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If Equilateral Triangles are erected externally on the sides of any Triangle, then the centers form an Equilateral Triangle (the outer Napoleon Triangle). Furthermore, the inner Napoleon Triangle is also Equilateral and the difference between the areas of the outer and inner Napoleon triangles equals the Area of the original Triangle.
See also Napoleon Points, Napoleon Triangles
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 60-65, 1967.
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 Mathematik 19, 15-29, 1990.
Wentzel, J. E. ``Converses of Napoleon's Theorem.'' Amer. Math. Monthly 99, 339-351, 1992.