Ptolemy's Theorem and Pablo Etchegoin

Today I visited Pablo Etchegoin in Wellington hospital. He had developed a painful bile duct infection and was admitted to be given intravenous antibiotics. He was much better when I saw him. Indeed he was sitting up in bed with a notebook open and a pen in hand. I immediately noticed what looked like a problem in Euclidean geometry on the pages. "Working?" I asked. Later in the conversation Pablo returned my attention to the pages. "I decided to solve Ptolemy's Theorem", he said.

I had to scratch my head for a bit. In my teens I was an expert in Euclidean geometry, but I struggled to remember it. Pablo reminded me "It's the relation between the four sides and two diagonals of a cyclic quadrilateral. For a quadrilateral inscribed in a circle, the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals.

Pablo finds hospital tedious. As a consequence he chose to solve an ancient problem worked out by the Greek Astronomer and mathematican Ptolemy, who lived in Alexandria from AD 90 to AD 168. He then explained to me his method. "I decided to use complex numbers", he said. "That means that the displacements between the vertices are just phase shifts in the Argand plane. It came out really nicely".

I was enthralled. "Poor Ptolemy didn't have complex numbers available to him", I said. "No, he had to do the geometric proof", said Pablo, who then went on to point out to me that Pythagoras's theorem was just a special case.

We had good conversation about our cancer treatments, the struggles we face, and the possible ways forward. But I walked out of Pablo's cubicle inspired again by his extraordinary intelligence, his curiosity, his physicist's love of mathematical beauty, and of course, his courage.


PS: Thanks to Wikipedia for the diagram. The entry on Ptolemy's theorem gives four different proofs, including the original geometric one.