On the Optimality of Napoleon Triangles

TL;DR

This paper explores the properties of Napoleon's theorem, demonstrating that iterative applications of inner and outer transformations produce triangles with optimality characteristics, including convergence to the centroid and minimal sum of squared distances.

Contribution

It reveals that two Napoleon iterations lead to a degenerate triangle at the centroid, while two outer iterations produce an equilateral triangle closest to the original in a least-squares sense.

Findings

Inner transformations converge to the centroid.

Outer transformations produce the closest equilateral triangle.

Two outer transformations minimize the sum of squared distances.

Abstract

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.