A generalisation of Napoleon's theorem. Is this result original?

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.

• 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 $x^2 + a^2 y^2 = a^2$. – octopus Jul 23 '15 at 14:48