On the Fermat problem for ellipse (Q2837903)

In the real Euclidean plane, the authors discuss the following problem which originates in a question of Pierre Fermat (probably 1607--1665):NEWLINENEWLINE(Fermat Problem.) Let \(P\) be a point on the semicircle that has the top side \(AB\) of the rectangle \(ABB'A'\) as diameter. Let \(|AB|:|AA'|=\sqrt{2}\). Let the segments \(PA'\) and \(PB'\) intersect the side \(AB\) in the points \(C\) and \(D\). Then \(|AD|^2+|BC|^2=|AB|^2\).'NEWLINENEWLINEThe authors adapt a proof of Lionnet where \(P\) is allowed to be on the complete circle with diameter \(AB\). They extend Fermat's problem such that the point \(P\) varies on an ellipse having \(AB\) as principal diameter. Furthermore, 31 (thirty one) invariants of the extended Fermat configuration are exhibited.