triangle = scene.nondegenerate_triangle(labels=('A', 'B', 'C'))
A, B, C = triangle.points
def napoleonic(A, B, C):
equilateral = scene.equilateral_triangle(A, B, '%s_1' % C.label)
_, _, C1 = equilateral.points
C1.comment = Comment(
'Third vertex of equilateral triangle with base $%{segment:AB}$',
{'AB': A.segment(B)})
line = A.line_through(B, layer='auxiliary')
comment = Comment(
'$%{triangle:equilateral}$ is facing away from $%{triangle:triangle}$',
{
'equilateral': equilateral,
'triangle': triangle
})
C1.opposite_side_constraint(C, line, comment=comment)
D = scene.circumcentre_point(equilateral, label='%s_2' % C.label)
napoleonic(A, B, C)
napoleonic(C, A, B)
napoleonic(B, C, A)
prop = EquilateralTriangleProperty(
(scene.get('A_2'), scene.get('B_2'), scene.get('C_2')))
run_sample(scene, prop)
scene = Scene()
triangle = scene.nondegenerate_triangle(labels=('A', 'B', 'C'))
D = scene.orthocentre_point(triangle, label='D')
D.inside_triangle_constraint(triangle)
A, B, C = triangle.points
H = A.line_through(D).intersection_point(B.line_through(C), label='H')
G = C.line_through(D).intersection_point(A.line_through(B), label='G')
A.segment(B).congruent_constraint(C.segment(D), comment='given')
#Proof
# +* 1* |AB| = |CD| *given
# +* 2* ∠ A H B = 90º *given (altitude)
# +* 3* ∠ A H C = 90º *given (altitude)
# +* 4* ∠ C G B = 90º *given (altitude)
# +* 5* ∠ A H B = ∠ A H C (a.k.a ∠ D H C) = 90º *2, *3 same arcs => equal angles
# +* 6* ∠ A H B = ∠ C G B *2, *4 same arcs => equal angles
# +* 7* ∠ A B H = ∠ C B G *same angle
# +* 8* △ A B H ∼ △ C B G *6, *7 two angles
# +* 9* ∠ H A B = ∠ G C B (a.k.a. ∠ D C H) *8 similar triangles => equal angles
# +*10* ∠ H A B = ∠ D C H *9 similar triangles => equal angles
# +*11* △ A B H ∼ △ C D H *5, *10 two anlges
# +*12* △ A B H = △ C D H *1, *11 similarity, side
# +*13* |AH| = |CH| *12 equal triangles => equal sides
# +*14* isosceles △ A H C *13 equal sides => isosceles
# +*15* ∠ A C H = (180º - ∠ A H C) / 2 = 45º *14 isocseles
# +*16* ∠ A C B = ∠ A C H *same angle
# +*17* ∠ A C B = ∠ A C H = 45º *15, *16 transitivity
run_sample(scene, C.angle(A, B))
from sandbox import Scene
from runner import run_sample
scene = Scene()
triangle = scene.nondegenerate_triangle(labels=('A', 'B', 'C'))
triangle.sides[0].congruent_constraint(triangle.sides[1])
triangle.sides[0].congruent_constraint(triangle.sides[2])
scene.incentre_point(triangle, label='D')
run_sample(scene)
altitudeA = scene.altitude(triangle, A)
A1 = altitudeA.intersection_point(B.line_through(C), label='A_1')
A2 = altitudeA.intersection_point(circ, label='A_2')
A2.not_equal_constraint(A)
altitudeB = scene.altitude(triangle, B)
B1 = altitudeB.intersection_point(A.line_through(C), label='B_1')
B2 = altitudeB.intersection_point(circ, label='B_2')
B2.not_equal_constraint(B)
altitudeC = scene.altitude(triangle, C)
C1 = altitudeC.intersection_point(B.line_through(A), label='C_1')
C2 = altitudeC.intersection_point(circ, label='C_2')
C2.not_equal_constraint(C)
#A.angle(B, C).is_obtuse_constraint()
#A.angle(B, C).is_right_constraint()
A.angle(B, C).is_acute_constraint(comment='assumption')
B.angle(A, C).is_acute_constraint(comment='assumption')
C.angle(B, A).is_acute_constraint(comment='assumption')
props = (
AngleRatioProperty(B.angle(B1, C), A.angle(B1, B2), 1),
AngleRatioProperty(A.angle(B1, H), B.angle(B1, C), 1),
AngleRatioProperty(A.angle(B1, H), A.angle(B1, B2), 1),
ProportionalLengthsProperty(B1.segment(H), B1.segment(B2), 1),
)
run_sample(scene, *props)
