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Problema 745 .

Dado un triángulo ABC, hallar una construcción con regla y compás
de una paralela A´B´ a AB tal que la distancia entre AB y A´B´ sea el doble del
radio de la circunferencia inscrita a A´B´C.

$(\mathcal H)$

C_{-k}

M_{-k}

D_{-k}

Z_{-k}

\ovalbox{\overrightarrow{AD_k}= \frac{a+b+c}{(1+k)a+b+c }\overrightarrow{AD}\

\overrightarrow{AD_{-k}}= \frac{a+b+c}{(1-k)a+b+c }\overrightarrow{AD}}

Animar Z ó k

U = (1, 0)
O = (0, 0)
j_1 = Line(O, U)
K = Point(j_1)
k: x(K)
C = (7.712384392615375, 5.323956998261902)
B = (2.6766234461214093, 5.330805817879793)
A = (6.3399004913498365, 9.671344347739339)
q = CircunferenciaInscrita(A, B, C)
I = Center(q)
polígono1, c, a, b = Polygon(A, B, C)
d = Line(A, C)
e = Line(A, B)
Z = Point(d)
f = Line(Z, a)
Y = Intersect(e, f)
g = CircunferenciaInscrita(A, Z, Y)
I_0 = Center(g)
h = OrthogonalLine(I_0, f)
D_0 = Intersect(h, f)
i = Segment(I_0, D_0)
j = OrthogonalLine(Z, f)
a_1 = Line(B, C)
Z_0 = Intersect(j, a_1)
l = Segment(Z, Z_0)
m = Line(I_0, Z)
n = Line(D_0, Z_0)
M = Intersect(m, n)
p = AngularBisector(B, A, C)
o = Distance(i, l)
C_k = puntoRazon(I, C, -1, k)
r = Line(A, C_k)
E = Midpoint(A, C)
a_2 = Line(C, I)
G = PuntoEnHiperbola(A, B, C, E)
t = ConicaPtPtP(G, p, A, a_1, C)
H = Midpoint(A, E)
J = Midpoint(E, C)
K1 = PuntoEnHiperbola(A, B, C, H)
L = PuntoEnHiperbola(A, B, C, J)
M_k = SegundoPuntoRectaConica(A, C, G, K1, L, r)
b_1 = Line(M_k, a_2)
B_a = Intersect(b_1, d)
c_1 = Line(B_a, a)
C_a = Intersect(e, c_1)
d_1 = CircunferenciaInscrita(A, C_a, B_a)
I' = Intersect(b_1, p)
s = OrthogonalLine(B_a, a)
e_1 = OrthogonalLine(I', a)
D_k = Intersect(c_1, e_1)
F = Intersect(a_1, s)
ρ_a = Segment(I', D_k)
d_a = Segment(B_a, F)
h_1 = Segment(M_k, F)
i_1 = Segment(C_a, B_a)
Z_k = puntoRazon(I_0, Z, -1, k)
texto2: "k=" + k + ""
C'_k = puntoRazon(I, C, 1, k)
f_1 = Line(A, C'_k)
M'_k = SegundoPuntoRectaConica(A, C, G, K1, L, f_1)
g_1 = Line(M'_k, a_2)
B'_a = Intersect(d, g_1)
k_1 = Line(B'_a, a)
C'_a = Intersect(e, k_1)
p_1 = CircunferenciaInscrita(A, C'_a, B'_a)
Q = Intersect(g_1, p)
l_1 = OrthogonalLine(Q, a)
m_1 = OrthogonalLine(B'_a, a)
R = Intersect(a_1, m_1)
S = Intersect(k_1, l_1)
ρ'_a = Segment(Q, S)
d'_a = Segment(R, B'_a)
n_1 = OrthogonalLine(I, a)
D = Intersect(a_1, n_1)
q_1 = Line(A, D)
texto3: "\frac{d_a}{\rho_a} = \frac{" + (LaTeX[d_a]) + "}{" + (LaTeX[ρ_a]) + "} =\frac{d'_a}{\rho'_a} = \frac{ " + (LaTeX[d'_a]) + "}{ " + (LaTeX[ρ'_a]) + "} = " + (LaTeX[k]) + ""
Z'_k = puntoRazon(I_0, Z, 1, k)

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: Circunferencia Inscrita (command CircunferenciaInscrita)
Select 3 points
Input: A, B, C
Output: k

poly1, c, a, b = Polygon(A, B, C)
f = AngularBisector(C, A, B)
h = AngularBisector(A, B, C)
D = Intersect(f, h)
i = OrthogonalLine(D, c)
I = Intersect(c, i)
k = Circle(D, I)

Tool: Punto en Conica:  P, tangente en Q, tangente en R y punto en QR. (command PuntoenConicaPtPtP)
Dar un punto P, una tangente en Q, una tangente R y un punto en QR.
Input: P, t_1, P_1, t_2, P_2, E
Output: X

P_2 = Point(t_2)
P_1 = Point(t_1)
a = Line(P_2, P_1)
E = Point(a)
b = Line(P, P_1)
c = Line(P, E)
F = Intersect(t_2, b)
RectaPascal = Line(F, E)
G = Intersect(t_1, RectaPascal)
e = Line(G, P_2)
X = Intersect(c, e)

Tool: Conica PtPtP (command ConicaPtPtP)
Dar P, tangente en Q y tangente en R.
Input: P, t_1, P_1, t_2, P_2
Output: d

P_2 = Point(t_2)
P_1 = Point(t_1)
H = Mirror(P_1, P_2)
I = Mirror(P_2, P_1)
J = PuntoenConicaPtPtP(P, t_1, P_1, t_2, P_2, H)
K = PuntoenConicaPtPtP(P, t_1, P_1, t_2, P_2, I)
d = Conic(P_2, P_1, P, K, J)

Tool: Segundo Punto Recta Conica (command SegundoPuntoRectaConica)
Dar ABCDE y recta por A
Input: A0, B0, C0, D0, E0, b
Output: A'

e = Line(A0, C0)
f = Line(B0, E0)
P0 = Intersect(e, f)
g = Line(B0, D0)
Q0 = Intersect(b, g)
h = Line(C0, D0)
i = Line(P0, Q0)
R0 = Intersect(h, i)
j = Line(E0, R0)
A' = Intersect(b, j)

Tool: Punto En Hiperbola (command PuntoEnHiperbola)
Dar ABC y punto en AB
Input: A, B, C, Y
Output: M

polígono1, c, a, b = Polygon(A, B, C)
d = Line(A, C)
e = Line(A, B)
f = Line(Y, a)
Z = Intersect(e, f)
g = CircunferenciaInscrita(A, Y, Z)
I' = Center(g)
h = OrthogonalLine(I', f)
F_1 = Intersect(h, f)
j = OrthogonalLine(Y, f)
k = Line(B, C)
E_1 = Intersect(j, k)
m = Line(I', Y)
n = Line(F_1, E_1)
M = Intersect(m, n)

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)