  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.
