X(18)    2nd NAPOLEON POINT

Let A'BC, AB'C, ABC' with centers U, V, W be the inward
equilateral triangles on sides BC, CA, AB  (cf X(17), using
outward equilateral triangles). The lines AU, BV, CW concur 
in X(18), labeled 18.

Sketch contributed by Michael Stairs, January 2018.

Drag A, B, C.

18

A = (2.4226088615798025, 3.600344237719717)
B = (0.24880715836865064, 0.2070733806574729)
C = (2.719090758816228, 0.2070733806574729)
f = Segment(B, A)
g = Segment(A, C)
h = Segment(C, B)
i = AngularBisector(B, C, A)
j = AngularBisector(C, B, A)
c = Circle(B, A)
d = Circle(A, B)
e = Circle(A, C)
p = Circle(C, A)
q = Circle(B, C)
r = Circle(C, B)
k = Circle(B, A)
s = Circle(A, B)
D = Intersect(k, s, 2)
l = Segment(A, D)
m = Segment(D, B)
t = Circle(A, C)
c_1 = Circle(C, A)
B' = Intersect(t, c_1, 2)
n = Segment(A, B')
a = Segment(B', C)
d_1 = Circle(B, C)
e_1 = Circle(C, B)
A' = Intersect(d_1, e_1, 1)
b = Segment(B, A')
f_1 = Segment(A', C)
G = Midpoint(A', C)
H = Midpoint(A', B)
I = Midpoint(B, C)
g_1 = Segment(A', I)
h_1 = Segment(G, B)
i_1 = Segment(H, C)
U = Intersect(g_1, h_1)
K = Midpoint(A, C)
L = Midpoint(C, B')
M = Midpoint(B', A)
j_1 = Segment(L, A)
k_1 = Segment(K, B')
l_1 = Segment(M, C)
V = Intersect(j_1, k_1)
E = Midpoint(A, D)
F = Midpoint(D, B)
J = Midpoint(B, A)
m_1 = Segment(D, J)
n_1 = Segment(E, B)
p_1 = Segment(F, A)
W = Intersect(m_1, n_1)
q_1 = Segment(V, W)
r_1 = Segment(W, U)
s_1 = Segment(U, V)
t1, w, v, u = Polygon(V, U, W)
t_1 = Line(U, A)
a_1 = Line(V, B)
b_1 = Line(W, C)
N = Intersect(t_1, a_1)

Tool: Centroid
Input: A, B, C
Output: D

2

f = Segment(B, A)
g = Segment(A, C)
C' = Midpoint(f)
B' = Midpoint(g)
k = Segment(C', C)
l = Segment(B, B')
D = Intersect(k, l)