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

**References:**

**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

Clark Kimberling Home Page