Ptolemy's equation and kin
Metric Geometry
2025-10-09 v2 Combinatorics
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.
Cite
@article{arxiv.2507.10444,
title = {Ptolemy's equation and kin},
author = {Katie Waddle},
journal= {arXiv preprint arXiv:2507.10444},
year = {2025}
}
Comments
7 pages, 7 figures, minor revisions, additional references, acknowledgements