This theorem is usually intended for equilateral triangles built outside of the base triangle,
but it also holds if the three triangles are built inward.
Napoléon's theorem
is one of the most rediscovered
results of elementary euclidean geometry. The French ruler
Napoléon Bonaparte
(17691821) certainly had the mathematical ability to discover this for himself,
but there's no evidence that he did so.
The theorem first appeared in print in 1825, in an article written for
The Ladies' Diary
by Dr. W. Rutherford.
It may well have been Rutherford himself who decided to name this theorem after the
recently deceased French emperor Napoléon I.
One easy way to prove this is to observe that properly rotating the figure by angles
of ± p/3 (successively)
about the centers of two of the equilateral triangles
brings the center of the third back to its original position.
This establishes the equality of two sides of the triangle
formed by the centers of the 3 equilateral triangles.
Since the same argument holds with any particular choice among such centers,
the aforementioned triangle is necessarily equilateral.
Napoleon Tiling :
Napoleon's theorem can be made visually obvious with a periodic tiling of the plane
like the one which serves as the background for this page.
The black triangles are congruent scalene triangles in three orientations.
The 3 families of equilateral
triangles are represented with 3 different colors.
Mathpages

CuttheKnot

MathWorld

Wikipedia
GeoGebra Institute
of Hong Kong (Napoleon Tiling)