Poncelet's theorem and dual billiards
A short proof, based on a construction of a special area form and a dual billiard transformation, of the classical Poncelet's theorem is established. The statement is as follows. Given two nested ellipses \(\Gamma_ 0\) and \(\Gamma_ 1\), \(\Gamma_ 0 \subset \text{interior}(\Gamma_ 1)\) we choose a point \(x \in \Gamma_ 1\), draw a tangent line to \(\Gamma_ 0\) from \(x\), find its intersection \(y\) with \(\Gamma_ 1\) and iterate, taking \(y\) as a new starting point. The claim is that, if \(x\) returns back to the initial position after a finite number of iterations, then every point of \(\Gamma_ 1\) will return back after the same number of iterations.
scientific article; zbMATH DE number 597687