Napoleon’s Theorem

Take any generic triangle, and construct equilateral triangles on each side whose side lengths are the same as the length of each side of the original triangle.

Surprise: the centers of the equilateral triangles form an equilateral triangle!

Presentation Suggestions:
Show the truth of the statement using some extreme cases for the initial triangle (a particularly instructive example is a triangle with one sidelength very close to zero).

The Math Behind the Fact:
This theorem is credited to Napoleon, who was fond of mathematics, though many doubt that he knew enough math to discover it!

How to Cite this Page:
Su, Francis E., et al. “Napoleon’s Theorem.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Michael Moody

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