Cúbica asociada a una recta
Author: Angel Montesdeoca
Date: 24 Noviembre 2014

$P^\bullet$

S

A = (0.16055672160126608, -0.48284795100079125)
B = (8.180556721601263, -0.5028479510007913)
M_c = Midpoint(A, B)
C = (5.840556721601266, 3.6571520489992086)
M_b = Midpoint(C, A)
X_2 = TriangleCenter(A, B, C, 2)
G = Mirror(M_b, X_2)
H = Mirror(M_c, X_2)
M_a = Midpoint(B, C)
p_1 = Conic(M_a, M_b, M_c, H, G)
p' = Dilate(p_1, 3.1622776601683795, X_2)
M_0 = Point(p')
h = Tangent(M_0, p')
k = Circle(M_0, 2)
M = Point(h)
p = Line(M_0, M)
polígono1, c, a, b = Polygon(A, B, C)
U = Point(p)
d = Tripolar(A, B, C, U)
e = Line(B, C)
f = Line(C, A)
g = Line(A, B)
D_1 = Intersect(d, e)
E_1 = Intersect(f, d)
F_1 = Intersect(d, g)
polígono2, D, E, F = TrianguloCeviano(A, B, C, U)
D_0 = Midpoint(F, E)
E_0 = Midpoint(F, D)
F_0 = Midpoint(D, E)
D_2 = Mirror(D_1, D_0)
F_2 = Mirror(F_1, F_0)
E_2 = Mirror(E_1, E_0)
d' = Line(E_2, F_2)
V = Intersect(d, d')
lugar1 = Locus(V, U)
I = Tripolo(A, B, C, p)
P = ConjugadoIsotomico(A, B, C, I)
S = Dilate(P, -5, X_2)
texto2, lista1 = CoordenadasBaricentrias(A, B, C, P)
pp = Element(lista1, 1)
qq = Element(lista1, 2)
rr = Element(lista1, 3)
B_a = Trilinear(A, B, C, 2 * c * (pp - qq - rr) * (qq + rr), 0, a * (-pp^(2) - 4 * pp * (qq + rr) + (qq + rr)^(2)))
C_a = Trilinear(A, B, C, -2 * b * (pp - qq - rr) * (qq + rr), a * (pp^(2) + 4 * pp * (qq + rr) - (qq + rr)^(2)), 0)
i = Line(B_a, C_a)
A_b = Trilinear(A, B, C, 0, 2 * c * (pp + rr) * (pp - qq + rr), b * (qq^(2) + 4 * qq * (pp + rr) - (pp + rr)^(2)))
C_b = Trilinear(A, B, C, b * (-qq^(2) - 4 * qq * (pp + rr) + (pp + rr)^(2)), -2 * a * (pp + rr) * (pp - qq + rr), 0)
j = Line(C_b, A_b)
B_c = Trilinear(A, B, C, c * (-(pp + qq)^(2) + 4 * (pp + qq) * rr + rr^(2)), 0, 2 * a * (pp + qq) * (pp + qq - rr))
A_c = Trilinear(A, B, C, 0, c * ((pp + qq)^(2) - 4 * (pp + qq) * rr - rr^(2)), -2 * b * (pp + qq) * (pp + qq - rr))
l = Line(B_c, A_c)

Tool: Tripolar
Dar los vértices de un triángulo y un punto
Input: A, B, C, P
Output: l

d = Line(B, C)
e = Line(C, A)
f = Line(A, B)
g = Line(P, A)
h = Line(P, B)
i = Line(P, C)
E = Intersect(h, e)
F = Intersect(d, g)
j = Line(E, F)
G = Intersect(f, i)
k = Line(G, F)
H = Intersect(e, k)
I = Intersect(j, f)
l = Line(H, I)

Tool: Triangulo Ceviano (command TrianguloCeviano)
Dar ABC y P
Input: A, B, C, P
Output: polígono2, A', B', C'

d = Line(B, C)
e = Line(C, A)
f = Line(A, B)
g = Line(A, P)
h = Line(B, P)
i = Line(C, P)
A' = Intersect(d, g)
B' = Intersect(e, h)
C' = Intersect(f, i)
polígono2, f_1, d_1, e_1 = Polygon(A', B', C')

Tool: Tripolo
Dados tres puntos y una recta, determina el tripolo de la recta respecto al triángulo de vértices los tres puntos.
Input: A, B, C, m
Output: U

n = Line(A, C)
p = Line(C, B)
K = Intersect(n, m)
N = Intersect(m, p)
A_1 = Midpoint(C, B)
c_1 = Circle(A_1, B)
a_1 = Polar(N, c_1)
R = Intersect(a_1, p)
r = Line(A, R)
S = Midpoint(A, C)
s = Circle(S, A)
t = Polar(K, s)
T = Intersect(n, t)
b_1 = Line(T, B)
U = Intersect(r, b_1)

Tool: Conjugado Isotomico (command ConjugadoIsotomico)
Dar los vértices de un triángulo y un punto
Input: A, B, C, P
Output: D

$P^\bullet$

polígono1, c, a, b = Polygon(A, B, C)
E = Midpoint(b)
F = Midpoint(c)
g = Line(C, A)
h = Line(A, B)
j = Line(P, B)
k = Line(P, C)
I = Intersect(g, j)
J = Intersect(h, k)
J' = Mirror(J, F)
m = Line(C, J')
I' = Mirror(I, E)
d = Line(B, I')
D = Intersect(m, d)

Tool: puntoRazon
Punto X que divide al segmento de extremos P y Q, según una razón  dada, PX : XQ = m : n
Input: P, Q, m, n
Output: X

X = Dilate(Q, m / (m + n), P)

Tool: Coordenadas Baricentrias (command CoordenadasBaricentrias)
Dar los vértices ABC y un punto P
Input: A, B, C, P
Output: texto2, CoordBaricentricas

q_1 = Segment(B, P)
APB, b_2, a_2, p_4 = Polygon(A, P, B)
e = Line(A, C)
E_1 = Intersect(e, q_1)
CPA, a_1, c_2, p_3 = Polygon(C, P, A)
BPC, c_1, b_1, p_2 = Polygon(B, P, C)
d = Line(C, B)
p_1 = Segment(A, P)
D_1 = Intersect(d, p_1)
f = Line(A, B)
r_1 = Segment(P, C)
F_1 = Intersect(f, r_1)
xx = If(Defined[D_1], -BPC, BPC)
yy = If(Defined[E_1], -CPA, CPA)
zz = If(Defined[F_1], -APB, APB)
CoordBaricentricas: {xx, yy, zz}
texto2: "" + CoordBaricentricas + ""