John Carlos Baez on Nostr: A theorem you'll never forget: Napoleon's theorem. Take any triangle. Erect ...

A theorem you'll never forget: Napoleon's theorem.

Take any triangle. Erect equilateral triangles on its three sides. The centers of these triangles are the vertices of another equilateral triangle!

While traditionally attributed to Napoleon, who *was* interested in geometry, there's no good evidence for this. The result first appears in 𝑇h𝑒 𝐿𝑎𝑑𝑖𝑒𝑠' 𝐷𝑖𝑎𝑟𝑦 in 1824, posed as a puzzle by one "Mr. W. Rutherford of Woodburn". William Rutherford is famous for having computed pi to 208 digits, of which the first 152 were correct.

This proof claims to be "geometrical" but is strategy is algebraic:

https://www.slideserve.com/hogan/a-geometric-proof-of-napoleon-s-theorem

Find a formula for the length of one side of the would-be equilateral triangle that depends symmetrically on the lengths of all 3 sides of the original triangle. This implies all sides of the would-be equilateral triangle have the same length! So it's indeed equilateral.

Napoleon's theorem has an excellent generalization to n-gons: the Napoleon-Barlotti theorem. The centers of regular n-gons constructed over the sides of an n-gon P form a regular n-gon if and only if P is an affine image of a regular n-gon! Roughly speaking, this means P is a regular n-gon that's been stretched or squashed or sheared in a linear way.

Since every triangle is an affine image of an equilateral triangle, this reduces to Napoleon's theorem when n = 3.

I wonder how Barlotti felt about sharing a result with the Emperor of the French.

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