## Napoleon's Theorem Grégoire Nicollier's Proof

### Grégoire Nicollier 13 October 2016

The simplest and deepest proof of Napoleon's theorem remains the following. Every triangle is the affine image of a positively oriented equilateral triangle, hence the sum $P + N\;$ of two equilateral triangles: $P\;$ positively and $N\;$ negatively oriented. The erection of right-hand (left-hand) isosceles ears with apex angle $120^{\circ}\;$ on the sides of a triangle is a linear operation. This operation kills $N(P)\;$ and rotates $P(N)\;$ by $60^{\circ}.$

For the framework of spectral theory that covers linear transformations, check Grégoire Nicollier's article, or its application to planar quadrilaterals, with an interactive illustration.