Reflexión de la recta de Euler y triángulo tangencial
Author: Angel Montesdeoca
Date: 5 Noviembre 2014

Euler line

$\Gamma$

$\Gamma'$

A = (6.9940296686456405, -3.1032157455020983)
B = (19.94030251722772, -3.585578987375419)
C = (15.147971187845151, 5.372423848797206)
X_{74} = TriangleCenter(A, B, C, 74)
d_2 = rectaEuler(A, B, C)
polígono1, c, a, b = Polygon(A, B, C)
d_c = Mirror(d_2, c)
d_b = Mirror(d_2, b)
X_{110} = Intersect(d_b, d_c)
d = Line(X_{110}, X_{74})
X_{155} = TriangleCenter(A, B, C, 155)
P_0 = Trilinear(A, B, C, -a * (a^(14) - 4 * a^(12) * (b^(2) + c^(2)) - 13 * a^(8) * b^(2) * c^(2) * (b^(2) + c^(2)) + b^(2) * c^(2) * (b^(2) - c^(2))^(4) * (b^(2) + c^(2)) + a^(10) * (5 * b^(4) + 11 * b^(2) * c^(2) + 5 * c^(4)) + a^(6) * (-5 * b^(8) + 12 * b^(6) * c^(2) + 2 * b^(4) * c^(4) + 12 * b^(2) * c^(6) - 5 * c^(8)) - a^(2) * (b^(2) - c^(2))^(2) * (b^(8) + b^(6) * c^(2) - 6 * b^(4) * c^(4) + b^(2) * c^(6) + c^(8)) + 2 * a^(4) * (2 * b^(10) - 4 * b^(8) * c^(2) + b^(6) * c^(4) + b^(4) * c^(6) - 4 * b^(2) * c^(8) + 2 * c^(10))), -b * (a^(12) * (-b^(2) + c^(2)) + b^(2) * (b^(2) - c^(2))^(5) * (b^(2) + c^(2)) + a^(10) * (4 * b^(4) + b^(2) * c^(2) - 3 * c^(4)) - a^(2) * (b^(2) - c^(2))^(3) * (4 * b^(6) + b^(4) * c^(2) + 4 * b^(2) * c^(4) + c^(6)) + a^(8) * (-5 * b^(6) - 8 * b^(4) * c^(2) + 7 * b^(2) * c^(4) + 2 * c^(6)) + 2 * a^(6) * (6 * b^(6) * c^(2) + b^(4) * c^(4) - 7 * b^(2) * c^(6) + c^(8)) + a^(4) * (5 * b^(10) - 13 * b^(8) * c^(2) + 2 * b^(6) * c^(4) + 2 * b^(4) * c^(6) + 7 * b^(2) * c^(8) - 3 * c^(10))), -c * (a^(12) * (b^(2) - c^(2)) + c^(2) * (-b^(2) + c^(2))^(5) * (b^(2) + c^(2)) + a^(10) * (-3 * b^(4) + b^(2) * c^(2) + 4 * c^(4)) + a^(8) * (2 * b^(6) + 7 * b^(4) * c^(2) - 8 * b^(2) * c^(4) - 5 * c^(6)) + a^(2) * (b^(2) - c^(2))^(3) * (b^(6) + 4 * b^(4) * c^(2) + b^(2) * c^(4) + 4 * c^(6)) + 2 * a^(6) * (b^(8) - 7 * b^(6) * c^(2) + b^(4) * c^(4) + 6 * b^(2) * c^(6)) + a^(4) * (-3 * b^(10) + 7 * b^(8) * c^(2) + 2 * b^(6) * c^(4) + 2 * b^(4) * c^(6) - 13 * b^(2) * c^(8) + 5 * c^(10))))
t = Circle(P_0, X_{155})
d_a = Mirror(d_2, a)
P = Point(t)
P_a = Mirror(P, d_a)
P_b = Mirror(P, d_b)
P_c = Mirror(P, d_c)
f = Circle(A, B, C)
g = Tangent(A, f)
h = Tangent(B, f)
i = Tangent(C, f)
polígono2, p_c, p_a, p_b = Polygon(P_a, P_b, P_c)
T_a = Intersect(h, i)
T_b = Intersect(g, i)
T_c = Intersect(g, h)
j = Line(T_a, P_a)
k = Line(T_b, P_b)
l = Line(T_c, P_c)
Q = Intersect(l, k)
X_{2931} = Mirror(X_{155}, X_{110})
X_{26} = TriangleCenter(A, B, C, 26)
n = Line(X_{26}, X_{110})
p = Circle(P_a, P_b, P_c)
polígono3, f_1, d_1, e_1 = Polygon(T_a, T_b, T_c)
S = Mirror(Q, n)
a_1 = Ray(P_0, X_{110})
G = Intersect(t, a_1)
G' = Mirror(G, P_0)
X_{1301} = TriangleCenter(A, B, C, 1301)
c_1 = Circle(T_a, T_b, T_c)
T_1 = Mirror(X_{155}, P_0)
b_1 = Line(S, P)
X_{2079} = TriangleCenter(A, B, C, 2079)
X_3 = TriangleCenter(A, B, C, 3)
X_{156} = TriangleCenter(T_a, T_b, T_c, 5)
d' = Line(X_{26}, X_{110})
q = Line(X_{26}, P_0)
T_2 = Mirror(T_1, q)
D, E = Intersect(c_1, d')

Tool: rectaEuler
Dar los vértices de un triángulo
Input: D, E, F
Output: m

G = TriangleCenter(D, E, F, 4)
H = TriangleCenter(D, E, F, 3)
m = Line(H, G)

Tool: centro semejanza (command centrosemejanza)
Dar dos segmentos AB y AC
Input: A, B, C, D
Output: E'

c = Line(A, B)
d = Line(C, D)
E = Intersect(c, d)
e = Circle(D, B, E)
f = Circle(C, E, A)
F = Center(f)
G = Center(e)
g = Segment(F, G)
E' = Mirror(E, g)