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?
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