A

B

C

A'

L_{CA}

L_{AB}

L_{BC}

L'_{BC}

L'_{AB}

L'_{CA}

Let F be X(13) or X(14). Let L1 and L2 be lines through F 
such that the angle between them is π/3. 
Let L_{BC} = L ∩ BC, and define L_{CA} and L_{AB} cyclically. 
Let L'_{BC} = L' ∩ BC, and define L'_{CA} and L'_{AB} cyclically. 
The lines L_{BC}L'_{CA}, L_{CA}L'_{AB}, L_{AB}L'_{BC} concur. 
(Dao Thanh Oai, 2014)

A = (21.399041354394686, -1.825322772508253)
B = (21.3609368130883, -1.9532696932536886)
f = Line(A, B)
C = (21.558104272224657, -1.9482027191136195)
g = Line(B, C)
h = Line(C, A)
M = TriangleCenter(A, B, C, 13)
t1, c, a_1, b = Polygon(A, B, C)
D = (21.525233751151486, -2.0406268791650417)
i = Line(M, D)
i' = Rotate(i, 60°, M)
E = Intersect(h, i)
F = Intersect(g, i')
j = Line(E, F)
G = Intersect(h, i')
H = Intersect(f, i)
k = Line(G, H)
I = Intersect(g, i)
J = Intersect(f, i')
l = Line(I, J)
α = Angle(I, M, G)
K = Intersect(j, k)