Gábor Kertész - The Dido problem on planes of constant curvature

What is the tract of land of maximum area that can be fenced by a given collection of fence segments? The segments should be the sides of a chord polygon. The proof is simple if the segments can not cross. However the proof of the general case was unknown for decades. Fifteen years ago I found a proof. Almost the same considerations show that this so called Dido type theorem for segments holds even in the hyperbolic plane.