A

B

C'

A'

B'

C

B_a

C_a

C_b

A_b

A_c

B_c

X(3602) 
Let ABC be a triangle, and let A'B'C' be the 2nd Morley triangle. Let Ba, Ca be points on BC such that
A'BaCa is an equilateral triangle having the same orientation as ABC.  
Define Cb, Ab, Ac, Bc cyclically. Let A'', B'', C'' be the midpoints of AbAc, BcBa, CaCb respectively. 
The triangle A''B''C'' is an equilateral triangle perspective to ABC and homothetic to the Morley triangle. 

A = (21.425440768011853, -1.7410547044991591)
B = (21.30963927725956, -1.955460996976369)
f = Line(A, B)
C = (21.695434343606955, -2.0701119207206657)
g = Line(B, C)
h = Line(C, A)
t1, c, a, b = Polygon(A, B, C)
α, β, γ = InteriorAngles(t1)
g' = Rotate(g, 60° + β / 3, B)
f' = Rotate(f, (-(60° + β / 3)), B)
f'_1 = Rotate(f, 60° + α / 3, A)
g'_1 = Rotate(g, (-(60° + γ / 3)), C)
h' = Rotate(h, (-(60° + α / 3)), A)
h'_1 = Rotate(h, 60° + γ / 3, C)
D = Intersect(g', g'_1)
E = Intersect(h', h'_1)
i = Segment(D, E)
F = Intersect(f', f'_1)
j = Segment(E, F)
k = Segment(F, D)
l = Segment(A, B)
m = Segment(B, C)
n = Segment(C, A)
p = OrthogonalLine(D, g)
q = OrthogonalLine(F, f)
r = OrthogonalLine(E, h)
p' = Rotate(p, 30°, D)
q' = Rotate(q, 30°, F)
r' = Rotate(r, 30°, E)
G = Intersect(g, p')
G' = Mirror(G, p)
t2, g_1, d, g'_2 = Polygon(D, G', G)
H = Intersect(f, q')
H' = Mirror(H, q)
t3, f_1, h'_2, h_1 = Polygon(H', H, F)
I = Intersect(h, r')
I' = Mirror(I, r)
t4, i', i_1, e = Polygon(I, E, I')
J = Midpoint(I, H')
K = Midpoint(G', H)
L = Midpoint(I', G)
t5, l_1, j_1, k_1 = Polygon(J, K, L)
s = Segment(J, K)
t = Segment(K, L)
m_1 = Segment(L, J)
n_1 = Line(J, A)
q_1 = Line(L, C)
r_1 = Line(K, B)
M = Intersect(n_1, q_1)