,ABC)
Problem: As shown in the diagram, AB=5, AB=AF, BF=6, 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=5，AB=AF，BF=6，AO平分∠FAB，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(AO,OF,FA)
Shape(AB,BO,OA)
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),5)
Equal(LengthOfLine(AB),LengthOfLine(AF))
Equal(LengthOfLine(BF),6)
IsBisectorOfAngle(AO,FAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AE))
Answer: 8
Theorem sequence:
line_addition(1,AO,OE)
line_addition(1,BO,OF)
parallel_property_collinear_extend(3,AD,BC,F)
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_collinear_extend(3,CB,FA,E)
isosceles_triangle_judgment_line_equal(1,ABF)
parallel_property_alternate_interior_angle(1,EB,DA)
isosceles_triangle_judgment_angle_equal(1,BEA)
parallelogram_judgment_parallel_and_equal(1,FABE)
parallelogram_property_diagonal_bisection(1,ABEF,O)
isosceles_triangle_property_line_coincidence(3,ABF,O)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
Problem: As shown in the diagram, AB=6, BC=8, CF bisects ∠BCD, quadrilateral ABCD is a parallelogram. Find the sum of the length of line AE and the length of line AF.
问题：如图所示，AB=6，BC=8，CF是∠BCD的角平分线，ABCD是平行四边形。求直线AE的长度与直线AF的长度之和。
Construction (CDL):
Shape(FA,AE,EF)
Shape(AB,BC,CE,EA)
Shape(EC,CD,DE)
Collinear(FAB)
Collinear(FEC)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(BC),8)
IsBisectorOfAngle(CF,BCD)
Parallelogram(ABCD)
Goal: Value(Add(LengthOfLine(AE),LengthOfLine(AF)))
Answer: 4
Theorem sequence:
parallel_property_collinear_extend(2,BA,CD,F)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_alternate_interior_angle(2,BF,CD)
line_addition(1,FA,AB)
parallel_property_corresponding_angle(2,CB,EA,F)
isosceles_triangle_judgment_angle_equal(1,AEF)
isosceles_triangle_judgment_angle_equal(1,BCF)
Problem: As shown in the diagram, AC=6, AD=8, BD=12, quadrilateral ABCD is a ▱. Find the perimeter of △OBC.
问题：如图所示，AC=6，AD=8，BD=12，ABCD是▱。求△OBC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),6)
Equal(LengthOfLine(AD),8)
Equal(LengthOfLine(BD),12)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(OBC))
Answer: 17
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,BO,OD)
triangle_perimeter_formula(1,OBC)
Problem: As shown in the diagram, ∠DAB=135°, BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠DCM.
问题：如图所示，∠DAB=135°，AD和BC是平行四边形ABCD的一组对边。求∠DCM的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Shape(DC,CM)
Collinear(BCM)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),135)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(DCM))
Answer: 45
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,ABCD)
adjacent_complementary_angle(1,BCD,DCM)
Problem: As shown in the diagram, AC=8, BC=10, BD=14, quadrilateral ABCD is a ▱. Find the perimeter of triangle COB.
问题：如图所示，AC=8，BC=10，BD=14，AD和BC是▱ABCD的一组对边。求△COB的周长。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(BC),10)
Equal(LengthOfLine(BD),14)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(COB))
Answer: 21
Theorem sequence:
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=11, BD=16, ABCD is a ▱. Find the perimeter of triangle COB.
问题：如图所示，AC=10，AD=11，BD=16，ABCD是▱。求三角形COB的周长。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AC),10)
Equal(LengthOfLine(AD),11)
Equal(LengthOfLine(BD),16)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(COB))
Answer: 24
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, AB=8, AD=6, CE bisects ∠BCD, AD and BC are opposite sides of the ▱ ABCD. Find the length of line AE.
问题：如图所示，AB=8，AD=6，CE是∠BCD的角平分线，ABCD是平行四边形。求直线AE的长度。
Construction (CDL):
Shape(AE,EC,CD,DA)
Shape(CE,EB,BC)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AD),6)
IsBisectorOfAngle(CE,BCD)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AE))
Answer: 2
Theorem sequence:
parallel_property_collinear_extend(3,BA,CD,E)
parallelogram_property_opposite_line_equal(1,BCDA)
parallel_property_alternate_interior_angle(2,BE,CD)
isosceles_triangle_judgment_angle_equal(1,BCE)
line_addition(1,AE,EB)
Problem: As shown in the diagram, AB=3, AD=7, AE bisects ∠DAB, ABCD is a ▱. Find the length of line EC.
问题：如图所示，AB=3，AD=7，AE平分∠DAB，BA和CD是平行四边形ABCD的一组对边。求直线EC的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EC,CD,DA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AD),7)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(EC))
Answer: 4
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,E)
parallelogram_property_opposite_line_equal(1,BCDA)
parallel_property_alternate_interior_angle(1,EB,DA)
isosceles_triangle_judgment_angle_equal(1,BEA)
line_addition(1,BE,EC)
Problem: As shown in the diagram, ∠DAB=50°, BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠BCD.
问题：如图所示，∠DAB=50°，AD和BC是平行四边形ABCD的一组对边。求∠BCD的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),50)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BCD))
Answer: 50
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,ABCD)
Problem: As shown in the diagram, the area of quadrilateral ABCD is 48, AE=6, AE is perpendicular to CE, quadrilateral ABCD is a rhombus. Find the length of line BC.
问题：如图所示，ABCD的面积为48，AE=6，AE⊥CE，ABCD是菱形。求直线BC的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EC,CD,DA)
Collinear(BEC)
Conditions (CDL):
Equal(AreaOfQuadrilateral(ABCD),48)
Equal(LengthOfLine(AE),6)
PerpendicularBetweenLine(AE,CE)
Rhombus(ABCD)
Goal: Value(LengthOfLine(BC))
Answer: 8
Theorem sequence:
adjacent_complementary_angle(1,BEA,AEC)
altitude_of_quadrilateral_judgment_left_vertex(1,AE,ABCD)
parallelogram_area_formula_common(1,ABCD)
Problem: As shown in the diagram, AB=3, AD=5, AE bisects ∠DAB, AD and BC are opposite sides of the parallelogram ABCD. Find the length of line EC.
问题：如图所示，AB=3，AD=5，AE是∠DAB的角平分线，AD和BC是▱ABCD的一组对边。求直线EC的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EC,CD,DA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AD),5)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(EC))
Answer: 2
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,E)
parallelogram_property_opposite_line_equal(1,BCDA)
parallel_property_alternate_interior_angle(1,EB,DA)
isosceles_triangle_judgment_angle_equal(1,BEA)
line_addition(1,BE,EC)
Problem: As shown in the diagram, AB=5, AB=AF, BF=6, AO is the angle bisector of ∠FAB, BA and CD are opposite sides of the parallelogram ABCD. Find the length of line AE.
问题：如图所示，AB=5，AB=AF，BF=6，AO是∠FAB的角平分线，BA和CD是平行四边形ABCD的一组对边。求直线AE的长度。
Construction (CDL):
Shape(AO,OF,FA)
Shape(AB,BO,OA)
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),5)
Equal(LengthOfLine(AB),LengthOfLine(AF))
Equal(LengthOfLine(BF),6)
IsBisectorOfAngle(AO,FAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AE))
Answer: 8
Theorem sequence:
line_addition(1,AO,OE)
line_addition(1,BO,OF)
parallel_property_collinear_extend(3,AD,BC,F)
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_collinear_extend(3,CB,FA,E)
isosceles_triangle_judgment_line_equal(1,ABF)
parallel_property_alternate_interior_angle(1,EB,DA)
isosceles_triangle_judgment_angle_equal(1,BEA)
parallelogram_judgment_parallel_and_equal(1,FABE)
parallelogram_property_diagonal_bisection(1,ABEF,O)
isosceles_triangle_property_line_coincidence(3,ABF,O)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
Problem: As shown in the diagram, Add(LengthOfLine(AC)=LengthOfLine(BD)), CD=4, AD and BC are opposite sides of the ▱ ABCD. Find the perimeter of △OAB.
问题：如图所示，Add(LengthOfLine(AC)=LengthOfLine(BD))，CD=4，BA和CD是平行四边形ABCD的一组对边。求△OAB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)),20)
Equal(LengthOfLine(CD),4)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(OAB))
Answer: 14
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,ABO)
Problem: As shown in the diagram, BE=2, the perimeter of ABCD is 16, AE bisects ∠DAB, AD and BC are opposite sides of the ▱ ABCD. Find the length of line EC.
问题：如图所示，BE=2，ABCD的周长为16，AE平分∠DAB，ABCD是▱。求直线EC的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EC,CD,DA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(BE),2)
Equal(PerimeterOfQuadrilateral(ABCD),16)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(EC))
Answer: 4
Theorem sequence:
quadrilateral_perimeter_formula(1,ABCD)
parallel_property_collinear_extend(3,CB,DA,E)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallel_property_alternate_interior_angle(1,EB,DA)
isosceles_triangle_judgment_angle_equal(1,BEA)
line_addition(1,BE,EC)
Problem: As shown in the diagram, ∠EAO=53°, ABCD is a ▱, OE is perpendicular to AE. Find the measure of ∠BCE.
问题：如图所示，∠EAO=53°，AD和BC是平行四边形ABCD的一组对边，OE⊥AE。求∠BCE的大小。
Construction (CDL):
Shape(AO,OE,EA)
Shape(AB,BC,CO,OA)
Shape(OC,CD,DO)
Collinear(EOC)
Collinear(AOD)
Collinear(EAB)
Conditions (CDL):
Equal(MeasureOfAngle(EAO),53)
Parallelogram(ABCD)
PerpendicularBetweenLine(OE,AE)
Goal: Value(MeasureOfAngle(BCE))
Answer: 37
Theorem sequence:
parallel_property_corresponding_angle(1,AD,BC,E)
triangle_property_angle_sum(1,EBC)
Problem: As shown in the diagram, EF=4, F bisects segment BD, EF∥AB, AD and BC are opposite sides of the ▱ ABCD. Find the length of line CD.
问题：如图所示，EF=4，F是线段BD的中点，EF平行于AB，AD和BC是平行四边形ABCD的一组对边。求直线CD的长度。
Construction (CDL):
Shape(DE,EF,FD)
Shape(EA,AB,BF,FE)
Shape(DF,FB,BC,CD)
Collinear(DEA)
Collinear(DFB)
Conditions (CDL):
Equal(LengthOfLine(EF),4)
IsMidpointOfLine(F,BD)
ParallelBetweenLine(EF,AB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(CD))
Answer: 8
Theorem sequence:
midsegment_of_triangle_judgment_parallel(2,EF,DAB)
midsegment_of_triangle_property_length(1,EF,DAB)
parallelogram_property_opposite_line_equal(1,ABCD)
Problem: As shown in the diagram, AC=20, BD=12, ∠ODA=90°, ABCD is a ▱. Find the length of line AD.
问题：如图所示，AC=20，BD=12，∠ODA=90°，ABCD是▱。求直线AD的长度。
Construction (CDL):
Shape(AO,OD,DA)
Shape(DO,OC,CD)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AC),20)
Equal(LengthOfLine(BD),12)
Equal(MeasureOfAngle(ODA),90)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AD))
Answer: 8
Theorem sequence:
line_addition(1,DO,OB)
line_addition(1,AO,OC)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
right_triangle_judgment_angle(1,ODA)
right_triangle_property_pythagorean(1,ODA)
Problem: As shown in the diagram, CE=1, the perimeter of quadrilateral ABCD is 10, AE is the angle bisector of ∠DAB, AD and BC are opposite sides of the ▱ ABCD. Find the length of line AB.
问题：如图所示，CE=1，ABCD的周长为10，AE是∠DAB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,BE,EA)
Shape(AE,EC,CD,DA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(CE),1)
Equal(PerimeterOfQuadrilateral(ABCD),10)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 2
Theorem sequence:
line_addition(1,BE,EC)
parallel_property_collinear_extend(3,CB,DA,E)
quadrilateral_perimeter_formula(1,ABCD)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallel_property_alternate_interior_angle(1,EB,DA)
isosceles_triangle_judgment_angle_equal(1,BEA)
Problem: As shown in the diagram, AD=5, BE=3, AE bisects ∠DAB, ABCD is a ▱. Find the length of line AB.
问题：如图所示，AD=5，BE=3，AE平分∠DAB，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),5)
Equal(LengthOfLine(BE),3)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 3
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_alternate_interior_angle(1,EB,DA)
isosceles_triangle_judgment_angle_equal(1,BEA)
Problem: As shown in the diagram, ∠DAB=65°, F bisects segment AD, BA and CD are opposite sides of the ▱ ABCD, DE is perpendicular to BE. Find the measure of ∠EFA.
问题：如图所示，∠DAB=65°，F平分线段AD，四边形ABCD是▱，DE垂直于BE。求∠EFA的大小。
Construction (CDL):
Shape(AE,EF,FA)
Shape(FE,ED,DF)
Shape(DE,EB,BC,CD)
Collinear(AFD)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),65)
IsMidpointOfLine(F,AD)
Parallelogram(ABCD)
PerpendicularBetweenLine(DE,BE)
Goal: Value(MeasureOfAngle(EFA))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,AED,DEB)
line_addition(1,AF,FD)
right_triangle_judgment_angle(1,AED)
median_of_triangle_judgment(1,EF,EDA)
right_triangle_property_length_of_median(1,AED,F)
isosceles_triangle_judgment_line_equal(1,FAE)
isosceles_triangle_property_angle_equal(1,FAE)
triangle_property_angle_sum(1,FAE)
Problem: As shown in the diagram, AB=4, AD=3, AE bisects ∠DAB, quadrilateral ABCD is a ▱. Find the length of line EC.
问题：如图所示，AB=4，AD=3，AE是∠DAB的角平分线，四边形ABCD是平行四边形。求直线EC的长度。
Construction (CDL):
Shape(AE,ED,DA)
Shape(AB,BC,CE,EA)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(AD),3)
IsBisectorOfAngle(AE,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(EC))
Answer: 1
Theorem sequence:
line_addition(1,DE,EC)
parallel_property_collinear_extend(3,DC,AB,E)
parallelogram_property_opposite_line_equal(1,ABCD)
parallel_property_alternate_interior_angle(2,DE,AB)
isosceles_triangle_judgment_angle_equal(1,DAE)
Problem: As shown in the diagram, the perimeter of triangle CDM is 40, ABCD is a parallelogram, AO is perpendicular to MO. Find the perimeter of quadrilateral ABCD.
问题：如图所示，三角形CDM的周长为40，ABCD是▱，AO⊥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(BC,CO,OB)
Shape(OC,CN,NO)
Shape(NC,CD,DN)
Collinear(AMD)
Collinear(AOC)
Collinear(BOND)
Collinear(MNC)
Conditions (CDL):
Equal(PerimeterOfTriangle(CDM),40)
Parallelogram(ABCD)
PerpendicularBetweenLine(AO,MO)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 80
Theorem sequence:
line_addition(1,AM,MD)
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,AOM,MOC)
mirror_congruent_triangle_judgment_sas(1,OMA,OCM)
mirror_congruent_triangle_property_line_equal(1,OMA,OCM)
triangle_perimeter_formula(1,DMC)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, BD=10, the perimeter of quadrilateral ABCD is 28, E is the midpoint of segment CD, BA and CD are opposite sides of the ▱ ABCD. Find the perimeter of △DOE.
问题：如图所示，BD=10，ABCD的周长为28，E平分线段CD，AD和BC是平行四边形ABCD的一组对边。求△DOE的周长。
Construction (CDL):
Shape(BO,OA,AB)
Shape(BC,CO,OB)
Shape(OC,CE,EO)
Shape(OE,ED,DO)
Shape(AO,OD,DA)
Collinear(AOC)
Collinear(BOD)
Collinear(CED)
Conditions (CDL):
Equal(LengthOfLine(BD),10)
Equal(PerimeterOfQuadrilateral(ABCD),28)
IsMidpointOfLine(E,CD)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(DOE))
Answer: 12
Theorem sequence:
line_addition(1,BO,OD)
line_addition(1,CE,ED)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
midsegment_of_triangle_judgment_midpoint(1,EO,CDA)
midsegment_of_triangle_property_length(1,EO,CDA)
triangle_perimeter_formula(1,OED)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, AB=6, BC=8, EO is the perpendicular bisector of AC, BA and CD are opposite sides of the parallelogram ABCD. Find the perimeter of △CDE.
问题：如图所示，AB=6，BC=8，EO垂直平分AB，AD和BC是平行四边形ABCD的一组对边。求△CDE的周长。
Construction (CDL):
Shape(BC,CO,OA,AB)
Shape(AO,OE,EA)
Shape(EO,OC,CE)
Shape(EC,CD,DE)
Collinear(AOC)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(BC),8)
IsPerpendicularBisectorOfLine(EO,AC)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(CDE))
Answer: 14
Theorem sequence:
line_addition(1,AE,ED)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
perpendicular_bisector_property_distance_equal(1,EO,AC)
triangle_perimeter_formula(1,ECD)