SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠CAB=42°, ∠DBA=35°, the center of ⊙O is O. Find the measure of ∠DPA.
问题：如图所示，∠CAB=42°，∠DBA=35°，⊙O的圆心为O。求∠DPA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AP,PC)
Shape(PA,OAD,DP)
Shape(O)
Shape(BP,PD,DB)
Shape(CP,PB,PBC)
Shape(BD,ODB)
Collinear(CPD)
Collinear(APB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),42)
Equal(MeasureOfAngle(DBA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DPA))
Answer: 77
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
triangle_property_angle_sum(1,APC)
adjacent_complementary_angle(1,DPA,APC)
Problem: As shown in the diagram, AC=4, ∠ACB=90°, ∠BAC=37°. Find the length of line BC.
问题：如图所示，AC=4，∠ACB=90°，∠BAC=37°。求直线BC的长度。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(BAC),37)
Goal: Value(LengthOfLine(BC))
Answer: 4*tan(37*pi/180)
Theorem sequence:
triangle_property_angle_sum(1,ACB)
sine_theorem(1,CBA)
Problem: As shown in the diagram, AB=10, BC=8, ∠ACB=90°. Find the value of sin(BA).
问题：如图所示，AB=10，BC=8，∠ACB=90°。求sin(BA)的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),8)
Equal(MeasureOfAngle(ACB),90)
Goal: Value(Sin(MeasureOfAngle(BAC)))
Answer: 4/5
Theorem sequence:
sine_theorem(1,ACB)
sine_theorem(1,CBA)
Problem: As shown in the diagram, AB=10, AC=8, ∠BCA=90°. Find the value of tan(AB).
问题：如图所示，AB=10，AC=8，∠BCA=90°。求tan(AB)的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),8)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(Tan(MeasureOfAngle(ABC)))
Answer: 4/3
Theorem sequence:
sine_theorem(1,ABC)
Problem: As shown in the diagram, AC=3, BC=4, ∠BCA=90°. Find the value of sin(AB).
问题：如图所示，AC=3，BC=4，∠BCA=90°。求sin(AB)的值。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(LengthOfLine(BC),4)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(Sin(MeasureOfAngle(ABC)))
Answer: 3/5
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
sine_theorem(1,BCA)
sine_theorem(1,CAB)
Problem: As shown in the diagram, ∠CBO=30°, ∠ODA=35°, the center of circle O is O. Find the measure of ∠DAB.
问题：如图所示，∠CBO=30°，∠ODA=35°，圆O的圆心为O。求∠DAB的大小。
Construction (CDL):
Shape(DA,AO,OD)
Shape(OA,AC,CO)
Shape(OC,CB,BO)
Shape(ODA,AD)
Shape(BC,CA,OAB)
Shape(DO,OB,OBD)
Collinear(ACB)
Collinear(DOC)
Cocircular(O,DAB)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),30)
Equal(MeasureOfAngle(ODA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 65
Theorem sequence:
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,OAB)
angle_addition(1,DAO,OAC)
Problem: As shown in the diagram, AB=8, the ratio of similar triangles ABC and DEF is 2/5, triangle ABC is similar to triangle DEF.. Find the length of line DE.
问题：如图所示，AB=8，三角形ABC和三角形DEF的相似比为2/5，三角形ABC相似于三角形DEF。求直线DE的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(RatioOfSimilarTriangle(ABC,DEF),2/5)
SimilarBetweenTriangle(ABC,DEF)
Goal: Value(LengthOfLine(DE))
Answer: 20
Theorem sequence:
similar_triangle_property_line_ratio(1,CAB,FDE)
Problem: As shown in the diagram, ∠CAO=55°, O is the center of ⊙O, BC is the diameter of ⊙O. Find the measure of ∠ADC.
问题：如图所示，∠CAO=55°，O是⊙O的圆心，BC是⊙O的直径。求∠ADC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OA)
Shape(AO,OM,MA)
Shape(AM,MC,CA)
Shape(AC,OCA)
Shape(MD,DC,CM)
Shape(CD,ODC)
Shape(MO,OB,OBD,DM)
Collinear(BOMC)
Collinear(AMD)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),55)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 35
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAB,O)
angle_addition(1,CAO,OAB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, AC=4, BC=3, ∠BCA=90°, AD is perpendicular to CD. Find the value of sin(BC).
问题：如图所示，AC=4，BC=3，∠BCA=90°，AD垂直于CD。求sin(BC)的值。
Construction (CDL):
Shape(CA,AD,DC)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(BCA),90)
PerpendicularBetweenLine(AD,CD)
Goal: Value(Sin(MeasureOfAngle(BCD)))
Answer: sin(pi*(-asin(4/5)/pi+1/2))
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
sine_theorem(1,CAD)
sine_theorem(1,CDB)
sine_theorem(1,ABC)
sine_theorem(1,BCA)
triangle_property_angle_sum(1,CDB)
Problem: As shown in the diagram, ∠CAB=40°, the center of ⊙O is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=40°，O是圆O的圆心。求∠COB的大小。
Construction (CDL):
Shape(MA,OAB,BO,OM)
Shape(OB,BM,MO)
Shape(AM,MC,CA)
Shape(AC,OCA)
Shape(CM,MB,OBC)
Collinear(OMC)
Collinear(AMB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, AC=3, BC=4, ∠BCA=90°, AD is perpendicular to CD. Find the value of cos(DC).
问题：如图所示，AC=3，BC=4，∠BCA=90°，AD⊥CD。求cos(DC)的值。
Construction (CDL):
Shape(AD,DC,CA)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(LengthOfLine(BC),4)
Equal(MeasureOfAngle(BCA),90)
PerpendicularBetweenLine(AD,CD)
Goal: Value(Cos(MeasureOfAngle(DCA)))
Answer: cos(pi*(-asin(4/5)/pi+1/2))
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
sine_theorem(1,BCA)
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, AB=AC, AD=AE, ∠EDC=20°. Find the measure of ∠DAB.
问题：如图所示，AB=AC，AD=AE，∠EDC=20°。求∠DAB的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(AD),LengthOfLine(AE))
Equal(MeasureOfAngle(EDC),20)
Goal: Value(MeasureOfAngle(DAB))
Answer: 40
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_judgment_line_equal(1,ADE)
isosceles_triangle_property_angle_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ADE)
triangle_property_angle_sum(1,ABD)
triangle_property_angle_sum(1,ADE)
triangle_property_angle_sum(1,EDC)
triangle_property_angle_sum(1,ADC)
adjacent_complementary_angle(1,CED,DEA)
adjacent_complementary_angle(1,BDA,ADC)
Problem: As shown in the diagram, ∠COB=50°, the center of circle O is O, BF is perpendicular to OF. Find the measure of ∠ADC.
问题：如图所示，∠COB=50°，O是⊙O的圆心，BF垂直于OF。求∠ADC的大小。
Construction (CDL):
Shape(OCA,AE,EC)
Shape(CE,EF,FC)
Shape(OBC,CF,FB)
Shape(OAD,DA)
Shape(AD,DE,EA)
Shape(OF,FE,ED,ODB,BO)
Shape(FO,OB,BF)
Collinear(AEFB)
Collinear(CED)
Collinear(CFO)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BF,OF)
Goal: Value(MeasureOfAngle(ADC))
Answer: 25
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OBA,OFC)
congruent_arc_judgment_length_equal(1,OBC,OCA)
congruent_arc_property_measure_equal(1,OBC,OCA)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠BCA=90°, Mul(3=LengthOfLine(CD)), CD is perpendicular to BD. Find the value of sin(BC).
问题：如图所示，∠BCA=90°，Mul(3=LengthOfLine(CD))，CD垂直于BD。求sin(BC)的值。
Construction (CDL):
Shape(CA,AD,DC)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),90)
Equal(Mul(3,LengthOfLine(CD)),Mul(2,LengthOfLine(AC)))
PerpendicularBetweenLine(CD,BD)
Goal: Value(Sin(MeasureOfAngle(BCD)))
Answer: sin(pi*(-acos(2/3)/pi+1/2))
Theorem sequence:
adjacent_complementary_angle(1,ADC,CDB)
angle_addition(1,BCD,DCA)
triangle_property_angle_sum(1,CAD)
sine_theorem(1,CAD)
Problem: As shown in the diagram, ∠CDA=20°, the center of ⊙O is O. Find the measure of ∠CBA.
问题：如图所示，∠CDA=20°，O是圆O的圆心。求∠CBA的大小。
Construction (CDL):
Shape(CP,PA,OAC)
Shape(PC,CB,BP)
Shape(AP,PD,DA)
Shape(OCB,BC)
Shape(ODA,AD)
Shape(PB,OBD,DP)
Shape(O)
Collinear(CPD)
Collinear(APB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, AC=1, the radius of circle O is 1, the center of ⊙O is O, CA is the tangent to ⊙O. Find the value of tan(DE).
问题：如图所示，AC=1，圆O的半径为1，圆O的圆心为O，CA是⊙O的切线。求tan(DE)的值。
Construction (CDL):
Shape(OEA,AE)
Shape(EA,AM,ME)
Shape(MA,OAD,DM)
Shape(OM,MD,DB,BO)
Shape(EM,MO,OB,OBE)
Shape(BD,ODB)
Shape(OAD,AC,CD)
Collinear(AMOB)
Collinear(EMD)
Collinear(CDB)
Cocircular(O,EADB)
Conditions (CDL):
Equal(LengthOfLine(AC),1)
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CA,O)
Goal: Value(Tan(MeasureOfAngle(DEA)))
Answer: 1/2
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
circle_property_length_of_radius_and_diameter(1,O)
diameter_of_circle_property_length_equal(1,AB,O)
tangent_of_circle_property_perpendicular(1,CA,O,O)
right_triangle_judgment_angle(1,BAC)
right_triangle_property_pythagorean(1,BAC)
sine_theorem(1,CBA)
arc_property_circumference_angle_external(1,OAD,E)
arc_property_circumference_angle_external(1,OAD,B)
Problem: As shown in the diagram, AB=4, DE=8, triangle ECD is mirror congruent to triangle EDN, ABCD is a rectangle. Find the value of sin(NE).
问题：如图所示，AB=4，DE=8，△ECD与△EDN是镜像全等三角形，ABCD是矩形。求sin(NE)的值。
Construction (CDL):
Shape(AB,BE,EM,MA)
Shape(NM,MD,DN)
Shape(ME,ED,DM)
Shape(DE,EC,CD)
Collinear(AMD)
Collinear(NME)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(DE),8)
MirrorCongruentBetweenTriangle(ECD,EDN)
Rectangle(ABCD)
Goal: Value(Sin(MeasureOfAngle(NED)))
Answer: 1/2
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,ECD,EDN)
parallelogram_property_opposite_line_equal(1,ABCD)
sine_theorem(1,DEC)
Problem: As shown in the diagram, AC=3, the center of ⊙O is O, AP⊥DP. Find the length of line DE.
问题：如图所示，AC=3，圆O的圆心为O，AP⊥DP。求直线DE的长度。
Construction (CDL):
Shape(OAC,CA)
Shape(AC,CP,PA)
Shape(AP,PN,NO,OM,MA)
Shape(AM,ME,EA)
Shape(AE,OEA)
Shape(PC,CB,BP)
Shape(OCB,BC)
Shape(PB,BN,NP)
Shape(EM,MD,DE)
Shape(ED,ODE)
Shape(MO,ON,ND,DM)
Shape(NB,BD,DN)
Shape(DB,OBD)
Collinear(AMD)
Collinear(APB)
Collinear(CPND)
Collinear(BNOME)
Cocircular(O,ACBDE)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AP,DP)
Goal: Value(LengthOfLine(DE))
Answer: 3
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,BOE,O)
diameter_of_circle_property_right_angle(1,EAB,O)
parallel_judgment_ipsilateral_internal_angle(1,AE,PD)
parallel_property_alternate_interior_angle(1,AE,PD)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,ODE,A)
congruent_arc_judgment_measure_equal(1,OAC,ODE)
congruent_arc_property_chord_equal(1,OAC,ODE)
Problem: As shown in the diagram, AC=4, BC=3, ∠BCA=90°, CD⊥BD. Find the value of sin(BC).
问题：如图所示，AC=4，BC=3，∠BCA=90°，CD垂直于BD。求sin(BC)的值。
Construction (CDL):
Shape(AD,DC,CA)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(BCA),90)
PerpendicularBetweenLine(CD,BD)
Goal: Value(Sin(MeasureOfAngle(BCD)))
Answer: 3/5
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
sine_theorem(1,BCA)
angle_addition(1,BCD,DCA)
adjacent_complementary_angle(1,ADC,CDB)
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, AC=4, BC=3, ∠ACB=90°, ∠CBA=x°. Find sin(x).
问题：如图所示，AC=4，BC=3，∠ACB=90°，∠CBA=x°。求sin(x)。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(CBA),x)
Goal: Value(Sin(x))
Answer: 4/5
Theorem sequence:
cosine_theorem(1,BAC)
cosine_theorem(1,CBA)
Problem: As shown in the diagram, AC=12, BC=5, ∠ACB=90°. Find the value of sin(BA).
问题：如图所示，AC=12，BC=5，∠ACB=90°。求sin(BA)的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BC),5)
Equal(MeasureOfAngle(ACB),90)
Goal: Value(Sin(MeasureOfAngle(BAC)))
Answer: 5/13
Theorem sequence:
cosine_theorem(1,ACB)
cosine_theorem(1,CBA)
Problem: As shown in the diagram, AB=3, BC=2, ∠BCA=90°. Find the value of cos(AB).
问题：如图所示，AB=3，BC=2，∠BCA=90°。求cos(AB)的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(BC),2)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(Cos(MeasureOfAngle(ABC)))
Answer: 2/3
Theorem sequence:
cosine_theorem(1,BCA)
cosine_theorem(1,CAB)
Problem: As shown in the diagram, AB=5, AC=3, ∠BCA=90°. Find the value of sin(AB).
问题：如图所示，AB=5，AC=3，∠BCA=90°。求sin(AB)的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(AC),3)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(Sin(MeasureOfAngle(ABC)))
Answer: 3/5
Theorem sequence:
cosine_theorem(1,BCA)
cosine_theorem(1,CAB)
Problem: As shown in the diagram, AB=3, OB=t, ∠OBA=90°, Tan(MeasureOfAngle(AOB))=2. Find the value of t.
问题：如图所示，AB=3，OB=t，∠OBA=90°，Tan(MeasureOfAngle(AOB))=2。求t的值。
Construction (CDL):
Shape(CO,OA,AC)
Shape(AO,OB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(OB),t)
Equal(MeasureOfAngle(OBA),90)
Equal(Tan(MeasureOfAngle(AOB)),2)
Goal: Value(t)
Answer: 3/2
Theorem sequence:
triangle_property_angle_sum(1,AOB)
sine_theorem(1,AOB)
sine_theorem(1,OBA)
Problem: As shown in the diagram, ∠CDB=35°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠ABC.
问题：如图所示，∠CDB=35°，O是⊙O的圆心，⊙O的直径为AB。求∠ABC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DE,EO,OA,OAD)
Shape(BD,ODB)
Shape(CB,OBC)
Shape(AO,OE,EC,CA)
Shape(CE,EB,BC)
Shape(ED,DB,BE)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 55
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, AC=8, BC=6, ∠BCA=90°, ∠CDB=90°. Find the value of cos(BC).
问题：如图所示，AC=8，BC=6，∠BCA=90°，∠CDB=90°。求cos(BC)的值。
Construction (CDL):
Shape(AD,DC,CA)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(BC),6)
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CDB),90)
Goal: Value(Cos(MeasureOfAngle(BCD)))
Answer: 4/5
Theorem sequence:
line_addition(1,AD,DB)
adjacent_complementary_angle(1,ADC,CDB)
right_triangle_judgment_angle(1,ADC)
right_triangle_judgment_angle(1,CDB)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,ADC)
right_triangle_property_pythagorean(1,CDB)
right_triangle_property_pythagorean(1,BCA)
sine_theorem(1,CDB)
sine_theorem(1,DBC)
Problem: As shown in the diagram, AB=1, OB=2, ∠AOB=α°, ∠OBA=90°. Find tan(a).
问题：如图所示，AB=1，OB=2，∠AOB=α°，∠OBA=90°。求tan(a)。
Construction (CDL):
Shape(AO,OB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),1)
Equal(LengthOfLine(OB),2)
Equal(MeasureOfAngle(AOB),α)
Equal(MeasureOfAngle(OBA),90)
Goal: Value(Tan(α))
Answer: 1/2
Theorem sequence:
cosine_theorem(1,OBA)
cosine_theorem(1,BAO)
Problem: As shown in the diagram, ∠ADC=70°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠ADC=70°，圆O的圆心为O，圆O的直径为AB。求∠CAB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DA,OAD)
Shape(CB,OBC)
Shape(BE,ED,ODB)
Shape(AD,DE,EO,OA)
Shape(AO,OE,EC,CA)
Shape(CE,EB,BC)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, AC=12, BC=5, ∠BCA=90°, ∠CDB=90°. Find the value of sin(BC).
问题：如图所示，AC=12，BC=5，∠BCA=90°，∠CDB=90°。求sin(BC)的值。
Construction (CDL):
Shape(AD,DC,CA)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BC),5)
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CDB),90)
Goal: Value(Sin(MeasureOfAngle(BCD)))
Answer: 5/13
Theorem sequence:
right_triangle_judgment_angle(1,CDB)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,CDB)
right_triangle_property_pythagorean(1,BCA)
cosine_theorem(1,CDB)
cosine_theorem(1,BCA)
sine_theorem(1,CDB)
Problem: As shown in the diagram, Cos(α)=3/5, AP=m, OA=4, ∠POA=α°. Find tan(a).
问题：如图所示，Cos(α)=3/5，AP=m，OA=4，∠POA=α°。求tan(a)。
Construction (CDL):
Shape(PO,OA,AP)
Conditions (CDL):
Equal(Cos(α),3/5)
Equal(LengthOfLine(AP),m)
Equal(LengthOfLine(OA),4)
Equal(MeasureOfAngle(POA),α)
Goal: Value(Tan(α))
Answer: 4/3
Theorem sequence:
Problem: As shown in the diagram, the length of arc OBC is equal to the length of ⌒OCD, ∠CAB=30°, the center of ⊙O is P, the diameter of circle O is AB. Find the measure of ∠DAC.
问题：如图所示，弧OBC的长度与弧OCD的长度相等，∠CAB=30°，P是圆O的圆心，AB是圆O的直径。求∠DAC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AC,OCD)
Shape(CA,AB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfArc(OBC),LengthOfArc(OCD))
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(P,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 30
Theorem sequence:
congruent_arc_judgment_length_equal(1,OBC,OCD)
congruent_arc_property_measure_equal(1,OBC,OCD)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OCD,A)
Problem: As shown in the diagram, ∠AEC=27°, ∠ECD=52°, AB∥CD. Find the measure of ∠EAB.
问题：如图所示，∠AEC=27°，∠ECD=52°，AB∥CD。求∠EAB的大小。
Construction (CDL):
Shape(EC,CF,FA,AE)
Shape(AF,FD,DB,BA)
Shape(EA,AB)
Collinear(EAF)
Collinear(CFD)
Conditions (CDL):
Equal(MeasureOfAngle(AEC),27)
Equal(MeasureOfAngle(ECD),52)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EAB))
Answer: 79
Theorem sequence:
parallel_property_collinear_extend(3,DC,BA,F)
triangle_property_angle_sum(1,ECF)
adjacent_complementary_angle(1,CFA,AFD)
parallel_property_corresponding_angle(1,AB,FD,E)
Problem: As shown in the diagram, ∠ABD=90°, ∠BDA=60°, ∠BEF=90°, ∠CAB=30°, ∠EFC=45°, ∠FBE=45°. Find the measure of ∠FCD.
问题：如图所示，∠ABD=90°，∠BDA=60°，∠BEF=90°，∠CAB=30°，∠EFC=45°，∠FBE=45°。求∠FCD的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(CB,BD,DC)
Shape(CD,DE,EF,FC)
Collinear(ACD)
Collinear(BCF)
Collinear(BDE)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),90)
Equal(MeasureOfAngle(BDA),60)
Equal(MeasureOfAngle(BEF),90)
Equal(MeasureOfAngle(CAB),30)
Equal(MeasureOfAngle(EFC),45)
Equal(MeasureOfAngle(FBE),45)
Goal: Value(MeasureOfAngle(FCD))
Answer: 105
Theorem sequence:
triangle_property_angle_sum(1,CBD)
adjacent_complementary_angle(1,FCD,DCB)
Problem: As shown in the diagram, ∠ADC=30°, ∠DOB=140°, O is the center of ⊙O. Find the measure of ∠AEC.
问题：如图所示，∠ADC=30°，∠DOB=140°，O是⊙O的圆心。求∠AEC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(DO,OB,OBD)
Shape(CD,ODC)
Shape(AE,EC,OCA)
Shape(AB,BE,EA)
Shape(CE,ED,DC)
Shape(EB,BO,OD,DE)
Collinear(AED)
Collinear(BEC)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),30)
Equal(MeasureOfAngle(DOB),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AEC))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,C)
triangle_property_angle_sum(1,CED)
adjacent_complementary_angle(1,AEC,CED)
Problem: As shown in the diagram, AB=6, ∠ACB=90°, Sin(MeasureOfAngle(BAC))=2/3. Find the length of line AC.
问题：如图所示，AB=6，∠ACB=90°，Sin(MeasureOfAngle(BAC))=2/3。求直线AC的长度。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(MeasureOfAngle(ACB),90)
Equal(Sin(MeasureOfAngle(BAC)),2/3)
Goal: Value(LengthOfLine(AC))
Answer: 2*sqrt(5)
Theorem sequence:
sine_theorem(1,BAC)
cosine_theorem(1,ACB)
Problem: As shown in the diagram, BC=4, CD=3, ∠BCA=90°, CD is the median of triangle CAB. Find the value of tan(AB).
问题：如图所示，BC=4，CD=3，∠BCA=90°，CD是△CAB的中线。求tan(AB)的值。
Construction (CDL):
Shape(AD,DC,CA)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(BC),4)
Equal(LengthOfLine(CD),3)
Equal(MeasureOfAngle(BCA),90)
IsMedianOfTriangle(CD,CAB)
Goal: Value(Tan(MeasureOfAngle(ABC)))
Answer: sqrt(5)/2
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_length_of_median(1,BCA,D)
right_triangle_property_pythagorean(1,BCA)
cosine_theorem(1,BCA)
Problem: As shown in the diagram, AC=6, BC=8, ∠ACB=90°. Find the value of tan(BA).
问题：如图所示，AC=6，BC=8，∠ACB=90°。求tan(BA)的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(BC),8)
Equal(MeasureOfAngle(ACB),90)
Goal: Value(Tan(MeasureOfAngle(BAC)))
Answer: 4/3
Theorem sequence:
cosine_theorem(1,ACB)
cosine_theorem(1,CBA)
Problem: As shown in the diagram, ∠CBO=30°, ∠ODA=30°, O is the center of ⊙O. Find the measure of ∠DAB.
问题：如图所示，∠CBO=30°，∠ODA=30°，圆O的圆心为O。求∠DAB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(BC,CA,OAB)
Shape(DO,OB,OBD)
Shape(AO,OD,DA)
Shape(OA,AC,CO)
Shape(OC,CB,BO)
Collinear(DOC)
Collinear(ACB)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),30)
Equal(MeasureOfAngle(ODA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 60
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,OAB)
angle_addition(1,DAO,OAC)
Problem: As shown in the diagram, AC=12, BC=5, ∠BCA=90°, CD is the median of △ CAB. Find the value of sin(DC).
问题：如图所示，AC=12，BC=5，∠BCA=90°，CD是△CAB的中线。求sin(DC)的值。
Construction (CDL):
Shape(BC,CD,DB)
Shape(DC,CA,AD)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BC),5)
Equal(MeasureOfAngle(BCA),90)
IsMedianOfTriangle(CD,CAB)
Goal: Value(Sin(MeasureOfAngle(DCA)))
Answer: 5/13
Theorem sequence:
line_addition(1,AD,DB)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
right_triangle_property_length_of_median(1,BCA,D)
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,DCA)
sine_theorem(1,BCA)
Problem: As shown in the diagram, AC=9, ∠CAB=90°, Sin(MeasureOfAngle(ABC))=3/5. Find the length of line AB.
问题：如图所示，AC=9，∠CAB=90°，Sin(MeasureOfAngle(ABC))=3/5。求直线AB的长度。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AC),9)
Equal(MeasureOfAngle(CAB),90)
Equal(Sin(MeasureOfAngle(ABC)),3/5)
Goal: Value(LengthOfLine(AB))
Answer: 12
Theorem sequence:
sine_theorem(1,CAB)
cosine_theorem(1,ABC)
Problem: As shown in the diagram, AC=2, BC=1, ∠ADC=90°, ∠BCA=90°. Find the value of sin(DC).
问题：如图所示，AC=2，BC=1，∠ADC=90°，∠BCA=90°。求sin(DC)的值。
Construction (CDL):
Shape(AD,DC,CA)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(BC),1)
Equal(MeasureOfAngle(ADC),90)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(Sin(MeasureOfAngle(DCA)))
Answer: 2*sqrt(5)/5
Theorem sequence:
right_triangle_judgment_angle(1,ADC)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,ADC)
right_triangle_property_pythagorean(1,BCA)
cosine_theorem(1,CAD)
cosine_theorem(1,ABC)
sine_theorem(1,CAD)