Suppose ABC is a triangle. Let points D,E,F be points, as in the figure, for which the three triangles DBC, CAE, ABF are equilateral. Let

G = center of triangle DBC,
H = center of triangle CAE,
I = center of triangle ABF

The lines AG, BH, CI meet in a point. Labeled N, it is called the first Napoleon point.

Trilinear coordinates for N are csc(A + π/6): csc(B + π/6): csc(C + π/6).

If the three equilateral triangles point inward instead of away from triangle ABC, the three lines AG, BH, CI meet in the second Napoleon point, with trilinears csc(A - π/6): csc(B - π/6): csc(C - π/6).


Dao Thanh Oai, Equilateral Triangles and Kiepert Perspectors in Complex Numbers, Forum Geometricorum 15 (2015) 105-114.

Dao Thanh Oai, A family of Napoleon triangles associated with the Kiepert configuration, The Mathematical Gazette 99 (March 2015) 151-153.

John Rigby, "Napoleon revisited," Journal of Geometry 33 (1988) 129-146.

Encyclopedia of Triangle Centers, X(17) and X(18).

Biographical Sketch of Napoleon
Triangle Centers
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