# Napoleon's Theorem by Plane Tessellation

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The given triangle is the one at whose vertices I placed yellow dots. (These are draggable so that you must be able to modify the triangle.) It appears that the triangle and the attached Napoleon's triangles form a starting configuration for a simple tessellation of the plane. Napoleon's triangles have different backgrounds. To clarify the argument, pick two colors and erase all other triangles. Then it becomes practically obvious that the centers of triangles of a single color form a *hexagonal lattice* (black lines).

Centers of all other triangles lie at the centers of the lattice triangles. In the diagram, all such points are connected by greenish lines. The greenish lines form a finer lattice (a lattice with smaller base region.) Centers of all Napoleon's triangles lie at the nodes of the latter lattice. This proves the theorem.

### 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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