Ptolemy's equation and kin
arXiv:2507.10444 · doi:10.1007/s00283-025-10453-7
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 Plücker'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.
7 pages, 7 figures, minor revisions, additional references, acknowledgements