If equilateral triangles 
, 
, and 
are erected externally on the sides of any triangle 
, then their centers 
, 
, and 
, respectively, form an equilateral
triangle (the outer Napoleon triangle)
. An additional
property of the externally erected triangles also attributed to Napoleon is that
their circumcircles concur in the first Fermat point 
(Coxeter 1969, p. 23; Eddy and
Fritsch 1994). Furthermore, the lines 
, 
, and 
connecting the vertices of 
with the opposite vectors of the erected triangles
also concur at 
.
This theorem is generally attributed to Napoleon Bonaparte (1769-1821), although it has also been traced back to 1825 (Schmidt 1990, Wentzel 1992, Eddy and Fritsch 1994).
Analogous theorems hold when equilateral triangles 
, 
, and 
are erected internally on the sides of
a triangle 
. Namely, the inner
Napoleon triangle 
is equilateral,
the circumcircles of the erected triangles intersect in the second
Fermat point 
,
and the lines connecting the vertices 
, 
, and 
concur at 
.
Amazingly, the difference between the areas of the outer and inner Napoleon triangles equals the area of the original triangle (Wells 1991, p. 156).
Drawing the centers of one equilateral triangle inwards and two outwards gives a 
-
-
triangle (Wells 1991,
p. 156).
Napoleon's theorem has a very beautiful generalization in the case of externally constructed triangles: If similar triangles of any shape are constructed externally on a triangle such that each is rotated relative to its neighbors and any three corresponding points of these triangles are connected, the result is a triangle which is similar to the external triangles (Wells 1991, pp. 156-157).
Napoleon's theorem is related to van Aubel's theorem and is a special case of the Petr-Neumann-Douglas theorem.
