In 1992 Schäfke and Volkmer showed that there is at most a finite number of values of excentricity for which the curve may exist. They outlined a feasible strategy for a computer-assisted proof. Their method consists of obtaining extremely accurate approximations to the hypothetical curve. In 1996 Rychlik fully solved the problem. Rychlik's proof Marek Rychlik's proof was published in the hard to read article. There is also an easy to read, freely available on-line, research announcement article, but it only hints at the ideas used in the proof. The proof does not use a computer. Instead it introduces a complexification of the original problem, and develops a generalization of the theory of normally hyperbolic invariant curves and stable manifolds to multi-valued maps . This method allows the use of global methods of complex analysis. The prototypical global theorem is the Liouville's theorem. Another global theorem is Chow's theorem. The global method was used in the proof of Ushiki's Theorem. See also Similar problems and their generalizations have also been studied. The equireciprocal point problem The general chordal problem of Gardner Equiproduct point problem References Convex geometry Dynamical systems