Every triangle has an infinite sequence of regular hexagons. Move any of the three red dots to change the gray triangle to any arbitrary triangle. This first theorem says there is an infinite sequence of regular hexagons intimately associated with each triangle, and centered on it (its centroid). Some of the hexagons you might think would be in the sequence aren't. Only those that are 2n3m times as large as the two smallest hexagons are in the sequence, for nonnegative integers n, m. You can also move the green point along one edge of the triangle. This changes the parameterization of the hexagons. See paper for full details, such as how this theorem is a generalization of Napoleon's Theorem. An even prettier theorem.
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