SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, BC=8, CG=12, GA=4, D is the midpoint of segment EF, AF∥EC. Find the length of line AF.
问题：如图所示，BC=8，CG=12，GA=4，D平分线段EF，AF平行于EC。求直线AF的长度。
Construction (CDL):
Shape(AG,GF,FA)
Shape(AD,DG,GA)
Shape(DE,EB,BD)
Shape(GD,DB,BC,CG)
Collinear(ADB)
Collinear(AGC)
Collinear(FGDE)
Collinear(EBC)
Conditions (CDL):
Equal(LengthOfLine(BC),8)
Equal(LengthOfLine(CG),12)
Equal(LengthOfLine(GA),4)
IsMidpointOfLine(D,EF)
ParallelBetweenLine(AF,EC)
Goal: Value(LengthOfLine(AF))
Answer: 4
Theorem sequence:
parallel_property_collinear_extend(3,CE,FA,B)
parallel_property_alternate_interior_angle(1,AF,EC)
parallel_property_alternate_interior_angle(2,AF,EC)
parallel_property_alternate_interior_angle(1,BE,FA)
similar_triangle_judgment_aa(1,GFA,GEC)
similar_triangle_judgment_aa(1,DFA,DEB)
similar_triangle_property_line_ratio(1,GFA,GEC)
similar_triangle_property_line_ratio(1,FAG,ECG)
similar_triangle_property_line_ratio(1,DFA,DEB)
similar_triangle_property_line_ratio(1,ADF,BDE)
line_addition(1,EB,BC)
Problem: As shown in the diagram, OM=12, the radius of circle O is 6, the center of ⊙O is O, AN⊥ON. Find the length of line OP.
问题：如图所示，OM=12，⊙O的半径为6，⊙O的圆心为O，AN垂直于ON。求直线OP的长度。
Construction (CDL):
Shape(OBA,AB)
Shape(AD,DB,BA)
Shape(OAE,EN,NA)
Shape(AN,NP,PA)
Shape(AP,PD,DA)
Shape(DP,PO,OD)
Shape(OEC,CN,NE)
Shape(NC,CP,PN)
Shape(PC,CO,OP)
Shape(AM,ME,OAE)
Shape(BD,DO,OC,OCB)
Collinear(MAB)
Collinear(MENPO)
Collinear(ANC)
Collinear(ADO)
Collinear(CPDB)
Cocircular(O,BAEC)
Conditions (CDL):
Equal(LengthOfLine(OM),12)
Equal(RadiusOfCircle(O),6)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AN,ON)
Goal: Value(LengthOfLine(OP))
Answer: 3
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,ON,AC)
mirror_congruent_triangle_judgment_sas(1,NOA,NCO)
mirror_congruent_triangle_property_angle_equal(1,OAN,ONC)
angle_addition(1,CON,NOA)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
triangle_property_angle_sum(1,AMO)
triangle_property_angle_sum(1,MPB)
mirror_congruent_triangle_judgment_sas(1,NPA,NCP)
mirror_congruent_triangle_property_angle_equal(1,PAN,PNC)
vertical_angle(1,CPN,DPO)
angle_addition(1,NPA,APD)
angle_addition(1,APD,DPO)
mirror_similar_triangle_judgment_aa(1,MOA,APO)
radius_of_circle_property_length_equal(1,OA,O)
mirror_similar_triangle_property_line_ratio(1,MOA,APO)
mirror_similar_triangle_property_line_ratio(1,AMO,POA)
Problem: As shown in the diagram, the area of triangle DOE is 12, E is the midpoint of segment DC, AD and BC are opposite sides of the ▱ DABC. Find the area of △OAB.
问题：如图所示，三角形DOE的面积为12，E是线段DC的中点，四边形DABC是平行四边形。求三角形OAB的面积。
Construction (CDL):
Shape(DO,OE,ED)
Shape(DA,AO,OD)
Shape(OA,AB,BO)
Shape(CE,EO,OB,BC)
Collinear(DEC)
Collinear(DOB)
Collinear(AOE)
Conditions (CDL):
Equal(AreaOfTriangle(DOE),12)
IsMidpointOfLine(E,DC)
Parallelogram(DABC)
Goal: Value(AreaOfTriangle(OAB))
Answer: 48
Theorem sequence:
parallel_property_collinear_extend(3,DC,AB,E)
parallel_property_alternate_interior_angle(1,DE,AB)
parallel_property_alternate_interior_angle(2,DE,AB)
similar_triangle_judgment_aa(1,OED,OAB)
parallelogram_property_opposite_line_equal(1,ABCD)
line_addition(1,DE,EC)
similar_triangle_property_line_ratio(1,OED,OAB)
similar_triangle_property_area_square_ratio(1,OED,OAB)
Problem: As shown in the diagram, AE=4, EB=6, EF=4, EF∥BC, AD and BC are opposite sides of the ▱ ABCD. Find the length of line AD.
问题：如图所示，AE=4，EB=6，EF=4，EF平行于BC，四边形ABCD是平行四边形。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(AE),4)
Equal(LengthOfLine(EB),6)
Equal(LengthOfLine(EF),4)
ParallelBetweenLine(EF,BC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AD))
Answer: 10
Theorem sequence:
line_addition(1,AE,EB)
parallel_property_corresponding_angle(1,EF,BC,A)
similar_triangle_judgment_aa(1,FAE,CAB)
similar_triangle_property_line_ratio(1,FAE,CAB)
similar_triangle_property_line_ratio(1,AEF,ABC)
parallelogram_property_opposite_line_equal(1,BCDA)
Problem: As shown in the diagram, BC=13, CD=5, ∠ACB=90°, ∠ACD=∠CBA. Find the length of line BD.
问题：如图所示，BC=13，CD=5，∠ACB=90°，∠ACD=∠CBA。求直线BD的长度。
Construction (CDL):
Shape(AC,CD,DA)
Shape(DC,CB,BD)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(BC),13)
Equal(LengthOfLine(CD),5)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(ACD),MeasureOfAngle(CBA))
Goal: Value(LengthOfLine(BD))
Answer: 12
Theorem sequence:
angle_addition(1,ACD,DCB)
triangle_property_angle_sum(1,DCB)
right_triangle_judgment_angle(1,BDC)
right_triangle_property_pythagorean(1,BDC)
Problem: As shown in the diagram, the area of △DMN is 3, M bisects segment DC, ADCB is a ▱. Find the area of △ANB.
问题：如图所示，三角形DMN的面积为3，M平分线段DC，DA和CB是平行四边形ADCB的一组对边。求三角形ANB的面积。
Construction (CDL):
Shape(AD,DN,NA)
Shape(AN,NB,BA)
Shape(ND,DM,MN)
Shape(NM,MC,CB,BN)
Collinear(DNB)
Collinear(ANM)
Collinear(DMC)
Conditions (CDL):
Equal(AreaOfTriangle(DMN),3)
IsMidpointOfLine(M,DC)
Parallelogram(ADCB)
Goal: Value(AreaOfTriangle(ANB))
Answer: 12
Theorem sequence:
parallel_property_collinear_extend(3,CD,BA,M)
parallel_property_alternate_interior_angle(1,AB,DM)
parallel_property_alternate_interior_angle(2,AB,DM)
similar_triangle_judgment_aa(1,NDM,NBA)
line_addition(1,DM,MC)
parallelogram_property_opposite_line_equal(1,DCBA)
similar_triangle_property_line_ratio(1,NDM,NBA)
similar_triangle_property_area_square_ratio(1,NDM,NBA)
Problem: As shown in the diagram, BE=2, EC=3, BD bisects ∠ABC, DE∥AB. Find the length of line AB.
问题：如图所示，BE=2，EC=3，BD是∠ABC的角平分线，DE平行于AB。求直线AB的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BE,ED)
Shape(DE,EC,CD)
Collinear(ADC)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(BE),2)
Equal(LengthOfLine(EC),3)
IsBisectorOfAngle(BD,ABC)
ParallelBetweenLine(DE,AB)
Goal: Value(LengthOfLine(AB))
Answer: 10/3
Theorem sequence:
parallel_property_alternate_interior_angle(1,DE,AB)
isosceles_triangle_judgment_angle_equal(1,EDB)
line_addition(1,BE,EC)
parallel_property_corresponding_angle(1,DE,AB,C)
similar_triangle_judgment_aa(1,ECD,BCA)
similar_triangle_property_line_ratio(1,CDE,CAB)
similar_triangle_property_line_ratio(1,DEC,ABC)
Problem: As shown in the diagram, AB=10, BC=16, BF is the angle bisector of ∠ABC, D is the midpoint of segment AB, BF⊥AF. Find the length of line EF.
问题：如图所示，AB=10，BC=16，BF平分∠ABC，D平分线段AB，BF⊥AF。求直线EF的长度。
Construction (CDL):
Shape(AD,DF,FA)
Shape(AF,FE,EA)
Shape(DB,BF,FD)
Shape(FB,BC,CE,EF)
Collinear(ADB)
Collinear(AEC)
Collinear(DFE)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),16)
IsBisectorOfAngle(BF,ABC)
IsMidpointOfLine(D,AB)
PerpendicularBetweenLine(BF,AF)
Goal: Value(LengthOfLine(EF))
Answer: 3
Theorem sequence:
line_addition(1,AD,DB)
right_triangle_judgment_angle(1,BFA)
median_of_triangle_judgment(1,FD,FAB)
right_triangle_property_length_of_median(1,BFA,D)
isosceles_triangle_judgment_line_equal(1,DBF)
isosceles_triangle_property_angle_equal(1,DBF)
parallel_judgment_alternate_interior_angle(2,DF,BC)
parallel_property_corresponding_angle(1,DF,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_line_ratio(1,ADE,ABC)
line_addition(1,DF,FE)
Problem: As shown in the diagram, AB=4, BC=3, EF=1, ∠ABC=90°, DEFG is a square. Find the length of line BN.
问题：如图所示，AB=4，BC=3，EF=1，∠ABC=90°，DEFG是正方形。求直线BN的长度。
Construction (CDL):
Shape(DA,AM,MD)
Shape(MA,AE,EM)
Shape(GD,DM,MG)
Shape(GM,ME,EF,FG)
Shape(CD,DG,GN,NC)
Shape(NG,GF,FB,BN)
Collinear(ADC)
Collinear(AMGN)
Collinear(AEFB)
Collinear(DME)
Collinear(CNB)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BC),3)
Equal(LengthOfLine(EF),1)
Equal(MeasureOfAngle(ABC),90)
Square(DEFG)
Goal: Value(LengthOfLine(BN))
Answer: 12/7
Theorem sequence:
adjacent_complementary_angle(1,AED,DEF)
similar_triangle_judgment_aa(1,DAE,CAB)
similar_triangle_property_line_ratio(1,DAE,CAB)
similar_triangle_property_line_ratio(1,AED,ABC)
line_addition(1,AE,EF)
similar_triangle_judgment_aa(1,GAF,NAB)
similar_triangle_property_line_ratio(1,GAF,NAB)
similar_triangle_property_line_ratio(1,AFG,ABN)
Problem: As shown in the diagram, AD=2, DB=4, DE=2, DE∥BC. Find the length of line BC.
问题：如图所示，AD=2，DB=4，DE=2，DE平行于BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),2)
Equal(LengthOfLine(DB),4)
Equal(LengthOfLine(DE),2)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 6
Theorem sequence:
line_addition(1,AD,DB)
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_line_ratio(1,ADE,ABC)
Problem: As shown in the diagram, AE=4, DE=4, EC=6, DE∥BC. Find the length of line BC.
问题：如图所示，AE=4，DE=4，EC=6，DE平行于BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AE),4)
Equal(LengthOfLine(DE),4)
Equal(LengthOfLine(EC),6)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 10
Theorem sequence:
line_addition(1,AE,EC)
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_property_line_ratio(1,ADE,ABC)
similar_triangle_property_line_ratio(1,DEA,BCA)
Problem: As shown in the diagram, ∠BAC=30°, O is the center of ⊙O, the diameter of ⊙O is AB, CD is the diameter of ⊙O, EB∥DC. Find the ratio of the length of line BE to the length of line AB.
问题：如图所示，∠BAC=30°，圆O的圆心为O，⊙O的直径为AB，圆O的直径为CD，EB∥DC。求直线BE的长度与直线AB的长度之比。
Construction (CDL):
Shape(ODA,AF,FD)
Shape(OED,DF,FE)
Shape(FA,AO,OF)
Shape(EF,FO,OB,BE)
Shape(OBE,EB)
Shape(OA,AC,CO)
Shape(OAC,CA)
Shape(BO,OC,OCB)
Collinear(DFOC)
Collinear(AFE)
Collinear(AOB)
Cocircular(O,DACBE)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(EB,DC)
Goal: Value(Div(LengthOfLine(BE),LengthOfLine(AB)))
Answer: 1/2
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
parallel_property_collinear_extend(3,CD,BE,O)
parallel_property_alternate_interior_angle(2,EB,OC)
diameter_of_circle_property_right_angle(1,BEA,O)
triangle_property_angle_sum(1,EAB)
sine_theorem(1,BEA)
Problem: As shown in the diagram, CE=1, ED=2, ABCD is a rhombus. Find the length of line CF.
问题：如图所示，CE=1，ED=2，ABCD是菱形。求直线CF的长度。
Construction (CDL):
Shape(AB,BC,CE,EA)
Shape(AE,ED,DA)
Shape(EC,CF,FE)
Collinear(BCF)
Collinear(AEF)
Collinear(CED)
Conditions (CDL):
Equal(LengthOfLine(CE),1)
Equal(LengthOfLine(ED),2)
Rhombus(ABCD)
Goal: Value(LengthOfLine(CF))
Answer: 3/2
Theorem sequence:
line_addition(1,CE,ED)
parallel_property_collinear_extend(1,CB,DA,F)
parallel_property_alternate_interior_angle(1,AD,CF)
parallel_property_alternate_interior_angle(2,AD,CF)
similar_triangle_judgment_aa(1,ECF,EDA)
similar_triangle_property_line_ratio(1,ECF,EDA)
similar_triangle_property_line_ratio(1,FEC,AED)
Problem: As shown in the diagram, the perimeter of △DEF is 1, E is the midpoint of segment AD, BA and CD are opposite sides of the ▱ ABCD. Find the perimeter of △BCF.
问题：如图所示，△DEF的周长为1，E是线段AD的中点，ABCD是▱。求三角形BCF的周长。
Construction (CDL):
Shape(AB,BF,FE,EA)
Shape(EF,FD,DE)
Shape(FB,BC,CF)
Shape(DF,FC,CD)
Collinear(AED)
Collinear(BFD)
Collinear(EFC)
Conditions (CDL):
Equal(PerimeterOfTriangle(DEF),1)
IsMidpointOfLine(E,AD)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(BCF))
Answer: 2
Theorem sequence:
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,AE,ED)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_alternate_interior_angle(1,ED,BC)
parallel_property_alternate_interior_angle(2,ED,BC)
similar_triangle_judgment_aa(1,FBC,FDE)
similar_triangle_property_line_ratio(1,FBC,FDE)
similar_triangle_property_perimeter_ratio(1,FBC,FDE)
Problem: As shown in the diagram, PA=PB, PB=4, PD=3, ∠BPA=Mul(2,MeasureOfAngle(BCA))°, P is the midpoint of segment EB. Find the value of AD*DC.
问题：如图所示，PA=PB，PB=4，PD=3，∠BPA=Mul(2,MeasureOfAngle(BCA))°，P是线段EB的中点。求AD*DC的值。
Construction (CDL):
Shape(EA,AP,PE)
Shape(PA,AD,DP)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Collinear(EPDB)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(PA),LengthOfLine(PB))
Equal(LengthOfLine(PB),4)
Equal(LengthOfLine(PD),3)
Equal(MeasureOfAngle(BPA),Mul(2,MeasureOfAngle(BCA)))
IsMidpointOfLine(P,EB)
Goal: Value(Mul(LengthOfLine(AD),LengthOfLine(DC)))
Answer: 7
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,PEA)
isosceles_triangle_property_angle_equal(1,PEA)
triangle_property_angle_sum(1,PEA)
adjacent_complementary_angle(1,BPA,APE)
vertical_angle(1,ADE,CDB)
mirror_similar_triangle_judgment_aa(1,ADE,BCD)
line_addition(1,PD,DB)
line_addition(1,EP,PD)
mirror_similar_triangle_property_line_ratio(1,ADE,BCD)
mirror_similar_triangle_property_line_ratio(1,EAD,CDB)
Problem: As shown in the diagram, Div(LengthOfLine(DE)=LengthOfLine(BC)), DE∥BC. Find the ratio of the area of triangle ADE to the area of △ABC.
问题：如图所示，Div(LengthOfLine(DE)=LengthOfLine(BC))，DE∥BC。求三角形ADE的面积与△ABC的面积之比。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(Div(LengthOfLine(DE),LengthOfLine(BC)),1/3)
ParallelBetweenLine(DE,BC)
Goal: Value(Div(AreaOfTriangle(ADE),AreaOfTriangle(ABC)))
Answer: 1/9
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_property_line_ratio(1,ADE,ABC)
similar_triangle_property_area_square_ratio(1,EAD,CAB)
Problem: As shown in the diagram, AC=4, AF=1, ∠EFD=60°, triangle CAB is an equilateral triangle, FD⊥ED. Find the length of line AD.
问题：如图所示，AC=4，AF=1，∠EFD=60°，△CAB为等边△，FD⊥ED。求直线AD的长度。
Construction (CDL):
Shape(CF,FE,EC)
Shape(FA,AD,DF)
Shape(FD,DE,EF)
Shape(ED,DB,BE)
Collinear(CFA)
Collinear(CEB)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(AF),1)
Equal(MeasureOfAngle(EFD),60)
EquilateralTriangle(CAB)
PerpendicularBetweenLine(FD,ED)
Goal: Value(LengthOfLine(AD))
Answer: 3/2
Theorem sequence:
equilateral_triangle_property_angle(1,CAB)
equilateral_triangle_property_angle(1,ABC)
triangle_property_angle_sum(1,FAD)
angle_addition(1,CFE,EFD)
adjacent_complementary_angle(1,CFD,DFA)
similar_triangle_judgment_aa(1,ECF,FAD)
line_addition(1,AF,FC)
similar_triangle_property_line_ratio(1,ECF,FAD)
similar_triangle_property_line_ratio(1,CFE,ADF)
triangle_property_angle_sum(1,EFD)
sine_theorem(1,FDE)
Problem: As shown in the diagram, AD=4, DB=2, DE=7/2, DE∥BC. Find the length of line BC.
问题：如图所示，AD=4，DB=2，DE=7/2，DE平行于BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),4)
Equal(LengthOfLine(DB),2)
Equal(LengthOfLine(DE),7/2)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 21/4
Theorem sequence:
line_addition(1,AD,DB)
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_line_ratio(1,ADE,ABC)
Problem: As shown in the diagram, the area of △ADE is 2, AD=1, DB=2, DE is parallel to BC. Find the area of △ABC minus the area of △ADE.
问题：如图所示，△ADE的面积为2，AD=1，DB=2，DE平行于BC。求三角形ABC的面积减去△ADE的面积。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(AreaOfTriangle(ADE),2)
Equal(LengthOfLine(AD),1)
Equal(LengthOfLine(DB),2)
ParallelBetweenLine(DE,BC)
Goal: Value(Sub(AreaOfTriangle(ABC),AreaOfTriangle(ADE)))
Answer: 16
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
line_addition(1,AD,DB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_area_square_ratio(1,EAD,CAB)
Problem: As shown in the diagram, AB=9, AD=3, DE=3, DE is parallel to BC. Find the length of line BC.
问题：如图所示，AB=9，AD=3，DE=3，DE∥BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),9)
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(DE),3)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 9
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_line_ratio(1,ADE,ABC)
Problem: As shown in the diagram, the area of △ADE is 3, D is the midpoint of segment AB, DE is parallel to BC. Find the area of △ABC.
问题：如图所示，△ADE的面积为3，D是线段AB的中点，DE平行于BC。求△ABC的面积。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(AreaOfTriangle(ADE),3)
IsMidpointOfLine(D,AB)
ParallelBetweenLine(DE,BC)
Goal: Value(AreaOfTriangle(ABC))
Answer: 12
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
line_addition(1,AD,DB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_area_square_ratio(1,EAD,CAB)
Problem: As shown in the diagram, AC=2, AD=1, ∠DCA=∠ABC. Find the length of line DB.
问题：如图所示，AC=2，AD=1，∠DCA=∠ABC。求直线DB的长度。
Construction (CDL):
Shape(AD,DC,CA)
Shape(DB,BC,CD)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(AD),1)
Equal(MeasureOfAngle(DCA),MeasureOfAngle(ABC))
Goal: Value(LengthOfLine(DB))
Answer: 3
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,DCA,CAB)
line_addition(1,AD,DB)
mirror_similar_triangle_property_line_ratio(1,DCA,CAB)
mirror_similar_triangle_property_line_ratio(1,CAD,BCA)
Problem: As shown in the diagram, AE=EB, AF=2, quadrilateral DABC is a ▱. Find the length of line FC.
问题：如图所示，AE=EB，AF=2，DABC是平行四边形。求直线FC的长度。
Construction (CDL):
Shape(DA,AF,FD)
Shape(DF,FC,CD)
Shape(FA,AE,EF)
Shape(CF,FE,EB,BC)
Collinear(DFE)
Collinear(AFC)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AE),LengthOfLine(EB))
Equal(LengthOfLine(AF),2)
Parallelogram(DABC)
Goal: Value(LengthOfLine(FC))
Answer: 4
Theorem sequence:
parallel_property_collinear_extend(3,BA,CD,E)
parallel_property_alternate_interior_angle(1,DC,AE)
parallel_property_alternate_interior_angle(2,DC,AE)
similar_triangle_judgment_aa(1,FAE,FCD)
similar_triangle_property_line_ratio(1,FAE,FCD)
similar_triangle_property_line_ratio(1,EFA,DFC)
line_addition(1,AE,EB)
parallelogram_property_opposite_line_equal(1,ABCD)
Problem: As shown in the diagram, AD=5, BD=2, AD is the angle bisector of ∠CAB. Find the length of line DE.
问题：如图所示，AD=5，BD=2，AD平分∠CAB。求直线DE的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(OCA,AC)
Shape(EB,BD,DE)
Shape(OBD,DB)
Shape(CE,ED,ODC)
Collinear(BEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(LengthOfLine(AD),5)
Equal(LengthOfLine(BD),2)
IsBisectorOfAngle(AD,CAB)
Goal: Value(LengthOfLine(DE))
Answer: 4/5
Theorem sequence:
arc_property_circumference_angle_external(1,ODC,B)
arc_property_circumference_angle_external(1,ODC,A)
mirror_similar_triangle_judgment_aa(1,BDA,EBD)
mirror_similar_triangle_property_line_ratio(1,BDA,EBD)
mirror_similar_triangle_property_line_ratio(1,ABD,BDE)
Problem: As shown in the diagram, AB=6, EC=2, quadrilateral ABCD is a ▱. Find the ratio of the length of line DF to the length of line AF.
问题：如图所示，AB=6，EC=2，四边形ABCD是▱。求直线DF的长度与直线AF的长度之比。
Construction (CDL):
Shape(FD,DE,EF)
Shape(DA,AB,BE,ED)
Shape(EB,BC,CE)
Collinear(FEB)
Collinear(FDA)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(EC),2)
Parallelogram(ABCD)
Goal: Value(Div(LengthOfLine(DF),LengthOfLine(AF)))
Answer: 2/3
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
line_addition(1,DE,EC)
parallel_property_corresponding_angle(1,DC,AB,F)
similar_triangle_judgment_aa(1,BFA,EFD)
similar_triangle_property_line_ratio(1,BFA,EFD)
similar_triangle_property_line_ratio(1,FAB,FDE)
Problem: As shown in the diagram, Div(LengthOfLine(AD)=LengthOfLine(DB)), DE=2, DE is parallel to BC. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(AD)=LengthOfLine(DB))，DE=2，DE∥BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(Div(LengthOfLine(AD),LengthOfLine(DB)),1/2)
Equal(LengthOfLine(DE),2)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 6
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
parallel_property_corresponding_angle(2,CB,ED,A)
line_addition(1,AD,DB)
similar_triangle_judgment_aa(1,ABC,ADE)
similar_triangle_property_line_ratio(1,ABC,ADE)
similar_triangle_property_line_ratio(1,CAB,EAD)
Problem: As shown in the diagram, Div(LengthOfLine(DE)=LengthOfLine(EA)), CD=7, EF is parallel to AB, ABCD is a ▱. Find the length of line EF.
问题：如图所示，Div(LengthOfLine(DE)=LengthOfLine(EA))，CD=7，EF∥AB，BA和CD是平行四边形ABCD的一组对边。求直线EF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(Div(LengthOfLine(DE),LengthOfLine(EA)),3/4)
Equal(LengthOfLine(CD),7)
ParallelBetweenLine(EF,AB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(EF))
Answer: 3
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
line_addition(1,DE,EA)
parallel_property_corresponding_angle(1,EF,AB,D)
parallel_property_corresponding_angle(2,BA,FE,D)
similar_triangle_judgment_aa(1,DAB,DEF)
similar_triangle_property_line_ratio(1,DAB,DEF)
similar_triangle_property_line_ratio(1,BDA,FDE)
Problem: As shown in the diagram, the area of triangle AED is 1, Div(LengthOfLine(AD)=LengthOfLine(AB)), Div(LengthOfLine(AD)=LengthOfLine(AB)). Find the area of triangle ABC minus the area of triangle AED.
问题：如图所示，三角形AED的面积为1，Div(LengthOfLine(AD)=LengthOfLine(AB))，Div(LengthOfLine(AD)=LengthOfLine(AB))。求△ABC的面积减去三角形AED的面积。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EB,BC,CD,DE)
Collinear(AEB)
Collinear(ADC)
Conditions (CDL):
Equal(AreaOfTriangle(AED),1)
Equal(Div(LengthOfLine(AD),LengthOfLine(AB)),1/2)
Equal(Div(LengthOfLine(AD),LengthOfLine(AB)),Div(LengthOfLine(AE),LengthOfLine(AC)))
Goal: Value(Sub(AreaOfTriangle(ABC),AreaOfTriangle(AED)))
Answer: 3
Theorem sequence:
mirror_similar_triangle_judgment_sas(1,AED,ABC)
mirror_similar_triangle_property_line_ratio(1,EDA,CAB)
mirror_similar_triangle_property_area_square_ratio(1,AED,ABC)
Problem: As shown in the diagram, Div(LengthOfLine(AD)=LengthOfLine(BD)), DE=2, DE∥BC. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(AD)=LengthOfLine(BD))，DE=2，DE平行于BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(Div(LengthOfLine(AD),LengthOfLine(BD)),1/2)
Equal(LengthOfLine(DE),2)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 6
Theorem sequence:
line_addition(1,AD,DB)
parallel_property_corresponding_angle(1,DE,BC,A)
parallel_property_corresponding_angle(2,CB,ED,A)
similar_triangle_judgment_aa(1,ABC,ADE)
similar_triangle_property_line_ratio(1,ABC,ADE)
similar_triangle_property_line_ratio(1,CAB,EAD)
Problem: As shown in the diagram, AB=2, CD=5, AB∥DC. Find the ratio of the perimeter of △AEB to the perimeter of △EDC.
问题：如图所示，AB=2，CD=5，AB∥DC。求三角形AEB的周长与△EDC的周长之比。
Construction (CDL):
Shape(AE,EB,BA)
Shape(ED,DC,CE)
Collinear(AEC)
Collinear(BED)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(CD),5)
ParallelBetweenLine(AB,DC)
Goal: Value(Div(PerimeterOfTriangle(AEB),PerimeterOfTriangle(EDC)))
Answer: 2/5
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,DC)
parallel_property_alternate_interior_angle(2,AB,DC)
similar_triangle_judgment_aa(1,EDC,EBA)
similar_triangle_property_line_ratio(1,EDC,EBA)
similar_triangle_property_perimeter_ratio(1,EDC,EBA)