A Short Proof of a Ptolemy-Like Relation for an Even number of Points on a Circle Discovered by Jane McDougall

TL;DR

This paper presents a concise proof of a Ptolemy-like relation for an even number of points on a circle, extending classical geometric relations to more complex configurations.

Contribution

It provides a simplified proof of a generalized Ptolemy-like relation for multiple points on a circle, building on McDougall's earlier discovery.

Findings

Established a new Ptolemy-like relation for even point configurations

Connected the relation to classical Plücker and Ptolemy theorems

Extended geometric relations to higher-order point arrangements

Abstract

We give a short proof of a Ptolemy-style result first discovered and proved by Jane McDougall. It may be viewed as a generalization to any even number of points of the cubic relation connecting the six joint distances of four points on a circle (whose quadratic relation is the famed Plucker relation, alias Ptolemy's theorem). This article is in fond memory of Andrei Zelevinsky (1953-2013) who loved Ptolemy's theorem.