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).

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).

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Coxeter, H. S. M.
and Greitzer, S. L. Geometry
Revisited. Washington, DC: Math. Assoc. Amer., pp. 60-65, 1967.Eddy,
R. H. and Fritsch, R. "The Conics of Ludwig Kiepert: A Comprehensive Lesson
in the Geometry of the Triangle." Math. Mag.67, 188-205, 1994.Pappas,
T. "Napoleon's Theorem." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 57, 1989.Schmidt,
F. "200 Jahre französische Revolution--Problem und Satz von Napoleon."
Didaktik der Mathematik19, 15-29, 1990.Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 74-75 and 156-158, 1991.Wentzel, J. E. "Converses
of Napoleon's Theorem." Amer. Math. Monthly99, 339-351, 1992.