Ptolemy's equation and kin

TL;DR

This paper explores the interconnectedness of three-term relations across various mathematical contexts, such as Ptolemy's theorem, Casey's theorem, and Plücker's identity, showing how they can be derived from one another.

Contribution

It provides a unified explanation of different three-term relations in geometry and algebra, revealing their underlying connections.

Findings

Demonstrates how Ptolemy's equation relates to Casey's theorem and Plücker's identity.

Shows that these relations can be derived from a common framework.

Highlights the interconnected nature of geometric and algebraic identities.

Abstract

Three-term relations of the form AB+CD=EF arise in multiple mathematical contexts, including the Ptolemy equation for a cyclic quadrilateral, Casey's theorem on bitangents, Penner's relation for lambda lengths, and Pl\"ucker's identity for the maximal minors of a 2x4-matrix. In this note, we explain how these different occurrences of the 3-term relation can be directly obtained from each other.