Outer Napoleon Triangle

The outer Napoleon triangle is the triangle formed by the centers of externally erected equilateral triangles , , and on the sides of a given triangle . By Napoleon's theorem, it is also an equilateral triangle. It has trilinear vertex matrix

(Kimberling 1998, p. 171; typos corrected) and area

where is the area of the original triangle.

The lines connecting the vertices of the outer Napoleon triangle with the opposite vertices of the original triangle concur in the first Napoleon point, and similarly for the inner Napoleon triangle, where the corresponding lines concur in the second Napoleon point.

The circumcircle of the outer Napoleon triangle is the outer Napoleon circle.

All triangle centers of the outer Napoleon triangle correspond to the triangle centroid of the reference triangle.