SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=5, AD=12, ∠AGD=90°, ABCD is a rectangle. Find the length of line AG.
问题：如图所示，AB=5，AD=12，∠AGD=90°，四边形ABCD是长方形。求直线AG的长度。
Construction (CDL):
Shape(AB,BG,GA)
Shape(OA,AG,GO)
Shape(AO,OD,DA)
Shape(OG,GB,BC,CO)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(DOGB)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(AD),12)
Equal(MeasureOfAngle(AGD),90)
Rectangle(ABCD)
Goal: Value(LengthOfLine(AG))
Answer: 60/13
Theorem sequence:
right_triangle_judgment_angle(1,DAB)
right_triangle_property_pythagorean(1,DAB)
mirror_similar_triangle_judgment_aa(1,AGD,BDA)
mirror_similar_triangle_property_line_ratio(1,GDA,ABD)
mirror_similar_triangle_property_line_ratio(1,DAG,DAB)
Problem: As shown in the diagram, AB=2, BC=3, ∠AEF=90°, ∠FEC=45°, AE bisects ∠DAB, ABCD is a rectangle. Find the length of line AF.
问题：如图所示，AB=2，BC=3，∠AEF=90°，∠FEC=45°，AE平分∠DAB，ABCD是长方形。求直线AF的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EF,FA)
Shape(AF,FD,DA)
Shape(FE,EC,CF)
Collinear(DFC)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(AEF),90)
Equal(MeasureOfAngle(FEC),45)
IsBisectorOfAngle(AE,DAB)
Rectangle(ABCD)
Goal: Value(LengthOfLine(AF))
Answer: sqrt(10)
Theorem sequence:
line_addition(1,DF,FC)
line_addition(1,BE,EC)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
triangle_property_angle_sum(1,FEC)
isosceles_triangle_judgment_angle_equal(1,CFE)
right_triangle_judgment_angle(1,ABE)
right_triangle_judgment_angle(1,AEF)
right_triangle_judgment_angle(1,FDA)
right_triangle_judgment_angle(1,ECF)
right_triangle_property_pythagorean(1,ABE)
right_triangle_property_pythagorean(1,AEF)
right_triangle_property_pythagorean(1,FDA)
right_triangle_property_pythagorean(1,ECF)
Problem: As shown in the diagram, AB=3, AD=4, ∠CEB=90°, AB is the altitude of triangle ABC, BE is the altitude of triangle BCA, ABCD is a rectangle. Find the length of line BE.
问题：如图所示，AB=3，AD=4，∠CEB=90°，AB是三角形ABC的高，BE是△BCA的高，ABCD是长方形。求直线BE的长度。
Construction (CDL):
Shape(EA,AB,BE)
Shape(EB,BC,CE)
Shape(DA,AE,EC,CD)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AD),4)
Equal(MeasureOfAngle(CEB),90)
IsAltitudeOfTriangle(AB,ABC)
IsAltitudeOfTriangle(BE,BCA)
Rectangle(ABCD)
Goal: Value(LengthOfLine(BE))
Answer: 12/5
Theorem sequence:
parallelogram_property_opposite_line_equal(1,BCDA)
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
mirror_similar_triangle_judgment_aa(1,BCE,ABC)
mirror_similar_triangle_property_line_ratio(1,CEB,CAB)
mirror_similar_triangle_property_line_ratio(1,EBC,BCA)
Problem: As shown in the diagram, AE=2, BE=EC, ∠EAB=30°, ABCD is a rectangle. Find the length of line AC.
问题：如图所示，AE=2，BE=EC，∠EAB=30°，四边形ABCD是长方形。求直线AC的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(AC,CD,DA)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AE),2)
Equal(LengthOfLine(BE),LengthOfLine(EC))
Equal(MeasureOfAngle(EAB),30)
Rectangle(ABCD)
Goal: Value(LengthOfLine(AC))
Answer: sqrt(7)
Theorem sequence:
triangle_property_angle_sum(1,ABE)
sine_theorem(1,ABE)
sine_theorem(1,EAB)
line_addition(1,BE,EC)
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
Problem: As shown in the diagram, ∠DAE=∠EAF, ∠FAB=60°, quadrilateral ABCD is a rectangle. Find the measure of ∠DAE.
问题：如图所示，∠DAE=∠EAF，∠FAB=60°，ABCD是长方形。求∠DAE的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EA,AF,FE)
Shape(EF,FC,CE)
Shape(FA,AB,BF)
Collinear(DEC)
Collinear(BFC)
Conditions (CDL):
Equal(MeasureOfAngle(DAE),MeasureOfAngle(EAF))
Equal(MeasureOfAngle(FAB),60)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(DAE))
Answer: 15
Theorem sequence:
angle_addition(1,DAE,EAF)
angle_addition(1,DAF,FAB)
Problem: As shown in the diagram, Add(LengthOfLine(GH)=LengthOfLine(GF)), GF and HE are opposite sides of the parallelogram GHEF, quadrilateral ABCD is a rectangle. Find the perimeter of quadrilateral GHEF.
问题：如图所示，Add(LengthOfLine(GH)=LengthOfLine(GF))，GF和HE是平行四边形GHEF的一组对边，四边形ABCD是长方形。求GHEF的周长。
Construction (CDL):
Shape(AH,HG,GA)
Shape(EH,HB,BE)
Shape(FE,EC,CF)
Shape(DG,GF,FD)
Shape(GH,HE,EF,FG)
Collinear(AHB)
Collinear(AGD)
Collinear(BEC)
Collinear(DFC)
Conditions (CDL):
Equal(Add(LengthOfLine(GH),LengthOfLine(GF)),2)
Parallelogram(GHEF)
Rectangle(ABCD)
Goal: Value(PerimeterOfQuadrilateral(GHEF))
Answer: 4
Theorem sequence:
parallelogram_property_opposite_line_equal(1,GHEF)
parallelogram_property_opposite_line_equal(1,HEFG)
quadrilateral_perimeter_formula(1,GHEF)
Problem: As shown in the diagram, AB=4, ∠AEO=90°, ∠COB=120°, quadrilateral ABCD is a rectangle. Find the length of line AE.
问题：如图所示，AB=4，∠AEO=90°，∠COB=120°，四边形ABCD是矩形。求直线AE的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EO,OA)
Shape(OE,EB,BC,CO)
Shape(AO,OD,DA)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(BEOD)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(MeasureOfAngle(AEO),90)
Equal(MeasureOfAngle(COB),120)
Rectangle(ABCD)
Goal: Value(LengthOfLine(AE))
Answer: 2*sqrt(3)
Theorem sequence:
flat_angle(1,BEO)
angle_addition(1,ABE,EBC)
angle_addition(1,BEA,AEO)
angle_addition(1,CDO,ODA)
rectangle_property_diagonal_equal(1,ABCD)
triangle_property_angle_sum(1,OBC)
triangle_property_angle_sum(1,BCD)
sine_theorem(1,ABE)
sine_theorem(1,BDA)
sine_theorem(1,ABC)
Problem: As shown in the diagram, AB=8, AD=6, EO=OF, EO perpendicular bisects DB, FO is the perpendicular bisector of BD, DABC is a rectangle. Find the length of line EF.
问题：如图所示，AB=8，AD=6，EO=OF，EO垂直平分AB，FO是BD的垂直平分线，DABC是矩形。求直线EF的长度。
Construction (CDL):
Shape(ED,DO,OE)
Shape(OF,FB,BO)
Shape(OD,DA,AF,FO)
Shape(EO,OB,BC,CE)
Collinear(DEC)
Collinear(AFB)
Collinear(DOB)
Collinear(EOF)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AD),6)
Equal(LengthOfLine(EO),LengthOfLine(OF))
IsPerpendicularBisectorOfLine(EO,DB)
IsPerpendicularBisectorOfLine(FO,BD)
Rectangle(DABC)
Goal: Value(LengthOfLine(EF))
Answer: 15/2
Theorem sequence:
right_triangle_judgment_angle(1,DAB)
right_triangle_property_pythagorean(1,DAB)
line_addition(1,DO,OB)
line_addition(1,EO,OF)
mirror_similar_triangle_judgment_aa(1,FBO,DAB)
mirror_similar_triangle_property_line_ratio(1,FBO,DAB)
mirror_similar_triangle_property_line_ratio(1,BOF,BDA)
Problem: As shown in the diagram, AC=2, BC=1, quadrilateral ABCD is a rectangle. Find the area of quadrilateral DABC.
问题：如图所示，AC=2，BC=1，四边形ABCD是矩形。求四边形DABC的面积。
Construction (CDL):
Shape(CD,DA,AC)
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(BC),1)
Rectangle(ABCD)
Goal: Value(AreaOfQuadrilateral(DABC))
Answer: sqrt(3)
Theorem sequence:
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
parallelogram_area_formula_sine(1,ABCD)
Problem: As shown in the diagram, ∠AOD=130°, AC∥DE, CA∥ED, quadrilateral ABCD is a rectangle. Find the measure of ∠CED.
问题：如图所示，∠AOD=130°，AC平行于DE，CA平行于ED，ABCD是矩形。求∠CED的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Shape(DC,CE,ED)
Collinear(AOC)
Collinear(BOD)
Collinear(BCE)
Conditions (CDL):
Equal(MeasureOfAngle(AOD),130)
ParallelBetweenLine(AC,DE)
ParallelBetweenLine(CA,ED)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(CED))
Answer: 25
Theorem sequence:
line_addition(1,AO,OC)
line_addition(1,BO,OD)
vertical_angle(1,AOD,COB)
rectangle_property_diagonal_equal(1,ABCD)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,BCO)
parallel_property_corresponding_angle(1,CA,ED,B)
Problem: As shown in the diagram, the height of quadrilateral DEBF is 12, AB=7, AE=5, BC=12, EB=2, quadrilateral DEBF is a ▱, quadrilateral ABCD is a rectangle. Find the area of quadrilateral DEBF.
问题：如图所示，DEBF的高为12，AB=7，AE=5，BC=12，EB=2，DF和EB是平行四边形DEBF的一组对边，ABCD是矩形。求DEBF的面积。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EB,BF,FD,DE)
Shape(FB,BC,CF)
Collinear(AEB)
Collinear(DFC)
Conditions (CDL):
Equal(HeightOfQuadrilateral(DEBF),12)
Equal(LengthOfLine(AB),7)
Equal(LengthOfLine(AE),5)
Equal(LengthOfLine(BC),12)
Equal(LengthOfLine(EB),2)
Parallelogram(DEBF)
Rectangle(ABCD)
Goal: Value(AreaOfQuadrilateral(DEBF))
Answer: 24
Theorem sequence:
parallelogram_area_formula_common(1,DEBF)
Problem: As shown in the diagram, AC=6, BD=8, ABCD is a rhombus. Find the length of line AB.
问题：如图所示，AC=6，BD=8，四边形ABCD是菱形。求直线AB的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(AO,OD,DA)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(BD),8)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 5
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
Problem: As shown in the diagram, EF=2, ∠FAE=60°, E bisects segment AB, F bisects segment AD, quadrilateral ABCD is a rhombus. Find the perimeter of quadrilateral ABCD.
问题：如图所示，EF=2，∠FAE=60°，E平分线段AB，F平分线段AD，ABCD是菱形。求ABCD的周长。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BD,DF,FE)
Shape(BC,CD,DB)
Collinear(AEB)
Collinear(AFD)
Conditions (CDL):
Equal(LengthOfLine(EF),2)
Equal(MeasureOfAngle(FAE),60)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,AD)
Rhombus(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 16
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,EF,ABD)
midsegment_of_triangle_property_length(1,EF,ABD)
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
triangle_property_angle_sum(1,ABD)
isosceles_triangle_judgment_angle_equal(1,BDA)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, AC=3, ∠DAB=120°, ABCD is a rhombus. Find the perimeter of quadrilateral ABCD.
问题：如图所示，AC=3，∠DAB=120°，四边形ABCD是菱形。求ABCD的周长。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(MeasureOfAngle(DAB),120)
Rhombus(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 12
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,BC)
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
angle_addition(1,DAC,CAB)
triangle_property_angle_sum(1,ABC)
isosceles_triangle_judgment_angle_equal(1,CAB)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, AC=8, BD=6, quadrilateral ABCD is a rhombus. Find the length of line AB.
问题：如图所示，AC=8，BD=6，四边形ABCD是菱形。求直线AB的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(DO,OC,CD)
Shape(OB,BC,CO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(BD),6)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 5
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
kite_property_diagonal_perpendicular_bisection(1,DABC,O)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
Problem: As shown in the diagram, AB=6, BC=4, ∠BOA=90°, E is the midpoint of segment BA, quadrilateral ADCB is a rectangle. Find the length of line OC.
问题：如图所示，AB=6，BC=4，∠BOA=90°，E是线段BA的中点，ADCB是长方形。求直线OC的长度。
Construction (CDL):
Shape(BO,OE,EB)
Shape(EO,OA,AE)
Shape(CB,BE,EC)
Shape(EA,AD,DC,CE)
Collinear(OEC)
Collinear(BEA)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(BC),4)
Equal(MeasureOfAngle(BOA),90)
IsMidpointOfLine(E,BA)
Rectangle(ADCB)
Goal: Value(LengthOfLine(OC))
Answer: 8
Theorem sequence:
line_addition(1,BE,EA)
right_triangle_judgment_angle(1,BOA)
median_of_triangle_judgment(1,OE,OAB)
right_triangle_property_length_of_median(1,BOA,E)
right_triangle_judgment_angle(1,CBE)
right_triangle_property_pythagorean(1,CBE)
line_addition(1,OE,EC)
Problem: As shown in the diagram, AB=5, BC=5, CD=5, CF=4, DA=5, BE⊥CE, CF is perpendicular to DF. Find the length of line CE.
问题：如图所示，AB=5，BC=5，CD=5，CF=4，DA=5，BE垂直于CE，CF⊥DF。求直线CE的长度。
Construction (CDL):
Shape(FD,DC,CF)
Shape(DA,AC,CD)
Shape(AB,BC,CA)
Shape(CB,BE,EC)
Collinear(ADF)
Collinear(ABE)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(CD),5)
Equal(LengthOfLine(CF),4)
Equal(LengthOfLine(DA),5)
PerpendicularBetweenLine(BE,CE)
PerpendicularBetweenLine(CF,DF)
Goal: Value(LengthOfLine(CE))
Answer: 4
Theorem sequence:
parallelogram_judgment_equal_and_equal(1,ABCD)
parallel_property_alternate_interior_angle(1,AD,BC)
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
mirror_congruent_triangle_judgment_aas(1,CAE,CFA)
mirror_congruent_triangle_property_line_equal(1,AEC,ACF)
Problem: As shown in the diagram, AC=6, BD=8, BE=EC, ABCD is a rhombus. Find the length of line OE.
问题：如图所示，AC=6，BD=8，BE=EC，四边形ABCD是菱形。求直线OE的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BE,EO)
Shape(EC,CO,OE)
Shape(OC,CD,DO)
Collinear(BOD)
Collinear(AOC)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(BD),8)
Equal(LengthOfLine(BE),LengthOfLine(EC))
Rhombus(ABCD)
Goal: Value(LengthOfLine(OE))
Answer: 5/2
Theorem sequence:
parallelogram_property_diagonal_bisection(1,DABC,O)
kite_property_diagonal_perpendicular_bisection(1,DABC,O)
line_addition(1,BO,OD)
line_addition(1,AO,OC)
right_triangle_judgment_angle(1,DOC)
right_triangle_property_pythagorean(1,DOC)
midsegment_of_triangle_judgment_midpoint(1,EO,BCD)
midsegment_of_triangle_property_length(1,EO,BCD)
Problem: As shown in the diagram, the perimeter of quadrilateral ABCD is 24, E bisects segment AD, ABCD is a rhombus. Find the length of line OE.
问题：如图所示，ABCD的周长为24，E是线段AD的中点，ABCD是菱形。求直线OE的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OE,EA)
Shape(EO,OD,DE)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(AED)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),24)
IsMidpointOfLine(E,AD)
Rhombus(ABCD)
Goal: Value(LengthOfLine(OE))
Answer: 3
Theorem sequence:
quadrilateral_perimeter_formula(1,ABCD)
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
midsegment_of_triangle_judgment_midpoint(1,EO,DAB)
midsegment_of_triangle_property_length(1,EO,DAB)
Problem: As shown in the diagram, ∠OAB=50°, ABCD is a rhombus. Find the measure of ∠ABC.
问题：如图所示，∠OAB=50°，四边形ABCD是菱形。求∠ABC的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(BC,CO,OB)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),50)
Rhombus(ABCD)
Goal: Value(MeasureOfAngle(ABC))
Answer: 80
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,BCA)
Problem: As shown in the diagram, ∠CDA=130°, AE⊥OE, ABCD is a rhombus. Find the measure of ∠EOA.
问题：如图所示，∠CDA=130°，AE⊥OE，ABCD是菱形。求∠EOA的大小。
Construction (CDL):
Shape(DA,AO,OD)
Shape(DO,OC,CD)
Shape(AE,EO,OA)
Shape(OE,EB,BO)
Shape(OB,BC,CO)
Collinear(AOC)
Collinear(DOB)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),130)
PerpendicularBetweenLine(AE,OE)
Rhombus(ABCD)
Goal: Value(MeasureOfAngle(EOA))
Answer: 65
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,DC,AB)
parallel_property_alternate_interior_angle(2,DC,AB)
isosceles_triangle_judgment_line_equal(1,DAC)
isosceles_triangle_property_angle_equal(1,DAC)
angle_addition(1,DAO,OAE)
triangle_property_angle_sum(1,AEO)
Problem: As shown in the diagram, AB=5, ∠BCD=120°, ABCD is a rhombus. Find the perimeter of triangle ABC.
问题：如图所示，AB=5，∠BCD=120°，ABCD是菱形。求三角形ABC的周长。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(MeasureOfAngle(BCD),120)
Rhombus(ABCD)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 15
Theorem sequence:
parallel_property_alternate_interior_angle(2,BA,CD)
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
angle_addition(1,BCA,ACD)
triangle_property_angle_sum(1,BCA)
isosceles_triangle_judgment_angle_equal(1,ABC)
triangle_perimeter_formula(1,ABC)
Problem: As shown in the diagram, OA=4, OB=3, quadrilateral ABCD is a rhombus. Find the perimeter of quadrilateral ABCD.
问题：如图所示，OA=4，OB=3，ABCD是菱形。求ABCD的周长。
Construction (CDL):
Shape(DA,AO,OD)
Shape(DO,OC,CD)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Collinear(AOC)
Collinear(DOB)
Conditions (CDL):
Equal(LengthOfLine(OA),4)
Equal(LengthOfLine(OB),3)
Rhombus(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 20
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,BCDA,O)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, BE=AF, ∠EBC=60°, quadrilateral ABCD is a rhombus. Find the sum of the measure of ∠AEC and the measure of ∠CFA.
问题：如图所示，BE=AF，∠EBC=60°，四边形ABCD是菱形。求∠AEC的大小与∠CFA的大小之和。
Construction (CDL):
Shape(AE,EC,CA)
Shape(AC,CF,FA)
Shape(EB,BC,CE)
Shape(FC,CD,DF)
Collinear(BEA)
Collinear(AFD)
Conditions (CDL):
Equal(LengthOfLine(BE),LengthOfLine(AF))
Equal(MeasureOfAngle(EBC),60)
Rhombus(ABCD)
Goal: Value(Add(MeasureOfAngle(AEC),MeasureOfAngle(CFA)))
Answer: 180
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,ABC)
parallelogram_property_opposite_angle_equal(1,DABC)
isosceles_triangle_judgment_line_equal(1,DAC)
isosceles_triangle_property_angle_equal(1,DAC)
triangle_property_angle_sum(1,DAC)
isosceles_triangle_judgment_angle_equal(1,CDA)
line_addition(1,BE,EA)
line_addition(1,AF,FD)
congruent_triangle_judgment_sas(1,AEC,DFC)
congruent_triangle_property_angle_equal(1,CAE,CDF)
angle_addition(1,ACF,FCD)
angle_addition(1,ECA,ACF)
angle_addition(1,FAC,CAE)
quadrilateral_property_angle_sum(1,AECF)
Problem: As shown in the diagram, AD=6, EO is parallel to CD, quadrilateral ABCD is a rhombus. Find the length of line OE.
问题：如图所示，AD=6，EO平行于CD，ABCD是菱形。求直线OE的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BE,EO)
Shape(OE,EC,CO)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(BOD)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AD),6)
ParallelBetweenLine(EO,CD)
Rhombus(ABCD)
Goal: Value(LengthOfLine(OE))
Answer: 3
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
midsegment_of_triangle_judgment_parallel(2,EO,BCD)
midsegment_of_triangle_property_length(1,EO,BCD)
Problem: As shown in the diagram, BC=10, ∠DAB=120°, quadrilateral ABCD is a rhombus. Find the length of line AC.
问题：如图所示，BC=10，∠DAB=120°，ABCD是菱形。求直线AC的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(LengthOfLine(BC),10)
Equal(MeasureOfAngle(DAB),120)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AC))
Answer: 10
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,BC)
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
angle_addition(1,DAC,CAB)
triangle_property_angle_sum(1,BCA)
isosceles_triangle_judgment_angle_equal(1,ABC)
Problem: As shown in the diagram, PF=3, AE is perpendicular to PE, PF is perpendicular to AF, ABCD is a rhombus. Find the length of line PE.
问题：如图所示，PF=3，AE垂直于PE，PF垂直于AF，ABCD是菱形。求直线PE的长度。
Construction (CDL):
Shape(FA,AP,PF)
Shape(AE,EP,PA)
Shape(FP,PC,CD,DF)
Shape(PE,EB,BC,CP)
Collinear(AFD)
Collinear(APC)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(PF),3)
PerpendicularBetweenLine(AE,PE)
PerpendicularBetweenLine(PF,AF)
Rhombus(ABCD)
Goal: Value(LengthOfLine(PE))
Answer: 3
Theorem sequence:
parallel_property_alternate_interior_angle(2,DC,AB)
isosceles_triangle_judgment_line_equal(1,DAC)
isosceles_triangle_property_angle_equal(1,DAC)
bisector_of_angle_judgment_angle_equal(1,AP,FAE)
bisector_of_angle_property_distance_equal(1,AP,FAE)
Problem: As shown in the diagram, AB=1, ∠DAB=60°, quadrilateral ABCD is a rhombus. Find the length of line BD.
问题：如图所示，AB=1，∠DAB=60°，ABCD是菱形。求直线BD的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(BC,CD,DB)
Conditions (CDL):
Equal(LengthOfLine(AB),1)
Equal(MeasureOfAngle(DAB),60)
Rhombus(ABCD)
Goal: Value(LengthOfLine(BD))
Answer: 1
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
triangle_property_angle_sum(1,ABD)
isosceles_triangle_judgment_angle_equal(1,BDA)
Problem: As shown in the diagram, AD=2, ∠ABC=120°, ABCD is a rhombus. Find the length of line AC.
问题：如图所示，AD=2，∠ABC=120°，四边形ABCD是菱形。求直线AC的长度。
Construction (CDL):
Shape(DA,AO,OD)
Shape(DO,OC,CD)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Collinear(AOC)
Collinear(DOB)
Conditions (CDL):
Equal(LengthOfLine(AD),2)
Equal(MeasureOfAngle(ABC),120)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AC))
Answer: 2*sqrt(3)
Theorem sequence:
parallel_property_alternate_interior_angle(2,AD,BC)
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
angle_addition(1,ABD,DBC)
sine_theorem(1,ABO)
kite_property_diagonal_perpendicular_bisection(1,BCDA,O)
line_addition(1,AO,OC)
Problem: As shown in the diagram, AB=17, AE=25, BD=16, quadrilateral ABCD is a rhombus. Find the length of line DE.
问题：如图所示，AB=17，AE=25，BD=16，四边形ABCD是菱形。求直线DE的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(BC,CO,OB)
Shape(OC,CD,DO)
Shape(AD,DE,EA)
Collinear(AOC)
Collinear(BODE)
Conditions (CDL):
Equal(LengthOfLine(AB),17)
Equal(LengthOfLine(AE),25)
Equal(LengthOfLine(BD),16)
Rhombus(ABCD)
Goal: Value(LengthOfLine(DE))
Answer: 12
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
line_addition(1,BO,OD)
right_triangle_judgment_angle(1,AOD)
right_triangle_property_pythagorean(1,AOD)
right_triangle_judgment_angle(1,AOE)
right_triangle_property_pythagorean(1,AOE)
line_addition(1,OD,DE)
Problem: As shown in the diagram, ∠BAD=60°, the perimeter of quadrilateral ADCB is 16, quadrilateral ADCB is a rhombus. Find the length of line BD.
问题：如图所示，∠BAD=60°，四边形ADCB的周长为16，ADCB是菱形。求直线BD的长度。
Construction (CDL):
Shape(AD,DB,BA)
Shape(BD,DC,CB)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),60)
Equal(PerimeterOfQuadrilateral(ADCB),16)
Rhombus(ADCB)
Goal: Value(LengthOfLine(BD))
Answer: 4
Theorem sequence:
quadrilateral_perimeter_formula(1,ADCB)
isosceles_triangle_judgment_line_equal(1,ADB)
isosceles_triangle_property_angle_equal(1,ADB)
triangle_property_angle_sum(1,ADB)
isosceles_triangle_judgment_angle_equal(1,BAD)
Problem: As shown in the diagram, the perimeter of ABCD is 8, E bisects segment AB, ABCD is a rhombus. Find the length of line OE.
问题：如图所示，ABCD的周长为8，E是线段AB的中点，ABCD是菱形。求直线OE的长度。
Construction (CDL):
Shape(DA,AO,OD)
Shape(DO,OC,CD)
Shape(AE,EO,OA)
Shape(OE,EB,BO)
Shape(OB,BC,CO)
Collinear(AOC)
Collinear(DOB)
Collinear(AEB)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),8)
IsMidpointOfLine(E,AB)
Rhombus(ABCD)
Goal: Value(LengthOfLine(OE))
Answer: 1
Theorem sequence:
quadrilateral_perimeter_formula(1,ABCD)
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
right_triangle_judgment_angle(1,BOA)
median_of_triangle_judgment(1,OE,OAB)
right_triangle_property_length_of_median(1,BOA,E)
Problem: As shown in the diagram, BE=1, BF=1, ∠BCD=60°, AD is parallel to BC, DC∥AB, BE is perpendicular to AE, CF⊥BF. Find the area of quadrilateral ABCD.
问题：如图所示，BE=1，BF=1，∠BCD=60°，AD平行于BC，DC平行于AB，BE⊥AE，CF垂直于BF。求四边形ABCD的面积。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BF,FD,DE)
Shape(BC,CF,FB)
Collinear(AED)
Collinear(DFC)
Conditions (CDL):
Equal(LengthOfLine(BE),1)
Equal(LengthOfLine(BF),1)
Equal(MeasureOfAngle(BCD),60)
ParallelBetweenLine(AD,BC)
ParallelBetweenLine(DC,AB)
PerpendicularBetweenLine(BE,AE)
PerpendicularBetweenLine(CF,BF)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 2*sqrt(3)/3
Theorem sequence:
parallelogram_judgment_parallel_and_parallel(1,ABCD)
parallelogram_property_opposite_angle_equal(1,ABCD)
sine_theorem(1,BEA)
sine_theorem(1,BCF)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
kite_judgment_equal_and_equal(1,ABCD)
parallel_property_ipsilateral_internal_angle(1,BA,CD)
kite_area_formula_sine(1,ABCD)
Problem: As shown in the diagram, ∠DAC=25°, ABCD is a rhombus. Find the measure of ∠ABC.
问题：如图所示，∠DAC=25°，四边形ABCD是菱形。求∠ABC的大小。
Construction (CDL):
Shape(DA,AC,CD)
Shape(AB,BC,CA)
Conditions (CDL):
Equal(MeasureOfAngle(DAC),25)
Rhombus(ABCD)
Goal: Value(MeasureOfAngle(ABC))
Answer: 130
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,BC)
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,BCA)