Regular polygons inscribed in an ellipse

Question

A regular $n$-gon is inscribed in an ellipse which is not a circle. what are the possible values for $n$? I know I can inscribe a square or even a equilateral triangle, but can we do it for all $n$? I think the answer is negative, but how to prove it?

Answer 1

A conic is fixed by five points in general position. Since the only conic through five (or more) concyclic points is a circle, the only regular $n$-agons that can be inscribed in a ellipse are a triangle or a square.

Answer 2

Five coplanar points (not in a line) determine a conic curve. Thus, for $n\ge 5$ the only conic is the circumscribed circle.