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.
Copyright © 1996-2018 Alexander Bogomolny