**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 2^{n}3^{m}
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.

**
Don't give up. **
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