There are a variety of proofs of the classical theorem about cyclic quadrilaterals (i.e. quadrilaterals whose vertices lie on a circle) that is attributed to Ptolemy. The theorem states that the product of the lengths of the diagonals of a cyclic quadrilateral is equal to the sum of the product of the sidelengths of opposite sides. The original proof that was used remains unknown, but one proof is found in a chapter in [A] by Dick Askey: "Completing Brahmagupta's Extension of Ptolemy's Theorem" This straightforward approach using the cosine rule leads to a proof both of Ptolemy's theorem, as well as Brahmagupta's relationship between the ratio of diagonal lengths and the side lengths of circular quadrilaterals. An extension of Ptolemy's theorem to hexagons is known as Furhmann's theorem, which appears in [F]. This theorem is a statement about the product of the lengths of three diagonals of a cyclic hexagon - see also the animated applet at MathWorld
About two years ago I realized that Brahmagupta's "ratio version" of Ptolemy's theorem extends to any circular polygon with an even number of sides. To my knowledge there has been no publication or discussion of any extension of Brahmaguptašs theorem on the ratios of diagonals of a circular polygon, not even to a hexagon. There were two ingredients to my proof of the Ptolemy like formula for which Marc and Doron have now found a short proof: (i) an existence theorem in a 21 page paper by Jenkins and Serrin [J], that utilizes the Perron process from PDE, and (ii) calculations that are in a paper I submitted last year, that utilize work by Terry Sheil-Small [S].
The existence of minimal surfaces that generalize the (doubly rather than singly periodic) Scherk surface is proved in [J]. For these surfaces (known as JS surfaces, and which are in fact graphs), one can examine polygonal projections to the plane. Terry Sheil-Small's work on harmonic mappings applies in particular to the harmonic mappings that arise from JS surfaces. The Ptolemy-like formula is a consequence of the existence of JS surfaces with complex dilatation equal to a power function. Although not explicitly stated there, in a paper to appear "Harmonic Mappings with Quadrilateral Image" [M] I used methods described in [S] and realized that the Ptolemy-like formula is a consequence. I communicated this problem to Marc during his visit to Colorado College, knowing that there should be a more direct proof, which happily has now been proved by Marc and Doron.
Bibliography
[A] Completing Brahmaguptašs extension of Ptolemyšs theorem. (English summary). The legacy of Alladi Ramakrishnan in the mathematical sciences, 197, Springer, New York, 2010.
[F] Furhmann's theorem, which appears in Synthetische Beweise Planimetrischer Satze, Berlin, p. 61, 1890.
[J] H. Jenkins and J. Serrin, Variational problems of minimal surface type II. Boundary value problems for the minimal surface equation, Arch. Rational Mech. Anal. 21 (1965/66), 321 - 342.
[M] Jane McDougall, "Harmonic Mappings with Quadrilateral Image", to appear.
[S] T. Sheil-Small, On the Fourier series of a step function, Mich. Math. J., 36 (1989), 459 - 475.