Ptolemys Theorem

Ptolemy's Theorem states that if a quadrilateral is cyclic, then the product of the lengths of the diagonals equals the sum of the products of the lengths of the two pairs of opposite sides. It appears in Ptolemy's Ìåãáëç Óõíôáèés and was important in his calculation of what we now recognize as the first table of trigonometric functions.

I remember the theorem from my 10th grade Geometry course, but it no longer shows up in most high school mathematics experiences. When there is a need to refer to it, the usual citation is Theorem D in book 6 of Euclid. This is at first sight strange, since Ptolemy came several hundred years after Euclid. How did it happen? Robert Simson, who published his great edition of Euclid in 1756, added the theorem to the text, and it has remained there ever since.

My interest in Ptomely's Theorem was boosted considerably when, in 1967, E.N. Gilbert, a fellow mathematician at Bell Labs, discovered how important the theorem was to the design of minimum cost communication networks. This is a harder problem that the design of minimum length commuincation networks, since it recognizes the likelihood that different links of a network may have different costs per mile. We shall first familiarize ourselves with Ptolemy's Theorem, and then look at minimum cost communication networks.