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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.

See also

Equilateral Triangle, Fermat Points, Inner Napoleon Triangle, Kiepert Hyperbola, Napoleon Points, Outer Napoleon Triangle, Petr-Neumann-Douglas Theorem, Similar, van Aubel's Theorem

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References

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.Dao Thanh Oai. "Equilateral Triangles and Kiepert Perspectors in Complex Numbers." Forum Geom. 15, 105-114, 2015a.Dao Thanh Oai. "99.09 A Family of Napoleon Triangles Associated with the Kiepert Configuration." Math. Gaz. 99, 151-153, 2015b. https://doi.org/10.1017/mag.2014.22.Dao Thanh Oai. "Two Generalizations of the Napoleon Theorem." Apr. 2, 2015c. https://dynamicmathematicslearning.com/Two-generalizations-of-Napoleon-theorem.pdf.Dao Thanh Oai. "Some New Equilateral Triangles in a Plane Geometry." Global J. Adv. Res. Class. Mod. Geom. 7, 73-91, 2018. https://geometry-math-journal.ro/pdf/Volume7-Issue2/5.pdf.de Villiers, M. D. and Meyer, J. H. "A Generalized Dual of Napoleon's Theorem and Some Further Extensions." Int. J. Math. Educ. Sci. Technol. 26, 233-241, 1995. https://doi.org/10.1080/0020739950260208.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.Humenberger, H.; Schuppar, B.; and de Villiers, M. "Geometric Proofs and Further Generalizations of Dao Than Oai's Napoleon Hexagon Theorem." Global J. Adv. Res. Class. Mod. Geom. 12, 158-168, 2023. https://geometry-math-journal.ro/pdf/Volume12-Issue1/10.pdf.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.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 74-75 and 156-158, 1991.Wetzel, J. E. "Converses of Napoleon's Theorem." Amer. Math. Monthly 99, 339-351, 1992.

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Napoleon's Theorem

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Weisstein, Eric W. "Napoleon's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NapoleonsTheorem.html

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