SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, OA=OB, OA=OC, ∠ADC=70°, ∠CBA=70°. Find the sum of the measure of ∠OAD and the measure of ∠DCO.
问题：如图所示，OA=OB，OA=OC，∠ADC=70°，∠CBA=70°。求∠OAD的大小与∠DCO的大小之和。
Construction (CDL):
Shape(AO,OB,BA)
Shape(BO,OC,CB)
Shape(OA,AD,DC,CO)
Conditions (CDL):
Equal(LengthOfLine(OA),LengthOfLine(OB))
Equal(LengthOfLine(OA),LengthOfLine(OC))
Equal(MeasureOfAngle(ADC),70)
Equal(MeasureOfAngle(CBA),70)
Goal: Value(Add(MeasureOfAngle(OAD),MeasureOfAngle(DCO)))
Answer: 150
Theorem sequence:
angle_addition(1,AOB,BOC)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,AOB)
triangle_property_angle_sum(1,BOC)
round_angle(1,COA,AOC)
quadrilateral_property_angle_sum(1,OADC)
quadrilateral_property_angle_sum(1,BAOC)
Problem: As shown in the diagram, AD=CD, ∠EAB=65°, AB∥CD. Find the measure of ∠CDA.
问题：如图所示，AD=CD，∠EAB=65°，AB平行于CD。求∠CDA的大小。
Construction (CDL):
Shape(EA,AB)
Shape(BA,AD)
Shape(AC,CD,DA)
Collinear(EAC)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(CD))
Equal(MeasureOfAngle(EAB),65)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(CDA))
Answer: 50
Theorem sequence:
parallel_property_corresponding_angle(1,AB,CD,E)
isosceles_triangle_judgment_line_equal(1,DAC)
isosceles_triangle_property_angle_equal(1,DAC)
triangle_property_angle_sum(1,ACD)
Problem: As shown in the diagram, AB=AC, ∠EDB=125°, DE is parallel to BC. Find the measure of ∠BCA.
问题：如图所示，AB=AC，∠EDB=125°，DE∥BC。求∠BCA的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(EDB),125)
ParallelBetweenLine(DE,BC)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
parallel_property_ipsilateral_internal_angle(1,DE,BC)
Problem: As shown in the diagram, AB=12, AC=18, BC=24, BO bisects ∠ABC, CO is the angle bisector of ∠BCA, MN is parallel to BC. Find the perimeter of △AMN.
问题：如图所示，AB=12，AC=18，BC=24，BO是∠ABC的角平分线，CO平分∠BCA，MN∥BC。求△AMN的周长。
Construction (CDL):
Shape(AM,MO,ON,NA)
Shape(MB,BO,OM)
Shape(OC,CN,NO)
Shape(OB,BC,CO)
Collinear(AMB)
Collinear(ANC)
Collinear(MON)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(AC),18)
Equal(LengthOfLine(BC),24)
IsBisectorOfAngle(BO,ABC)
IsBisectorOfAngle(CO,BCA)
ParallelBetweenLine(MN,BC)
Goal: Value(PerimeterOfTriangle(AMN))
Answer: 30
Theorem sequence:
line_addition(1,AM,MB)
line_addition(1,AN,NC)
line_addition(1,MO,ON)
parallel_property_collinear_extend(3,MN,BC,O)
parallel_property_alternate_interior_angle(1,ON,BC)
parallel_property_alternate_interior_angle(2,MO,BC)
isosceles_triangle_judgment_angle_equal(1,MBO)
isosceles_triangle_judgment_angle_equal(1,NOC)
triangle_perimeter_formula(1,AMN)
Problem: As shown in the diagram, AB=9, AD=3, AN=5, CA=CB, ANMD is a ▱. Find the perimeter of triangle BMN.
问题：如图所示，AB=9，AD=3，AN=5，CA=CB，四边形ANMD是▱。求△BMN的周长。
Construction (CDL):
Shape(CD,DM,MC)
Shape(DA,AN,NM,MD)
Shape(MN,NB,BM)
Collinear(ADC)
Collinear(ANB)
Collinear(CMB)
Conditions (CDL):
Equal(LengthOfLine(AB),9)
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(AN),5)
Equal(LengthOfLine(CA),LengthOfLine(CB))
Parallelogram(ANMD)
Goal: Value(PerimeterOfTriangle(BMN))
Answer: 10
Theorem sequence:
line_addition(1,AN,NB)
parallelogram_property_opposite_line_equal(1,NMDA)
parallel_property_corresponding_angle(2,AD,NM,B)
isosceles_triangle_judgment_line_equal(1,CAB)
isosceles_triangle_property_angle_equal(1,CAB)
isosceles_triangle_judgment_angle_equal(1,MNB)
triangle_perimeter_formula(1,BMN)
Problem: As shown in the diagram, AO=BO, DO=BO, ∠OAC=28°. Find the measure of ∠BOD.
问题：如图所示，AO=BO，DO=BO，∠OAC=28°。求∠BOD的大小。
Construction (CDL):
Shape(OA,AC,CO)
Shape(BO,OD)
Shape(OC,CD,DO)
Collinear(AOB)
Collinear(ACD)
Conditions (CDL):
Equal(LengthOfLine(AO),LengthOfLine(BO))
Equal(LengthOfLine(DO),LengthOfLine(BO))
Equal(MeasureOfAngle(OAC),28)
Goal: Value(MeasureOfAngle(BOD))
Answer: 56
Theorem sequence:
triangle_property_angle_sum(1,OAD)
isosceles_triangle_judgment_line_equal(1,OAD)
isosceles_triangle_property_angle_equal(1,OAD)
adjacent_complementary_angle(1,BOD,DOA)
Problem: As shown in the diagram, BA=BC, BA=BD, BA=BE, ∠EAB=60°, AE∥BC. Find the measure of ∠CDE.
问题：如图所示，BA=BC，BA=BD，BA=BE，∠EAB=60°，AE∥BC。求∠CDE的大小。
Construction (CDL):
Shape(EA,AB,BE)
Shape(EB,BD,DE)
Shape(CD,DB,BC)
Conditions (CDL):
Equal(LengthOfLine(BA),LengthOfLine(BC))
Equal(LengthOfLine(BA),LengthOfLine(BD))
Equal(LengthOfLine(BA),LengthOfLine(BE))
Equal(MeasureOfAngle(EAB),60)
ParallelBetweenLine(AE,BC)
Goal: Value(MeasureOfAngle(CDE))
Answer: 150
Theorem sequence:
angle_addition(1,EBD,DBC)
angle_addition(1,CDB,BDE)
triangle_property_angle_sum(1,EBD)
triangle_property_angle_sum(1,CDB)
parallel_property_alternate_interior_angle(2,AE,BC)
isosceles_triangle_judgment_line_equal(1,BEA)
isosceles_triangle_judgment_line_equal(1,BDE)
isosceles_triangle_judgment_line_equal(1,BCD)
isosceles_triangle_property_angle_equal(1,BEA)
isosceles_triangle_property_angle_equal(1,BDE)
isosceles_triangle_property_angle_equal(1,BCD)
Problem: As shown in the diagram, AB=AC, ∠DAC=65°, AD is parallel to BC. Find the measure of ∠CAB.
问题：如图所示，AB=AC，∠DAC=65°，AD∥BC。求∠CAB的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DA,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(DAC),65)
ParallelBetweenLine(AD,BC)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,BC)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, CF=FE, ∠BAF=48°, AB is parallel to CD. Find the measure of ∠FCE.
问题：如图所示，CF=FE，∠BAF=48°，AB平行于CD。求∠FCE的大小。
Construction (CDL):
Shape(BA,AF)
Shape(AF,FD)
Shape(DF,FE)
Shape(EF,FC,CE)
Shape(CF,FA)
Collinear(AFE)
Collinear(CFD)
Conditions (CDL):
Equal(LengthOfLine(CF),LengthOfLine(FE))
Equal(MeasureOfAngle(BAF),48)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(FCE))
Answer: 24
Theorem sequence:
parallel_property_collinear_extend(3,DC,BA,F)
triangle_property_angle_sum(1,EFC)
isosceles_triangle_judgment_line_equal(1,FCE)
isosceles_triangle_property_angle_equal(1,FCE)
adjacent_complementary_angle(1,DFE,EFC)
parallel_property_corresponding_angle(2,AB,FD,E)
Problem: As shown in the diagram, AB=AC, ∠CAB=30°, BA∥CD. Find the measure of ∠DCE.
问题：如图所示，AB=AC，∠CAB=30°，BA∥CD。求∠DCE的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD)
Shape(DC,CE)
Collinear(BCE)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),30)
ParallelBetweenLine(BA,CD)
Goal: Value(MeasureOfAngle(DCE))
Answer: 75
Theorem sequence:
triangle_property_angle_sum(1,ABC)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
parallel_property_corresponding_angle(2,BA,CD,E)
Problem: As shown in the diagram, AB=AC, ∠CAB=50°, AE bisects ∠CAB, FD is the perpendicular bisector of AB. Find the measure of ∠BCF.
问题：如图所示，AB=AC，∠CAB=50°，AE是∠CAB的角平分线，FD是AB的垂直平分线。求∠BCF的大小。
Construction (CDL):
Shape(AD,DF,FA)
Shape(FD,DB,BF)
Shape(AF,FC,CA)
Shape(FB,BE,EF)
Shape(FE,EC,CF)
Collinear(ADB)
Collinear(AFE)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),50)
IsBisectorOfAngle(AE,CAB)
IsPerpendicularBisectorOfLine(FD,AB)
Goal: Value(MeasureOfAngle(BCF))
Answer: 40
Theorem sequence:
angle_addition(1,CAE,EAB)
angle_addition(1,ABF,FBC)
perpendicular_bisector_property_distance_equal(1,FD,AB)
triangle_property_angle_sum(1,ABE)
isosceles_triangle_judgment_line_equal(1,FAB)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_line_coincidence(3,ABC,E)
isosceles_triangle_property_angle_equal(1,FAB)
perpendicular_bisector_judgment_per_and_mid(1,FE,BC)
perpendicular_bisector_property_distance_equal(1,FE,BC)
isosceles_triangle_judgment_line_equal(1,FBC)
isosceles_triangle_property_angle_equal(1,FBC)
Problem: As shown in the diagram, AB=BC, ∠ACD=110°, EA∥BC. Find the measure of ∠BAE.
问题：如图所示，AB=BC，∠ACD=110°，EA∥BC。求∠BAE的大小。
Construction (CDL):
Shape(BA,AE)
Shape(AB,BC,CA)
Shape(AC,CD)
Collinear(BCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(BC))
Equal(MeasureOfAngle(ACD),110)
ParallelBetweenLine(EA,BC)
Goal: Value(MeasureOfAngle(BAE))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,ABC)
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
adjacent_complementary_angle(1,BCA,ACD)
parallel_property_alternate_interior_angle(2,EA,BC)
Problem: As shown in the diagram, AB=13, AB=AC, AE=ED, BC=10, AD bisects ∠BAC. Find the perimeter of triangle DEC.
问题：如图所示，AB=13，AB=AC，AE=ED，BC=10，AD平分∠BAC。求△DEC的周长。
Construction (CDL):
Shape(AD,DB,BA)
Shape(AE,ED,DA)
Shape(DE,EC,CD)
Collinear(AEC)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(AB),13)
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(AE),LengthOfLine(ED))
Equal(LengthOfLine(BC),10)
IsBisectorOfAngle(AD,BAC)
Goal: Value(PerimeterOfTriangle(DEC))
Answer: 18
Theorem sequence:
line_addition(1,AE,EC)
line_addition(1,CD,DB)
isosceles_triangle_judgment_line_equal(1,ACB)
isosceles_triangle_property_line_coincidence(3,ACB,D)
triangle_perimeter_formula(1,DEC)
Problem: As shown in the diagram, AB=AC, ∠CAB=58°, AE bisects ∠CAB, OD perpendicular bisects AB. Find the measure of ∠AOC.
问题：如图所示，AB=AC，∠CAB=58°，AE是∠CAB的角平分线，OD垂直平分AB。求∠AOC的大小。
Construction (CDL):
Shape(AD,DO,OA)
Shape(AO,OC,CA)
Shape(BO,OD,DB)
Shape(BE,EO,OB)
Shape(OE,EC,CO)
Collinear(AOE)
Collinear(BEC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),58)
IsBisectorOfAngle(AE,CAB)
IsPerpendicularBisectorOfLine(OD,AB)
Goal: Value(MeasureOfAngle(AOC))
Answer: 122
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,OD,AB)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_line_coincidence(3,ABC,E)
isosceles_triangle_property_angle_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,OAB)
triangle_property_angle_sum(1,CAE)
triangle_property_angle_sum(1,ABC)
mirror_congruent_triangle_judgment_sas(1,AOC,ABO)
mirror_congruent_triangle_property_angle_equal(1,OCA,OAB)
Problem: As shown in the diagram, AB=AC, ∠ABC=70°, AF is the angle bisector of ∠CAB, ED perpendicular bisects AB. Find the measure of ∠BCE.
问题：如图所示，AB=AC，∠ABC=70°，AF平分∠CAB，ED是AB的垂直平分线。求∠BCE的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(AE,EC,CA)
Shape(ED,DB,BE)
Shape(EB,BF,FE)
Shape(EF,FC,CE)
Collinear(ADB)
Collinear(AEF)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(ABC),70)
IsBisectorOfAngle(AF,CAB)
IsPerpendicularBisectorOfLine(ED,AB)
Goal: Value(MeasureOfAngle(BCE))
Answer: 50
Theorem sequence:
angle_addition(1,ABE,EBC)
perpendicular_bisector_property_distance_equal(1,ED,AB)
isosceles_triangle_judgment_line_equal(1,EAB)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_line_coincidence(3,ABC,F)
isosceles_triangle_property_angle_equal(1,EAB)
triangle_property_angle_sum(1,ABF)
mirror_congruent_triangle_judgment_sas(1,FEB,FCE)
mirror_congruent_triangle_property_angle_equal(1,BFE,CEF)
Problem: As shown in the diagram, AB=AC, BC=9, BD=5, BE=6, BEFD is a parallelogram. Find the perimeter of △FEC.
问题：如图所示，AB=AC，BC=9，BD=5，BE=6，BD和EF是▱BEFD的一组对边。求△FEC的周长。
Construction (CDL):
Shape(AD,DF,FA)
Shape(DB,BE,EF,FD)
Shape(FE,EC,CF)
Collinear(ADB)
Collinear(AFC)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(BC),9)
Equal(LengthOfLine(BD),5)
Equal(LengthOfLine(BE),6)
Parallelogram(BEFD)
Goal: Value(PerimeterOfTriangle(FEC))
Answer: 13
Theorem sequence:
parallelogram_property_opposite_line_equal(1,EFDB)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
parallel_property_corresponding_angle(2,BD,EF,C)
isosceles_triangle_judgment_angle_equal(1,FEC)
line_addition(1,BE,EC)
triangle_perimeter_formula(1,FEC)
Problem: As shown in the diagram, ∠BCD=120°, ABCD is a ▱. Find the sum of the measure of ∠AEF and the measure of ∠EFA.
问题：如图所示，∠BCD=120°，AD和BC是▱ABCD的一组对边。求∠AEF的大小与∠EFA的大小之和。
Construction (CDL):
Shape(EF,FA,AE)
Shape(AB,BC,CD,DA)
Collinear(FAD)
Collinear(EAB)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),120)
Parallelogram(ABCD)
Goal: Value(Add(MeasureOfAngle(AEF),MeasureOfAngle(EFA)))
Answer: 60
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,ABCD)
vertical_angle(1,FAE,DAB)
triangle_property_angle_sum(1,EFA)
Problem: As shown in the diagram, AD=12, BA=AF, BE=EC, the perimeter of ABCD is 34, AD and BC are opposite sides of the ▱ ABCD, BE is perpendicular to FE. Find the length of line AE.
问题：如图所示，AD=12，BA=AF，BE=EC，四边形ABCD的周长为34，AD和BC是平行四边形ABCD的一组对边，BE⊥FE。求直线AE的长度。
Construction (CDL):
Shape(FA,AG,GF)
Shape(AE,EG,GA)
Shape(AB,BE,EA)
Shape(FG,GH,HF)
Shape(DH,HC,CD)
Shape(HG,GE,EC,CH)
Collinear(AGHD)
Collinear(BAF)
Collinear(BEC)
Collinear(FGE)
Collinear(FHC)
Conditions (CDL):
Equal(LengthOfLine(AD),12)
Equal(LengthOfLine(BA),LengthOfLine(AF))
Equal(LengthOfLine(BE),LengthOfLine(EC))
Equal(PerimeterOfQuadrilateral(ABCD),34)
Parallelogram(ABCD)
PerpendicularBetweenLine(BE,FE)
Goal: Value(LengthOfLine(AE))
Answer: 5
Theorem sequence:
line_addition(1,BA,AF)
quadrilateral_perimeter_formula(1,ABCD)
perpendicular_bisector_judgment_per_and_mid(1,FE,BC)
perpendicular_bisector_property_distance_equal(1,FE,BC)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
midsegment_of_triangle_judgment_midpoint(1,EA,BCF)
midsegment_of_triangle_property_length(1,EA,BCF)
Problem: As shown in the diagram, AB=1, ∠ABC=45°, AE∥BD, AD and BC are opposite sides of the ▱ ABCD, BF is perpendicular to EF. Find the length of line EF.
问题：如图所示，AB=1，∠ABC=45°，AE∥BD，ABCD是平行四边形，BF垂直于EF。求直线EF的长度。
Construction (CDL):
Shape(EA,AD,DE)
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Shape(ED,DC,CF,FE)
Collinear(CDE)
Collinear(BCF)
Conditions (CDL):
Equal(LengthOfLine(AB),1)
Equal(MeasureOfAngle(ABC),45)
ParallelBetweenLine(AE,BD)
Parallelogram(ABCD)
PerpendicularBetweenLine(BF,EF)
Goal: Value(LengthOfLine(EF))
Answer: sqrt(2)
Theorem sequence:
parallel_property_collinear_extend(1,DC,AB,E)
parallelogram_judgment_parallel_and_parallel(1,ABDE)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,ABDE)
parallel_property_corresponding_angle(2,BA,CD,F)
line_addition(1,CD,DE)
sine_theorem(1,ECF)
Problem: As shown in the diagram, AB=10, BF=12, AE is the angle bisector of ∠DAB, BF is the angle bisector of ∠ABC, AD and BC are opposite sides of the ▱ ABCD. Find the length of line AE.
问题：如图所示，AB=10，BF=12，AE平分∠DAB，BF平分∠ABC，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(AB,BO,OA)
Shape(AO,OF,FA)
Shape(BE,EO,OB)
Shape(FO,OE,EF)
Shape(FE,EC,CD,DF)
Collinear(AFD)
Collinear(BEC)
Collinear(AOE)
Collinear(BOF)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BF),12)
IsBisectorOfAngle(AE,DAB)
IsBisectorOfAngle(BF,ABC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AE))
Answer: 16
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,F)
parallel_property_collinear_extend(3,CB,DA,E)
parallel_property_ipsilateral_internal_angle(1,AD,BC)
parallel_property_alternate_interior_angle(1,EB,DA)
parallel_property_alternate_interior_angle(2,AF,BC)
isosceles_triangle_judgment_angle_equal(1,BEA)
isosceles_triangle_judgment_angle_equal(1,ABF)
cosine_theorem(1,BEA)
cosine_theorem(1,ABF)
Problem: As shown in the diagram, the area of ABCD is 24, the area of quadrilateral ABCD is Add(AreaOfTriangle(APD),AreaOfTriangle(PBC),AreaOfTriangle(PAB)), the height of quadrilateral DABC is equal to the length of line PE, PE is the altitude of triangle PAB, AD and BC are opposite sides of the parallelogram ABCD. Find the sum of the area of triangle APD and the area of △PBC.
问题：如图所示，四边形ABCD的面积为24，ABCD的面积为Add(AreaOfTriangle(APD),AreaOfTriangle(PBC),AreaOfTriangle(PAB))，四边形DABC的高与直线PE的长度相等，PE是△PAB的高，AD和BC是▱ABCD的一组对边。求△APD的面积与△PBC的面积之和。
Construction (CDL):
Shape(DA,AP,PD)
Shape(PB,BC,CP)
Shape(PA,AE,EP)
Shape(PE,EB,BP)
Collinear(DPC)
Collinear(AEB)
Conditions (CDL):
Equal(AreaOfQuadrilateral(ABCD),24)
Equal(AreaOfQuadrilateral(ABCD),Add(AreaOfTriangle(APD),AreaOfTriangle(PBC),AreaOfTriangle(PAB)))
Equal(HeightOfQuadrilateral(DABC),LengthOfLine(PE))
IsAltitudeOfTriangle(PE,PAB)
Parallelogram(ABCD)
Goal: Value(Add(AreaOfTriangle(APD),AreaOfTriangle(PBC)))
Answer: 12
Theorem sequence:
parallelogram_area_formula_common(1,DABC)
triangle_area_formula_common(1,PAB)
Problem: As shown in the diagram, AB=8, AD=10, E is the midpoint of segment AB, F is the midpoint of segment AP, G bisects segment DP, H is the midpoint of segment DC, AD and BC are opposite sides of the parallelogram ABCD. Find the sum of the length of line EF and the length of line GH.
问题：如图所示，AB=8，AD=10，E是线段AB的中点，F平分线段AP，G是线段DP的中点，H平分线段DC，四边形ABCD是平行四边形。求直线EF的长度与直线GH的长度之和。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BP,PF,FE)
Shape(AF,FP,PG,GD,DA)
Shape(DG,GH,HD)
Shape(GP,PC,CH,HG)
Collinear(AEB)
Collinear(AFP)
Collinear(DGP)
Collinear(DHC)
Collinear(BPC)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AD),10)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,AP)
IsMidpointOfLine(G,DP)
IsMidpointOfLine(H,DC)
Parallelogram(ABCD)
Goal: Value(Add(LengthOfLine(EF),LengthOfLine(GH)))
Answer: 5
Theorem sequence:
line_addition(1,BP,PC)
parallelogram_property_opposite_line_equal(1,BCDA)
midsegment_of_triangle_judgment_midpoint(1,EF,ABP)
midsegment_of_triangle_judgment_midpoint(1,GH,DPC)
midsegment_of_triangle_property_length(1,EF,ABP)
midsegment_of_triangle_property_length(1,GH,DPC)
Problem: As shown in the diagram, EF=3, E is the midpoint of segment AB, F bisects segment AO, BA and CD are opposite sides of the ▱ ABCD. Find the length of line BD.
问题：如图所示，EF=3，E是线段AB的中点，F是线段AO的中点，ABCD是▱。求直线BD的长度。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BO,OF,FE)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Shape(AF,FO,OD,DA)
Collinear(AFOC)
Collinear(AEB)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(EF),3)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,AO)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BD))
Answer: 12
Theorem sequence:
line_addition(1,BO,OD)
parallelogram_property_diagonal_bisection(1,BCDA,O)
midsegment_of_triangle_judgment_midpoint(1,EF,ABO)
midsegment_of_triangle_property_length(1,EF,ABO)
Problem: As shown in the diagram, BC=12, BF=Mul(4,LengthOfLine(AF)), CF is the angle bisector of ∠BCD, quadrilateral ABCD is a parallelogram. Find the length of line AF.
问题：如图所示，BC=12，BF=Mul(4,LengthOfLine(AF))，CF是∠BCD的角平分线，BA和CD是平行四边形ABCD的一组对边。求直线AF的长度。
Construction (CDL):
Shape(CD,DE,EC)
Shape(EF,FA,AE)
Shape(CE,EA,AB,BC)
Collinear(DEA)
Collinear(CEF)
Collinear(FAB)
Conditions (CDL):
Equal(LengthOfLine(BC),12)
Equal(LengthOfLine(BF),Mul(4,LengthOfLine(AF)))
IsBisectorOfAngle(CF,BCD)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AF))
Answer: 3
Theorem sequence:
parallel_property_collinear_extend(2,BA,CD,F)
parallel_property_alternate_interior_angle(2,BF,CD)
isosceles_triangle_judgment_angle_equal(1,BCF)
Problem: As shown in the diagram, AD=5, OE=2, E is the midpoint of segment BC, AD and BC are opposite sides of the parallelogram ABCD. Find the perimeter of quadrilateral ABCD.
问题：如图所示，AD=5，OE=2，E是线段BC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(AB,BO,OA)
Shape(OB,BE,EO)
Shape(OE,EC,CO)
Shape(AO,OD,DA)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(BEC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AD),5)
Equal(LengthOfLine(OE),2)
IsMidpointOfLine(E,BC)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 18
Theorem sequence:
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,OE,CAB)
midsegment_of_triangle_judgment_midpoint(1,EO,BCD)
midsegment_of_triangle_property_length(1,OE,CAB)
midsegment_of_triangle_property_length(1,EO,BCD)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, EF=6, E is the midpoint of segment AB, F is the midpoint of segment AC, AD and BC are opposite sides of the ▱ ABCD. Find the length of line AD.
问题：如图所示，EF=6，E是线段AB的中点，F平分线段AC，AD和BC是平行四边形ABCD的一组对边。求直线AD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)
IsMidpointOfLine(F,AC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AD))
Answer: 12
Theorem sequence:
parallelogram_property_opposite_line_equal(1,BCDA)
midsegment_of_triangle_judgment_midpoint(1,EF,ABC)
midsegment_of_triangle_property_length(1,EF,ABC)
Problem: As shown in the diagram, the perimeter of quadrilateral ABCD is 20, AD and BC are opposite sides of the ▱ ABCD, AO⊥EO. Find the perimeter of △CDE.
问题：如图所示，ABCD的周长为20，四边形ABCD是▱，AO⊥EO。求△CDE的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OE,EA)
Shape(EO,OF,FE)
Shape(EF,FD,DE)
Shape(DF,FC,CD)
Shape(FO,OC,CF)
Shape(OB,BC,CO)
Collinear(BOFD)
Collinear(AOC)
Collinear(EFC)
Collinear(AED)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),20)
Parallelogram(ABCD)
PerpendicularBetweenLine(AO,EO)
Goal: Value(PerimeterOfTriangle(CDE))
Answer: 10
Theorem sequence:
line_addition(1,AE,ED)
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)
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, AC=4, AE=AB, CE=BC, ∠AOD=120°, ACED is a parallelogram. Find the length of line CD.
问题：如图所示，AC=4，AE=AB，CE=BC，∠AOD=120°，AD和CE是平行四边形ACED的一组对边。求直线CD的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CO,OA)
Shape(AO,OD,DA)
Shape(DO,OE,ED)
Shape(EO,OC,CE)
Collinear(BCE)
Collinear(COD)
Collinear(AOE)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(AE),LengthOfLine(AB))
Equal(LengthOfLine(CE),LengthOfLine(BC))
Equal(MeasureOfAngle(AOD),120)
Parallelogram(ACED)
Goal: Value(LengthOfLine(CD))
Answer: 8
Theorem sequence:
line_addition(1,CO,OD)
line_addition(1,AO,OE)
median_of_triangle_judgment(1,AC,ABE)
isosceles_triangle_judgment_line_equal(1,ABE)
isosceles_triangle_property_line_coincidence(2,ABE,C)
parallelogram_property_diagonal_bisection(1,ACED,O)
parallelogram_property_diagonal_bisection(1,CEDA,O)
rectangle_judgment_right_angle(1,ACED)
rectangle_property_diagonal_equal(1,ACED)
vertical_angle(1,AOD,EOC)
isosceles_triangle_judgment_line_equal(1,OCE)
isosceles_triangle_property_angle_equal(1,OCE)
triangle_property_angle_sum(1,OCE)
sine_theorem(1,ACE)
Problem: As shown in the diagram, AB=7, AD=4, AF bisects ∠DAB, quadrilateral ABCD is a ▱. Find the length of line CF.
问题：如图所示，AB=7，AD=4，AF平分∠DAB，AD和BC是平行四边形ABCD的一组对边。求直线CF的长度。
Construction (CDL):
Shape(ED,DA,AE)
Shape(AB,BC,CE,EA)
Shape(FE,EC,CF)
Collinear(DEC)
Collinear(AEF)
Collinear(BCF)
Conditions (CDL):
Equal(LengthOfLine(AB),7)
Equal(LengthOfLine(AD),4)
IsBisectorOfAngle(AF,DAB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(CF))
Answer: 3
Theorem sequence:
line_addition(1,BC,CF)
parallel_property_collinear_extend(1,CB,DA,F)
parallel_property_alternate_interior_angle(1,FC,DA)
parallelogram_property_opposite_line_equal(1,BCDA)
isosceles_triangle_judgment_angle_equal(1,BFA)
Problem: As shown in the diagram, AB=10, AD=15, ABCD is a parallelogram, BO⊥EO. Find the perimeter of △ABE.
问题：如图所示，AB=10，AD=15，BA和CD是▱ABCD的一组对边，BO垂直于EO。求△ABE的周长。
Construction (CDL):
Shape(AF,FE,EA)
Shape(EF,FO,OE)
Shape(AB,BF,FA)
Shape(FB,BO,OF)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Shape(EO,OD,DE)
Collinear(AFOC)
Collinear(BOD)
Collinear(AED)
Collinear(BFE)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AD),15)
Parallelogram(ABCD)
PerpendicularBetweenLine(BO,EO)
Goal: Value(PerimeterOfTriangle(ABE))
Answer: 25
Theorem sequence:
line_addition(1,AE,ED)
parallelogram_property_diagonal_bisection(1,BCDA,O)
perpendicular_bisector_judgment_per_and_mid(1,EO,BD)
perpendicular_bisector_property_distance_equal(1,EO,BD)
triangle_perimeter_formula(1,ABE)