$P^*$

paralell to BP

paralell to CP

$co(P,P^*)$

PK(P)

trilinear polar of P

trilinear polar of P*

$P^*$

A = (-4.059532304386977, -0.1446577074016191)
B = (18.034606669734774, -0.4309748431353799)
C = (12.240711573178727, 9.028956042074618)
polígono1, c, a, b = Polygon(A, B, C)
d = Circle(A, B, C)
P = Point(d)
e = Segment(A, P)
f = AngularBisector(B, A, C)
A'_1 = Mirror(A, f)
P' = Mirror(P, f)
e' = Segment(A'_1, P')
k = LineBisector(P', A)
M = Point(k)
l = Line(M, e')
g = Line(A, B)
h = Line(A, C)
P_b = Intersect(h, l)
P_c = Intersect(l, g)
i = Segment(P, B)
j = Segment(P, C)
m = Line(P_c, i)
n = Line(P_b, j)
A' = Intersect(m, n)
p = Line(A', P)
D = Intersect(l, p)
u = Vector(A, P')
q, E = IPPPP(u, A, B, C, P)
F = PerspectorConica(A, B, C, q)
lugar2 = Locus(F, P)
r = Tripolar(A, B, C, P)
s = Line(A, P')
t = Line(C, B)
G = Intersect(s, t)
H = ConjugadoArmonico(C, B, G)
b_1 = Line(B, s)
I = Intersect(h, b_1)
J = ConjugadoArmonico(C, A, I)
a_1 = Line(H, J)
B' = PuntoA'(B, C, A, P, l)
C' = PuntoA'(C, A, B, P, l)
X_6 = TriangleCenter(A, B, C, 6)
polígono2, D_1, K, L = TrianguloCeviano(A, B, C, X_6)
X_1 = TriangleCenter(A, B, C, 1)
polígono3, N, O, R = TrianguloCeviano(A, B, C, X_1)
A_0 = Trilinear(A, B, C, 0, -1, 1)
c_1 = Tripolar(A, B, C, X_1)
B_0 = Intersect(c_1, h)
C_0 = Intersect(g, c_1)
d_1, e_1 = Asymptote(q)
v = Vector(A, A')
X_{2086} = TriangleCenter(A, B, C, 2086)
X_{2087} = TriangleCenter(A, B, C, 2087)
X_{2088} = TriangleCenter(A, B, C, 2088)
X_{99} = TriangleCenter(A, B, C, 99)
X_{100} = TriangleCenter(A, B, C, 100)
X_{101} = TriangleCenter(A, B, C, 101)

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: IPPPP
Conica dada la dirección de un punto del infinto y cuatro puntos más
Input: u, P_1, P_2, P_3, P_4
Output: h, X

a = Line(P_4, u)
M = Point(a)
b = Line(P_1, M)
d = Line(P_1, P_2)
V = Intersect(a, d)
c = Line(P_3, P_4)
U = Intersect(b, c)
e = Line(V, U)
f = Line(P_2, P_3)
W = Intersect(e, f)
g = Line(W, u)
X = Intersect(b, g)
h = Conic(P_1, P_2, P_3, P_4, X)

Tool: Perspector Conica (command PerspectorConica)
Dar ABC y una cónica (no necesariamente circunscrita)
Input: A, B, C, d
Output: L

e = Polar(A, d)
f = Polar(B, d)
g = Polar(C, d)
I = Intersect(e, g)
h = Line(B, I)
J = Intersect(e, f)
i = Line(C, J)
L = Intersect(i, h)

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: Conjugado Armonico (command ConjugadoArmonico)
Dar dos puntos A, B y un punto sobre AB, determina D, tal que (CD;AB)=-1
Input: A, B, C
Output: D

a = Line(A, B)
O = Midpoint(A, B)
c = Circle(O, A)
b = Polar(C, c)
D = Intersect(a, b)

Tool: Punto A' (command PuntoA')
Dar ABC, P en la circunferencia circunscrita y una recta en la direccion de P*
Input: A, B, C, P, l
Output: A'

$P^\bullet$

paralell to BP

paralell to CP

d = Circle(A, B, C)
f = AngularBisector(B, A, C)
A'_1 = Mirror(A, f)
P' = Mirror(P, f)
e' = Segment(A'_1, P')
k = LineBisector(P', A)
M = Point(k)
g = Line(A, B)
h = Line(A, C)
P_b = Intersect(h, l)
P_c = Intersect(l, g)
i = Segment(P, B)
j = Segment(P, C)
m = Line(P_c, i)
n = Line(P_b, j)
A' = Intersect(m, n)

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')