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I've found a generalisation of Napoleon's theorem to general polygons.

Take any regular $n$-gon inscribed in a circle and stretch it (in any direction) so that the circle becomes an ellipse and the $n$-gon is no longer regular. Then construct regular $n$-gons on the sides of the original $n$-gon. The centroids of these regular $n$-gons make another regular $n$-gon. This is not too hard to prove using vectors and some trigonometric identities. Is this result well-known? If so, is there a nice geometrical reason why it is true?

Hexagons

The case $n=3$ gives Napoleon's theorem because you can get any triangle by stretching an equilateral triangle. The regular $n$-gons in the picture are constructed on the outside. The result applies if they are all constructed on the inside too.

Geogebra link: https://tube.geogebra.org/m/1432065

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  • $\begingroup$ Does it work when you project a circle onto a parabola or a hyperbola? (I think it should, but maybe, you can slightly change your geogebra worksheet a bit to test my question.) $\endgroup$ – Batominovski Jul 23 '15 at 11:51
  • $\begingroup$ A quick tests suggest that the side length of the final pentagon might be invariant under rotations of the original pentagon inside a circle and depends only on the transformation applied. Have you checked this? $\endgroup$ – g.kov Jul 23 '15 at 14:34
  • $\begingroup$ Yes, that's definitely true. The final $n$-gon lies on a circle with radius $\cos\left(\frac{\pi}{n}\right) (a+1)$, where the ellipse is $x^2 + a^2 y^2 = a^2$. $\endgroup$ – octopus Jul 23 '15 at 14:48
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This should be the Napoleon-Barlotti theorem.

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