Another generalisation of Napoleon's theorem

Abstract

First let us remind ourselves of the original (classical) theorem. Starting from any triangle, adjoin to each of its edges an equilateral triangle. By this we mean construct three equilateral triangles each of which has an edge in common with the original triangle. Clearly there are two ways to adjoin an equilateral triangle: either outwardly (in which the centres of the original triangle and the equilateral triangle lie on opposite sides of their common edge) or inwardly if the centres lie on the same side of the edge. Napoleon’s theorem states that if all the equilateral triangles are adjoined outwardly (Fig. 1(a)), or inwardly (Fig. 1(b)), then their centres are vertices of another equilateral triangle (a Napoleon triangle). Most people find this result extremely surprising. It seems that the symmetries of the Napoleon triangles have “mysteriously” appeared from nowhere!