## 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.

### Napoleon's Theorem

- Napoleon's Theorem
- A proof with complex numbers
- A second proof with complex numbers
- A third proof with complex numbers
- Napoleon's Theorem, Two Simple Proofs
- Napoleon's Theorem via Inscribed Angles
- A Generalization
- Douglas' Generalization
- Napoleon's Propeller
- Napoleon's Theorem by Plane Tessellation
- Fermat's point
- Kiepert's theorem
- Lean Napoleon's Triangles
- Napoleon's Theorem by Transformation
- Napoleon's Theorem via Two Rotations
- Napoleon on Hinges
- Napoleon on Hinges in GeoGebra
- Napoleon's Relatives
- Napoleon-Barlotti Theorem
- Some Properties of Napoleon's Configuration
- Fermat Points and Concurrent Euler Lines I
- Fermat Points and Concurrent Euler Lines II
- Escher's Theorem
- Circle Chains on Napoleon Triangles
- Napoleon's Theorem by Vectors and Trigonometry
- An Extra Triple of Equilateral Triangles for Napoleon
- Joined Common Chords of Napoleon's Circumcircles
- Napoleon's Hexagon
- Fermat's Hexagon
- Lighthouse at Fermat Points
- Midpoint Reciprocity in Napoleon's Configuration
- Another Equilateral Triangle in Napoleon's Configuration
- Yet Another Analytic Proof of Napoleon's Theorem
- Leo Giugiuc's Proof of Napoleon's Theorem
- Gregoie Nicollier's Proof of Napoleon's Theorem
- Fermat Point Several Times Over

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