Given two conics α\alpha and β\beta in the plane we call a Poncelet’s polygon for (α,β)(\alpha,\beta) a poligon which is inscribed in α\alpha and circumscribed around β\beta. Poncelet’s closure theorem (or Poncelet’s porism) states that
If there is a nn-sided Poncelet’s polygon for a pair of plane conics (α,β)(\alpha,\beta), then there are infinitely many of them and each point of α\alpha or β\beta is a vertex or tangency (respectively) of one such polygon.
A proof of this theorem is way complicated. However, I’m interested in a weaker version of it, namely when α\alpha and β\beta are confocal ellipses. Is it possible to prove this in a simple way?
I don’t know if this can help, but a nn-sided Poncelet’s polygon for a pair of confocal ellipses has maximal (r. minimal) perimeter among the nn-sided polygons inscribed in the first ellipse (r. circumscribed around the second).
Is it possible to find explicit equations for the inner ellipse given nn and the outer ellipse?