Thread by @AlgebraFact

Napolean's theorem

Start with any triangle and draw equilateral triangles on each side. Then connect the centroids of the added triangles. The resulting triangle will be equilateral.

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The added triangles don't have to face outward. They could also face inward, though this makes a messier image.

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Napolean's theorem starts with an arbitrary triangle and adds equilateral triangles on the sides.

So a natural question is what happens if you start with a quadrilateral and draw squares on each side. If you connect the centers of the squares, do you get a square?

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No, but Van Aubel’s theorem says that if you connect the centers of squares on opposite faces of the quadrilateral (dashed lines below), the two line segments are the same length and perpendicular to each other.

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There is another variation on Napolean's theorem discovered recently. It takes a little more than 280 characters to state, so I'll link to the statement. You can get an idea of the theorem from the illustration.



5/n johndcook.com/blog/2023/02/2…

This thread was a condensed version of blog posts. See the posts for more information and references.

Napolean's theorem
johndcook.com/blog/2022/11/3…

Van Aubel's theorem
johndcook.com/blog/2023/01/2…

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