Parallelograms inscribed in a pair of confocal ellipses
In [Am. Math. Mon. 114, No. 10, 909--914 (2007; Zbl 1140.51010)], \textit{A. Connes} and \textit{D. Zagier} proved the following result for a pair of confocal ellipses \(E\) and \(E'\): Given any diameter \(AC\) of \(E\), there is a unique diameter \(BD\) of \(E'\), such that the parallelogram \(ABCD\) has maximum girth, and the value of this girth is independent of the diameter \(AC\). The original proof relies on analytical ideas, while the present work is devoted to provide a purely elementary geometrical proof. In passing, the authors introduce some other interesing results such as the Coxeter-Greitzer lemma or a generalization of a classical theorem by Monge.
scientific article; zbMATH DE number 7225566