SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AC=26, BD=10, quadrilateral ABCD is a rhombus. Find the area of quadrilateral ABCD.
问题：如图所示，AC=26，BD=10，ABCD是菱形。求四边形ABCD的面积。
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(LengthOfLine(AC),26)
Equal(LengthOfLine(BD),10)
Rhombus(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 130
Theorem sequence:
kite_area_formula_diagonal(1,ABCD)
Problem: As shown in the diagram, AB=6, ∠DAB=60°, ABCD is a rhombus. Find the length of line BD.
问题：如图所示，AB=6，∠DAB=60°，ABCD是菱形。求直线BD的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(MeasureOfAngle(DAB),60)
Rhombus(ABCD)
Goal: Value(LengthOfLine(BD))
Answer: 6
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, ABCD is congruent to OGHI, the perimeter of quadrilateral ABCD is 4, the perimeter of quadrilateral OGHI is 4, C is the midpoint of segment OH, O is the midpoint of segment AC, AD∥OI, BA is parallel to GO, BC∥GH, CD∥HI, ABCD is a rhombus, OGHI is a rhombus. Find the perimeter of quadrilateral OECF.
问题：如图所示，四边形ABCD与四边形OGHI全等，ABCD的周长为4，OGHI的周长为4，C是线段OH的中点，O是线段AC的中点，AD∥OI，BA平行于GO，BC平行于GH，CD∥HI，四边形ABCD是菱形，OGHI是菱形。求OECF的周长。
Construction (CDL):
Shape(DA,AO,OF,FD)
Shape(AB,BE,EO,OA)
Shape(FO,OC,CF)
Shape(OE,EC,CO)
Shape(FC,CH,HI,IF)
Shape(CE,EG,GH,HC)
Collinear(AOCH)
Collinear(DFC)
Collinear(OFI)
Collinear(OEG)
Collinear(BEC)
Conditions (CDL):
CongruentBetweenQuadrilateral(ABCD,OGHI)
Equal(PerimeterOfQuadrilateral(ABCD),4)
Equal(PerimeterOfQuadrilateral(OGHI),4)
IsMidpointOfLine(C,OH)
IsMidpointOfLine(O,AC)
ParallelBetweenLine(AD,OI)
ParallelBetweenLine(BA,GO)
ParallelBetweenLine(BC,GH)
ParallelBetweenLine(CD,HI)
Rhombus(ABCD)
Rhombus(OGHI)
Goal: Value(PerimeterOfQuadrilateral(OECF))
Answer: 2
Theorem sequence:
quadrilateral_perimeter_formula(1,ABCD)
quadrilateral_perimeter_formula(1,OGHI)
parallel_property_collinear_extend(3,IO,DA,F)
midsegment_of_triangle_judgment_parallel(2,FO,CDA)
midsegment_of_triangle_property_length(1,FO,CDA)
parallel_property_collinear_extend(3,OG,AB,E)
midsegment_of_triangle_judgment_parallel(1,OE,CAB)
midsegment_of_triangle_property_length(1,OE,CAB)
parallel_property_collinear_extend(3,CD,HI,F)
midsegment_of_triangle_judgment_parallel(1,CF,OHI)
midsegment_of_triangle_property_length(1,CF,OHI)
parallel_property_collinear_extend(3,BC,GH,E)
midsegment_of_triangle_judgment_parallel(2,EC,OGH)
midsegment_of_triangle_property_length(1,EC,OGH)
quadrilateral_perimeter_formula(1,OECF)
Problem: As shown in the diagram, AB=5, ∠ABC=60°, ABCD is a rhombus. Find the length of line AC.
问题：如图所示，AB=5，∠ABC=60°，四边形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(AB),5)
Equal(MeasureOfAngle(ABC),60)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AC))
Answer: 5
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,BCA)
isosceles_triangle_judgment_angle_equal(1,ABC)
Problem: As shown in the diagram, EF=4, E is the midpoint of segment AB, F is the midpoint of segment AC, ABCD is a rhombus. Find the length of line AD.
问题：如图所示，EF=4，E平分线段AB，F平分线段AC，ABCD是菱形。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(EF),4)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,AC)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AD))
Answer: 8
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,EF,ABC)
midsegment_of_triangle_property_length(1,EF,ABC)
Problem: As shown in the diagram, AO=3, BO=4, quadrilateral ABCD is a rhombus. Find the area of ABCD.
问题：如图所示，AO=3，BO=4，ABCD是菱形。求四边形ABCD的面积。
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(AO),3)
Equal(LengthOfLine(BO),4)
Rhombus(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 24
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
kite_property_diagonal_perpendicular_bisection(1,BCDA,O)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
kite_area_formula_diagonal(1,ABCD)
Problem: As shown in the diagram, AB=15, ∠CDA=120°, quadrilateral ABCD is a rhombus. Find the length of line BD.
问题：如图所示，AB=15，∠CDA=120°，ABCD是菱形。求直线BD的长度。
Construction (CDL):
Shape(DA,AB,BD)
Shape(DB,BC,CD)
Conditions (CDL):
Equal(LengthOfLine(AB),15)
Equal(MeasureOfAngle(CDA),120)
Rhombus(ABCD)
Goal: Value(LengthOfLine(BD))
Answer: 15
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,DC,AB)
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, OE=3, E is the midpoint of segment AB, ABCD is a rhombus. Find the perimeter of ABCD.
问题：如图所示，OE=3，E平分线段AB，ABCD是菱形。求ABCD的周长。
Construction (CDL):
Shape(AE,EO,OA)
Shape(EB,BO,OE)
Shape(AO,OD,DA)
Shape(BC,CO,OB)
Shape(OC,CD,DO)
Collinear(AEB)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(OE),3)
IsMidpointOfLine(E,AB)
Rhombus(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 24
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
midsegment_of_triangle_judgment_midpoint(1,OE,BDA)
midsegment_of_triangle_property_length(1,OE,BDA)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, the perimeter of quadrilateral ABCD is 32, E bisects segment AB, F is the midpoint of segment AC, quadrilateral ABCD is a rhombus. Find the length of line EF.
问题：如图所示，ABCD的周长为32，E是线段AB的中点，F是线段AC的中点，四边形ABCD是菱形。求直线EF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(PerimeterOfQuadrilateral(ABCD),32)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,AC)
Rhombus(ABCD)
Goal: Value(LengthOfLine(EF))
Answer: 4
Theorem sequence:
quadrilateral_perimeter_formula(1,ABCD)
midsegment_of_triangle_judgment_midpoint(1,EF,ABC)
midsegment_of_triangle_property_length(1,EF,ABC)
Problem: As shown in the diagram, AC=12, BD=16, quadrilateral ABCD is a rhombus. Find the length of line AB.
问题：如图所示，AC=12，BD=16，四边形ABCD是菱形。求直线AB的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(BC,CO,OB)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BD),16)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 10
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
line_addition(1,BO,OD)
kite_property_diagonal_perpendicular_bisection(1,BCDA,O)
line_addition(1,AO,OC)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
Problem: As shown in the diagram, AC=4, ∠DAB=120°, ABCD is a rhombus. Find the perimeter of quadrilateral ABCD.
问题：如图所示，AC=4，∠DAB=120°，四边形ABCD是菱形。求四边形ABCD的周长。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(MeasureOfAngle(DAB),120)
Rhombus(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 16
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,BCA)
isosceles_triangle_judgment_angle_equal(1,CAB)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, EF=4, E is the midpoint of 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=4，E平分线段AB，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),4)
IsMidpointOfLine(E,AB)
ParallelBetweenLine(EF,BC)
Rhombus(ABCD)
Goal: Value(LengthOfLine(CD))
Answer: 8
Theorem sequence:
midsegment_of_triangle_judgment_parallel(1,EF,ABC)
midsegment_of_triangle_property_length(1,EF,ABC)
Problem: As shown in the diagram, PQ=3, P is the midpoint of segment AD, Q bisects segment AC, ADCB is a rhombus. Find the perimeter of quadrilateral ADCB.
问题：如图所示，PQ=3，P是线段AD的中点，Q平分线段AC，四边形ADCB是菱形。求四边形ADCB的周长。
Construction (CDL):
Shape(AP,PQ,QA)
Shape(AQ,QC,CB,BA)
Shape(PD,DC,CQ,QP)
Collinear(APD)
Collinear(AQC)
Conditions (CDL):
Equal(LengthOfLine(PQ),3)
IsMidpointOfLine(P,AD)
IsMidpointOfLine(Q,AC)
Rhombus(ADCB)
Goal: Value(PerimeterOfQuadrilateral(ADCB))
Answer: 24
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,PQ,ADC)
midsegment_of_triangle_property_length(1,PQ,ADC)
quadrilateral_perimeter_formula(1,ADCB)
Problem: As shown in the diagram, AC=8, BD=6, ABCD is mirror congruent to EBDA, ABCD is a rhombus, AEBD is a rhombus. Find the area of quadrilateral AEBD.
问题：如图所示，AC=8，BD=6，四边形ABCD镜像全等于四边形EBDA，ABCD是菱形，AEBD是菱形。求四边形AEBD的面积。
Construction (CDL):
Shape(AE,EB,BA)
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(DO,OC,CD)
Shape(OB,BC,CO)
Collinear(AOC)
Collinear(BOD)
Collinear(EBC)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(BD),6)
MirrorCongruentBetweenQuadrilateral(ABCD,EBDA)
Rhombus(ABCD)
Rhombus(AEBD)
Goal: Value(AreaOfQuadrilateral(AEBD))
Answer: 24
Theorem sequence:
kite_area_formula_diagonal(1,ABCD)
mirror_congruent_quadrilateral_property_area_equal(1,ABCD,EBDA)
Problem: As shown in the diagram, ∠DAB=80°, FE is the perpendicular bisector of AB, quadrilateral ABCD is a rhombus. Find the measure of ∠CDF.
问题：如图所示，∠DAB=80°，FE是AB的垂直平分线，四边形ABCD是菱形。求∠CDF的大小。
Construction (CDL):
Shape(DA,AF,FD)
Shape(DF,FG,GD)
Shape(DG,GC,CD)
Shape(FA,AE,EF)
Shape(FE,EB,BF)
Shape(FB,BG,GF)
Shape(GB,BC,CG)
Collinear(AFGC)
Collinear(AEB)
Collinear(DGB)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),80)
IsPerpendicularBisectorOfLine(FE,AB)
Rhombus(ABCD)
Goal: Value(MeasureOfAngle(CDF))
Answer: 60
Theorem sequence:
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,DAF,FAE)
perpendicular_bisector_property_distance_equal(1,FE,AB)
kite_property_diagonal_perpendicular_bisection(1,ABCD,G)
perpendicular_bisector_judgment_per_and_mid(1,FG,BD)
perpendicular_bisector_property_distance_equal(1,FG,BD)
isosceles_triangle_judgment_line_equal(1,FDA)
isosceles_triangle_property_angle_equal(1,FDA)
parallel_property_ipsilateral_internal_angle(1,DC,AB)
angle_addition(1,CDF,FDA)
Problem: As shown in the diagram, AB=2, ∠ABE=60°, E is the midpoint of segment BC, F bisects segment CD, quadrilateral ABCD is a rhombus. Find the perimeter of △AEF.
问题：如图所示，AB=2，∠ABE=60°，E平分线段BC，F是线段CD的中点，四边形ABCD是菱形。求△AEF的周长。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EG,GA)
Shape(AG,GF,FA)
Shape(AF,FD,DA)
Shape(EC,CG,GE)
Shape(GC,CF,FG)
Collinear(AGC)
Collinear(EGF)
Collinear(BEC)
Collinear(CFD)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(MeasureOfAngle(ABE),60)
IsMidpointOfLine(E,BC)
IsMidpointOfLine(F,CD)
Rhombus(ABCD)
Goal: Value(PerimeterOfTriangle(AEF))
Answer: 3*sqrt(3)
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
line_addition(1,BE,EC)
line_addition(1,CF,FD)
mirror_congruent_triangle_judgment_sas(1,BEA,DAF)
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,BCA)
isosceles_triangle_judgment_angle_equal(1,ABC)
median_of_triangle_judgment(1,AE,ABC)
isosceles_triangle_property_line_coincidence(2,ABC,E)
right_triangle_judgment_angle(1,BEA)
right_triangle_property_pythagorean(1,BEA)
triangle_property_angle_sum(1,ABE)
parallel_property_ipsilateral_internal_angle(1,AD,BC)
mirror_congruent_triangle_property_angle_equal(1,ABE,AFD)
angle_addition(1,DAE,EAB)
angle_addition(1,DAF,FAE)
mirror_congruent_triangle_property_line_equal(1,BEA,DAF)
cosine_theorem(1,AEF)
triangle_perimeter_formula(1,AEF)
Problem: As shown in the diagram, ∠DAB=110°, FE is the perpendicular bisector of AB, quadrilateral ABCD is a rhombus. Find the measure of ∠CDF.
问题：如图所示，∠DAB=110°，FE是AB的垂直平分线，四边形ABCD是菱形。求∠CDF的大小。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BF,FE)
Shape(FB,BC,CF)
Shape(AF,FD,DA)
Shape(FC,CD,DF)
Collinear(AEB)
Collinear(AFC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),110)
IsPerpendicularBisectorOfLine(FE,AB)
Rhombus(ABCD)
Goal: Value(MeasureOfAngle(CDF))
Answer: 15
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,DAF,FAB)
perpendicular_bisector_property_distance_equal(1,FE,AB)
isosceles_triangle_judgment_line_equal(1,FAB)
isosceles_triangle_property_angle_equal(1,FAB)
mirror_congruent_triangle_judgment_sas(1,ABF,AFD)
mirror_congruent_triangle_property_angle_equal(1,BFA,DAF)
parallel_property_ipsilateral_internal_angle(1,DC,AB)
angle_addition(1,CDF,FDA)
Problem: As shown in the diagram, OE=3, E is the midpoint of segment AD, ABCD is a rhombus. Find the length of line AD.
问题：如图所示，OE=3，E是线段AD的中点，ABCD是菱形。求直线AD的长度。
Construction (CDL):
Shape(DE,EO,OD)
Shape(EA,AO,OE)
Shape(DO,OC,CD)
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Collinear(AED)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(OE),3)
IsMidpointOfLine(E,AD)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AD))
Answer: 6
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
right_triangle_judgment_angle(1,AOD)
median_of_triangle_judgment(1,OE,ODA)
right_triangle_property_length_of_median(1,AOD,E)
Problem: As shown in the diagram, EF=4, E is the midpoint of segment AB, EF∥BC, quadrilateral ABCD is a rhombus. Find the perimeter of quadrilateral ABCD.
问题：如图所示，EF=4，E是线段AB的中点，EF∥BC，四边形ABCD是菱形。求ABCD的周长。
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),4)
IsMidpointOfLine(E,AB)
ParallelBetweenLine(EF,BC)
Rhombus(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 32
Theorem sequence:
midsegment_of_triangle_judgment_parallel(1,EF,ABC)
midsegment_of_triangle_property_length(1,EF,ABC)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, BD=6, ∠BAD=60°, quadrilateral ADCB is a rhombus. Find the length of line AB.
问题：如图所示，BD=6，∠BAD=60°，四边形ADCB是菱形。求直线AB的长度。
Construction (CDL):
Shape(BA,AO,OB)
Shape(BO,OC,CB)
Shape(AD,DO,OA)
Shape(OD,DC,CO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(BD),6)
Equal(MeasureOfAngle(BAD),60)
Rhombus(ADCB)
Goal: Value(LengthOfLine(AB))
Answer: 6
Theorem sequence:
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, AB=6, ∠ABC=60°, ABCD is a rhombus. Find the length of line AC.
问题：如图所示，AB=6，∠ABC=60°，四边形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(AB),6)
Equal(MeasureOfAngle(ABC),60)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AC))
Answer: 6
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,BCA)
isosceles_triangle_judgment_angle_equal(1,ABC)
Problem: As shown in the diagram, AC=6, ∠ABC=60°, ABCD is a rhombus. Find the length of line AB.
问题：如图所示，AC=6，∠ABC=60°，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(BC,CO,OB)
Shape(OC,CD,DO)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(MeasureOfAngle(ABC),60)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 6
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,BCA)
isosceles_triangle_judgment_angle_equal(1,ABC)
Problem: As shown in the diagram, PE=4, BF⊥PF, PE is perpendicular to BE, ABCD is a rhombus. Find the length of line PF.
问题：如图所示，PE=4，BF⊥PF，PE垂直于BE，ABCD是菱形。求直线PF的长度。
Construction (CDL):
Shape(EB,BP,PE)
Shape(AE,EP,PD,DA)
Shape(BF,FP,PB)
Shape(PF,FC,CD,DP)
Collinear(BEA)
Collinear(BPD)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(PE),4)
PerpendicularBetweenLine(BF,PF)
PerpendicularBetweenLine(PE,BE)
Rhombus(ABCD)
Goal: Value(LengthOfLine(PF))
Answer: 4
Theorem sequence:
parallel_property_alternate_interior_angle(2,AD,BC)
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
bisector_of_angle_judgment_angle_equal(1,BP,EBF)
bisector_of_angle_property_distance_equal(1,BP,EBF)
Problem: As shown in the diagram, AB=2, ∠ABC=120°, CE is perpendicular to PE, PF is perpendicular to AF, quadrilateral ABCD is a rhombus. Find the sum of the length of line PE and the length of line PF.
问题：如图所示，AB=2，∠ABC=120°，CE⊥PE，PF⊥AF，四边形ABCD是菱形。求直线PE的长度与直线PF的长度之和。
Construction (CDL):
Shape(FA,AP,PF)
Shape(AB,BO,OP,PA)
Shape(OB,BC,CO)
Shape(FP,PG,GD,DF)
Shape(GP,PO,OG)
Shape(DG,GE,ED)
Shape(EG,GO,OC,CE)
Collinear(AFD)
Collinear(DEC)
Collinear(DGOB)
Collinear(APOC)
Collinear(PGE)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(MeasureOfAngle(ABC),120)
PerpendicularBetweenLine(CE,PE)
PerpendicularBetweenLine(PF,AF)
Rhombus(ABCD)
Goal: Value(Add(LengthOfLine(PE),LengthOfLine(PF)))
Answer: sqrt(3)
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
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,FAP,PAB)
kite_property_diagonal_perpendicular_bisection(1,BCDA,O)
right_triangle_judgment_angle(1,BOA)
sine_theorem(1,BOA)
right_triangle_property_pythagorean(1,BOA)
line_addition(1,AO,OC)
sine_theorem(1,PFA)
sine_theorem(1,PCE)
line_addition(1,AP,PC)
Problem: As shown in the diagram, AB=6, ∠DAB=30°, AE is perpendicular to DE, ABCD is a rhombus. Find the area of ABCD.
问题：如图所示，AB=6，∠DAB=30°，AE垂直于DE，四边形ABCD是菱形。求四边形ABCD的面积。
Construction (CDL):
Shape(DA,AE,ED)
Shape(DE,EB,BC,CD)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(MeasureOfAngle(DAB),30)
PerpendicularBetweenLine(AE,DE)
Rhombus(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 18
Theorem sequence:
parallelogram_area_formula_sine(1,DABC)
Problem: As shown in the diagram, AB=10, AC=16, quadrilateral ABCD is a rhombus. Find the area of quadrilateral ABCD.
问题：如图所示，AB=10，AC=16，ABCD是菱形。求ABCD的面积。
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(AB),10)
Equal(LengthOfLine(AC),16)
Rhombus(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 96
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
line_addition(1,AO,OC)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
kite_property_diagonal_perpendicular_bisection(1,DABC,O)
line_addition(1,DO,OB)
kite_area_formula_diagonal(1,ABCD)
Problem: As shown in the diagram, AB=4, AD∥GH, BA is parallel to EF, EF is parallel to CD, GH∥BC, ABCD is a rhombus. Find the sum of the perimeter of quadrilateral BEPG and the perimeter of quadrilateral PHDF.
问题：如图所示，AB=4，AD∥GH，BA∥EF，EF平行于CD，GH平行于BC，ABCD是菱形。求BEPG的周长与PHDF的周长之和。
Construction (CDL):
Shape(AG,GP,PF,FA)
Shape(GB,BP,PG)
Shape(FP,PD,DF)
Shape(BE,EP,PB)
Shape(PE,EC,CH,HP)
Shape(PH,HD,DP)
Collinear(BGA)
Collinear(BPD)
Collinear(BEC)
Collinear(GPH)
Collinear(EPF)
Collinear(AFD)
Collinear(CHD)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
ParallelBetweenLine(AD,GH)
ParallelBetweenLine(BA,EF)
ParallelBetweenLine(EF,CD)
ParallelBetweenLine(GH,BC)
Rhombus(ABCD)
Goal: Value(Add(PerimeterOfQuadrilateral(BEPG),PerimeterOfQuadrilateral(PHDF)))
Answer: 16
Theorem sequence:
parallel_property_collinear_extend(3,AD,GH,F)
parallel_property_collinear_extend(3,HG,FA,P)
parallel_property_collinear_extend(3,BA,EF,G)
parallel_property_collinear_extend(3,FE,AG,P)
parallelogram_judgment_parallel_and_parallel(1,GPFA)
parallel_property_collinear_extend(3,GH,BC,P)
parallel_property_collinear_extend(3,CB,HP,E)
parallel_property_collinear_extend(3,EF,CD,P)
parallel_property_collinear_extend(3,DC,PE,H)
parallelogram_judgment_parallel_and_parallel(1,ECHP)
parallelogram_property_opposite_line_equal(1,GPFA)
parallelogram_property_opposite_line_equal(1,AGPF)
parallelogram_property_opposite_line_equal(1,ECHP)
parallelogram_property_opposite_line_equal(1,PECH)
line_addition(1,BG,GA)
line_addition(1,AF,FD)
line_addition(1,BE,EC)
line_addition(1,CH,HD)
quadrilateral_perimeter_formula(1,BEPG)
quadrilateral_perimeter_formula(1,PHDF)
Problem: As shown in the diagram, ∠BCD=110°, EF is the perpendicular bisector of BC, quadrilateral ABCD is a rhombus. Find the measure of ∠DAE.
问题：如图所示，∠BCD=110°，EF垂直平分AB，四边形ABCD是菱形。求∠DAE的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EO,OA)
Shape(AO,OD,DA)
Shape(BF,FE,EB)
Shape(EF,FC,CE)
Shape(EC,CO,OE)
Shape(OC,CD,DO)
Collinear(BEOD)
Collinear(BFC)
Collinear(AOC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),110)
IsPerpendicularBisectorOfLine(EF,BC)
Rhombus(ABCD)
Goal: Value(MeasureOfAngle(DAE))
Answer: 75
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,BCDA,O)
perpendicular_bisector_judgment_per_and_mid(1,EO,CA)
perpendicular_bisector_property_distance_equal(1,EO,CA)
perpendicular_bisector_property_distance_equal(1,EF,BC)
isosceles_triangle_judgment_line_equal(1,EAB)
isosceles_triangle_property_angle_equal(1,EAB)
parallel_property_ipsilateral_internal_angle(1,BA,CD)
perpendicular_bisector_property_bisector(1,BO,CA)
angle_addition(1,ABE,EBF)
parallelogram_property_opposite_angle_equal(1,ABCD)
angle_addition(1,DAE,EAB)
Problem: As shown in the diagram, ∠ABD=20°, ABCD is a rhombus. Find the measure of ∠BCD.
问题：如图所示，∠ABD=20°，四边形ABCD是菱形。求∠BCD的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(BC,CD,DB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),20)
Rhombus(ABCD)
Goal: Value(MeasureOfAngle(BCD))
Answer: 140
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
triangle_property_angle_sum(1,ABD)
parallelogram_property_opposite_angle_equal(1,ABCD)
Problem: As shown in the diagram, ∠BCE=42°, quadrilateral ABCD is a ▱, CE is perpendicular to BE. Find the measure of ∠CDA.
问题：如图所示，∠BCE=42°，ABCD是▱，CE⊥BE。求∠CDA的大小。
Construction (CDL):
Shape(EA,AF,FE)
Shape(AB,BC,CF,FA)
Shape(FC,CD,DF)
Collinear(AFD)
Collinear(EFC)
Collinear(BAE)
Conditions (CDL):
Equal(MeasureOfAngle(BCE),42)
Parallelogram(ABCD)
PerpendicularBetweenLine(CE,BE)
Goal: Value(MeasureOfAngle(CDA))
Answer: 48
Theorem sequence:
triangle_property_angle_sum(1,CEB)
parallelogram_property_opposite_angle_equal(1,BCDA)
Problem: As shown in the diagram, ∠ABE=70°, DE is the angle bisector of ∠CDA, quadrilateral ABCD is a parallelogram. Find the measure of ∠CDE.
问题：如图所示，∠ABE=70°，DE平分∠CDA，四边形ABCD是平行四边形。求∠CDE的大小。
Construction (CDL):
Shape(AB,BE,ED,DA)
Shape(DE,EC,CD)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),70)
IsBisectorOfAngle(DE,CDA)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(CDE))
Answer: 35
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
angle_addition(1,CDE,EDA)
Problem: As shown in the diagram, BC=BD, ∠DCB=65°, quadrilateral DCBA is a parallelogram. Find the measure of ∠ADB.
问题：如图所示，BC=BD，∠DCB=65°，DA和CB是平行四边形DCBA的一组对边。求∠ADB的大小。
Construction (CDL):
Shape(DC,CB,BD)
Shape(DB,BA,AD)
Conditions (CDL):
Equal(LengthOfLine(BC),LengthOfLine(BD))
Equal(MeasureOfAngle(DCB),65)
Parallelogram(DCBA)
Goal: Value(MeasureOfAngle(ADB))
Answer: 50
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BDC)
isosceles_triangle_property_angle_equal(1,BDC)
parallel_property_ipsilateral_internal_angle(1,DA,CB)
angle_addition(1,ADB,BDC)