1/3

$\dfrac{5-\sqrt{5}}{10}$

A = (0.5921475730130006, 3.737543779322428)
B = (0, 0)
C = (5, 0)
bc = Line(B, C)
ab = Line(A, B)
ac = Line(A, C)
AB = Segment(A, B)
BC = Segment(B, C)
AC = Segment(C, A)
M = Midpoint(B, C)
m = Point(BC)
f = Segment(A, M)
G: (A + B + C) / 3
r = AffineRatio(B, C, m)
m_0 = Dilate(C, 1 / 3, B)
m' = Mirror(m, M)
n = Dilate(A, r, C)
p = Dilate(B, r, A)
g = Segment(B, n)
h = Segment(p, C)
i = Segment(A, m)
a = Intersect(g, h)
b = Intersect(h, i)
c = Intersect(g, i)
N = Midpoint(A, C)
P = Midpoint(A, B)
D = Midpoint(B, G)
E = Midpoint(G, C)
d = Conic(P, N, M, D, E)
d' = Dilate(d, 2, G)
n' = Mirror(n, N)
p' = Mirror(p, P)
j = Segment(B, n')
k = Segment(C, p')
l = Segment(A, m')
b' = Intersect(k, l)
c' = Intersect(j, l)
a' = Intersect(j, k)
e = Conic(b, c', a, b', c)
q = Conic(p, n', n, m', m)
s = Segment(B, N)
t = Segment(C, P)
A' = Rotate(C, 60°, B)
f_1 = Segment(A', B)
g_1 = Segment(A', C)
m_1 = Dilate(C, (5 - sqrt(5)) / 10, B)
texte1: "k = " + r + ""