X(17)    1st NAPOLEON POINT

Equilateral triangles A'BC, AB'C, ABC' are 
erected outward from sides BC, CA, AB, 
and U, V, W are their centers.  Lines AU, BV, 
CW concur in X(17), labeled 17.

Triangle UVW, known as the Napoleon triangle,
 is equilateral.

Sketch contributed by Michael Stairs, January 2018.

Drag A, B, C.

2

2

2

17

A = (2.963246696069768, 3.8263260975922924)
B = (0.24880715836865064, 0.2070733806574729)
C = (3.652628165962115, 0.1783491527452918)
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)
C' = Intersect(c, d, 1)
k = Segment(B, C')
l = Segment(C', A)
e = Circle(A, C)
p = Circle(C, A)
B' = Intersect(e, p, 1)
m = Segment(A, B')
n = Segment(B', C)
q = Circle(B, C)
r = Circle(C, B)
A' = Intersect(q, r, 2)
s = Segment(B, A')
t = Segment(C, A')
W = Centroid(C', B, A)
V = Centroid(A, C, B')
U = Centroid(C, B, A')
g_1 = Segment(B, V)
h_1 = Segment(C, W)
i_1 = Segment(A, U)
D = Intersect(g_1, h_1)
triangle1, v, w, u = Polygon(W, U, V)

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)