Publication Date
9-27-2025
Document Type
Conference Proceeding
Publication Title
Lecture Notes on Data Engineering and Communications Technologies
Volume
218
DOI
10.1007/978-3-031-71013-1_23
First Page
247
Last Page
258
Abstract
In The Almagest, we found the original Ptolemy’s theorem and its proof, and we compared it to “Ptolemy’s theorem” written in two modern geometry textbooks. They are different in that the new ones include the “converse” of the original one. We noticed the hypothesis of the “converse” can be interpreted in two ways. Using this as our motivation, we establish a new theorem related to Ptolemy’s theorem in this note as follows: If four distinct points A, B, C, D are given in the three-dimensional space so that ABC is a triangle, and if |AB||CD|+|AD||BC|=|AC||BD|, then the points A, B, C, D are not only on the same plane, but ABCD is also a cyclic quadrilateral.
Keywords
Crelle’s theorem, Law of Sines, Ptolemy’s theorem, quadrilateral, skew quadrilateral, The Almagest
Department
Mathematics and Statistics
Recommended Citation
Hidefumi Katsuura. "Converse of Ptolemy’s Theorem" Lecture Notes on Data Engineering and Communications Technologies (2025): 247-258. https://doi.org/10.1007/978-3-031-71013-1_23
Comments
This is a post-peer-review, pre-copy edit version of a conference paper published in ICGG 2024 - Proceedings of the 21st International Conference on Geometry and Graphics, 2025. The final authenticated version is available online at: https://doi.org/10.1007/978-3-031-71013-1_23