Variational aspects of Poncelet's theorem (Q1337103)

If \(E\) is an ellipse, and \(M\) is a point on \(E\), it is known that there is exactly one \(n\)-gon, inscribed in \(E\) and having \(M\) as one vertex, whose length is maximal. Chasles proved that every maximal polygon is circumscribed about an ellipse \(E'\) having the same foci as \(E\), and that all maximal \(n\)-gons have the same length (that is, the length is independent of \(M\)). In order to prove this, Chasles had to rely on a previous proof of Poncelet's theorem. There is also the `dual' problem. Let \(D\) be a tangent to \(E\); then there is exactly one \(n\)-sided convex polygon, circumscribed about \(E\) and having a side lying on \(D\), whose length is minimal. Darboux proved that every minimal polygon is inscribed in an ellipse \(E'\) with the same foci as \(E\), and all minimal \(n\)-gons have the same length, but again he had to rely on a previous proof of Poncelet's theorem. The author of the present paper proves both these results directly.