Napoleon's theorem states that 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). The lines 
, 
, and 
join each vertex of 
to the third vertex
of the equilateral triangle erected on the
opposite side. These three lines also concur at 
.
This theorem is generally attributed to Napoleon Bonaparte (1769-1821), although it has also been traced back to 1825 (Schmidt 1990, Wetzel 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 corresponding lines 
, 
, 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).
A generalization allows an arbitrary nondegenerate shape for the three exterior triangles. On 
, 
, and 
, erect 
, 
, and 
, respectively, so that these ordered triples of vertices give directly similar triangles. Any three corresponding points
of the erected triangles either coincide or form a triangle directly similar
to them (Wells 1991, pp. 156-157; de Villiers and Meyer 1995).
Dao Thanh Oai (2015a) extended the equilateral construction to a hexagon 
. Construct equilateral
triangles on the same side of each directed side and let 
be the triangle centroid
of the 
th
such triangle. The midpoints
of 
, 
, and 
form an equilateral
triangle or coincide. If the midpoints of 
, 
, and 
form a nondegenerate triangle,
the first three midpoints are the triangle
centroids in Napoleon's construction on that triangle.
Its circumcenter is a perspector
of the inner Napoleon triangle and outer
Napoleon triangle when both are nondegenerate.
For a hexagon 
, construct equilateral
triangles 
, 
, and 
on alternate sides using a consistent orientation. The triangle centroids of 
, 
, and 
form an equilateral
triangle or coincide, as do the triangle centroids
of 
, 
, and 
(Dao Thanh Oai 2015c). A third such triple comes from 
, 
, and 
(Humenberger et al. 2023).
In another generalization, construct three similar isosceles triangles with base angle 
consistently on the sides of 
, with 
, 
, and 
the apices opposite 
, 
, and 
. Take 
, with 
and 
defined cyclically. For either finite value 
, 
, 
, and 
form an equilateral
triangle or coincide (Dao Thanh Oai 2015b).
Let 
be either Fermat point of 
, 
a point of its Kiepert
hyperbola distinct from 
, and 
a point on 
. The line through 
perpendicular to 
meets 
at 
. Defining 
and 
cyclically gives an equilateral
triangle 
when the three intersections are unique and distinct
(Dao Thanh Oai 2018).
Napoleon's theorem is related to van Aubel's theorem and is a special case of the Petr-Neumann-Douglas theorem.
