NAPOLEON POINTS

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 + pi/6): csc(B + pi/6): csc(C + pi/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 - pi/6): csc(B - pi/6): csc(C - pi/6).

For a geometric discussion, see

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

Biographical Sketch of Napoleon
Triangle Centers
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