On the Optimality of Napoleon Triangles

An elementary geometric construction known as Napoleon's theorem produces an equilateral triangle built on the sides of any initial triangle: the centroids of each equilateral triangle meeting the original sides, all outward or all inward, comprise the vertices of the new equilateral triangle. In this note we observe that two Napoleon iterations yield triangles with useful optimality properties. Two inner transformations result in a (degenerate) triangle whose vertices coincide at the original centroid. Two outer transformations yield an equilateral triangle whose vertices are closest to the original in the sense of minimizing the sum of the three squared distances.