Centros X_{7145} y X_{6146}

Sondat's line

Let A'B'C' be the orthic triangle  of ABC. Let Ab be the orthogonal projection of A' onto line AB, and define Bc and Ca cyclically. 
Let Ac be the orthogonal projection of A' onto line AC, and define Ba and Cb cyclically. 
Let A'b be the reflection of A in Ab, and let A'c be the reflection of A in Ac .
 Let Na be the nine-point center of the triangle AA'bA'c, and define Nb and Nc cyclically. 
The triangle NaNbNc, here named the Hatzipolakis-Moses triangle, is orthologic to ABC and perpective to ABC.

The ABC-to-NaNbNc center of orthology is X(6445), and the NaNbNc-to-ABC center of orthology is X(6446). 
The perspector of ABC and NaNbNc is X(54). 
(Based on notes , Antreas Hatzipolakis and Peter Moses, November 19, 2014; 
see   https://groups.yahoo.com/neo/groups/Hyacinthos/conversations/messages/22774) 

Motrar descripción

Jerabek hyperbola

A = (-4, -8)
B = (19.955200356391185, -8.286317135733759)
C = (14.686120795328417, 9.676295879106165)
polígono1, c, a, b = Polygon(A, B, C)
d = Line(C, B)
e = Line(C, A)
f = Line(A, B)
g = OrthogonalLine(A, a)
h = OrthogonalLine(B, b)
i = OrthogonalLine(C, c)
A' = Intersect(g, d)
B' = Intersect(e, h)
C' = Intersect(f, i)
polígono2, c', a', b' = Polygon(A', B', C')
j = OrthogonalLine(A', f)
k = OrthogonalLine(A', e)
A_b = Intersect(f, j)
A_c = Intersect(k, e)
l = OrthogonalLine(B', a)
m = OrthogonalLine(B', f)
B_c = Intersect(l, d)
B_a = Intersect(f, m)
n = OrthogonalLine(C', d)
p = OrthogonalLine(C', e)
C_b = Intersect(d, n)
C_a = Intersect(p, e)
A'_b = Mirror(A, A_b)
A'_c = Mirror(A, A_c)
N_a = TriangleCenter(A, A'_b, A'_c, 5)
N_b = PuntoNa(B, C, A)
N_c = PuntoNa(C, A, B)
polígono3, n_c, n_a, n_b = Polygon(N_a, N_b, N_c)
q = Line(N_a, A)
r = Line(B, N_b)
s = Line(C, N_c)
t = OrthogonalLine(A, n_a)
a_1 = OrthogonalLine(B, n_b)
b_1 = OrthogonalLine(C, n_c)
X_{6145} = Intersect(t, a_1)
c_1 = OrthogonalLine(N_a, a)
d_1 = OrthogonalLine(N_b, b)
e_1 = OrthogonalLine(N_c, c)
X_{6146} = Intersect(d_1, e_1)
X_{54} = TriangleCenter(A, B, C, 54)
f_1 = Line(X_{6145}, X_{6146})
o = false
X_4 = TriangleCenter(A, B, C, 4)
g_1 = Conic(A, B, C, X_4, X_{54})

Tool: TrianguloETC
Dar el vértice A
Input: A_{etc}
Output: B_{etc}, C_{etc}, a, polígono1

c = Circle(A_{etc}, 13)
B_{etc} = Point(c)
d = Circle(A_{etc}, 9)
e = Circle(B_{etc}, 6)
C_{etc} = Intersect(d, e, 1)
polígono1, c_{etc}, a_{etc}, b_{etc} = Polygon(A_{etc}, B_{etc}, C_{etc})
a = Line(B_{etc}, C_{etc})

Tool: Coordenadas Trilineales (command CoordenadasTrilineales)
Dar los vértices de ABC y un  punto P
Input: A, B, C, P
Output: texto2, CoordTrilineales

q_1 = Segment(B, P)
e = Line(A, C)
E_1 = Intersect(e, q_1)
d = Line(C, B)
p_1 = Segment(A, P)
D_1 = Intersect(d, p_1)
ABC, c, a, b = Polygon(A, B, C)
e_1 = OrthogonalLine(P, a)
D = Intersect(e_1, d)
p = Segment(P, D)
α = If(Defined[D_1], -p, p)
f_1 = OrthogonalLine(P, b)
g = OrthogonalLine(P, c)
f = Line(A, B)
E = Intersect(f_1, e)
F = Intersect(g, f)
r_1 = Segment(P, C)
q = Segment(P, E)
r = Segment(P, F)
F_1 = Intersect(f, r_1)
β = If(Defined[E_1], -q, q)
γ = If(Defined[F_1], -r, r)
CoordTrilineales: {α, β, γ}
texto2: "" + CoordTrilineales + ""

Tool: Punto Na (command PuntoNa)
Dar ABC
Input: A, B, C
Output: N_a

polígono1, c, a, b = Polygon(A, B, C)
d = Line(C, B)
e = Line(C, A)
f = Line(A, B)
g = OrthogonalLine(A, a)
A' = Intersect(g, d)
j = OrthogonalLine(A', f)
k = OrthogonalLine(A', e)
A_b = Intersect(f, j)
A_c = Intersect(k, e)
A'_b = Mirror(A, A_b)
A'_c = Mirror(A, A_c)
N_a = TriangleCenter(A, A'_b, A'_c, 5)