$\Gamma_a$

$\Gamma_F$

Problema 737 (trianguloscabri)

Dados tres puntos B, C y P, hallar el lugar geométrico del 
punto A tal que la circunferencia de los nueve puntos del 
triángulo ABC pasa por P. 

Mover F

B = (5.768993299117157, -1.0228284329756974)
C = (7.077692581647222, 4.949064274011411)
a = Segment(B, C)
M_a = Midpoint(a)
P = (1.173186643644867, 4.86780960513982)
Γ_a = Circle(M_a, C)
F = Point(Γ_a)
m = Line(B, F)
Γ_F = Circle(P, M_a, F)
A_3 = Center(Γ_F)
e = Segment(A_3, M_a)
E = Mirror(F, e)
n = Line(C, E)
A = Intersect(m, n)
g = Line(P, M_a)
D, G = Intersect(Γ_a, g)
h = Line(B, G)
i = Line(C, D)
j = Line(B, D)
k = Line(C, G)
l = Line(P, h)
m_3 = Line(P, j)
p, H, I, J, K = HiperbolaPuntoDosAsintotas(B, l, m_3)
m_1 = Line(B, P)
q = OrthogonalLine(M_a, m_1)
F_1 = Mirror(B, q)
r = Circle(P, F_1, M_a)
L = Center(r)
s = Segment(M_a, L)
E_1 = Mirror(F_1, s)
n_1 = Line(E_1, C)
A_1 = Intersect(n_1, m_1)
n_2 = Line(C, P)
b_1 = OrthogonalLine(M_a, n_2)
E_2 = Mirror(C, b_1)
c_1 = Circle(M_a, E_2, P)
M = Center(c_1)
d_1 = Segment(M, M_a)
F_2 = Mirror(E_2, d_1)
m_2 = Line(B, F_2)
A_2 = Intersect(n_2, m_2)
f_1 = Segment(A_1, A_2)
polígono1, c_2, a_2, b_2 = Polygon(A, B, C)
N = Midpoint(A, B)
O = Midpoint(A, C)
g_1 = Segment(E_1, B)
h_1 = Segment(C, F_2)
α = Angle(A_1, E_1, B)
β = Angle(C, F_2, A_2)
i_1 = Line(E, F)
R = Intersect(i_1, g)

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: Afinidad
Dado ABC, A'B'C' y X, determiana el punto fijo propio y la imagen de X
Input: A, B, C, A', B', C', X
Output: F, X'

polígono1, c, a, b = Polygon(A, B, C)
polígono2, f, d, e = Polygon(A', B', C')
g = Line(A, a)
h = Line(C, c)
i = Line(A', d)
j = Line(C', f)
k = Line(A, B)
l = Line(A', B')
L = Intersect(k, l)
m = Line(B, C)
n = Line(B', C')
M = Intersect(m, n)
N = Intersect(j, h)
P = Intersect(g, i)
p = Line(L, N)
q = Line(M, P)
F = Intersect(p, q)
r = Line(X, c)
U = Intersect(p, r)
s = Line(X, g)
V = Intersect(q, s)
t = Line(U, f)
a_1 = Line(V, i)
X' = Intersect(a_1, t)

Tool: Hiperbola Punto Dos Asintotas (command HiperbolaPuntoDosAsintotas)
Dar un punto y las dos asíntotas
Input: P, h_1, h_2
Output: p, P', P'_1, P_1, P_2

E = Intersect(h_1, h_2)
P' = Mirror(P, E)
b = Line(P, h_2)
c = Line(P', h_1)
d = Line(P, h_1)
e = Line(P', h_2)
F = Intersect(e, d)
G = Midpoint(P', F)
f = Line(P, G)
H = Intersect(b, c)
g = Line(H, G)
h = Line(P', g)
P_1 = Intersect(f, h)
P'_1 = Mirror(P_1, E)
J = Midpoint(G, F)
i = Line(P, J)
j = Line(H, i)
K = Intersect(h_2, j)
k = Line(K, P')
P_2 = Intersect(i, k)
p = Conic(P', P_1, P_2, P, P'_1)