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(It's my first time using a math forum for help so please bear with me) I was working on a relatively simple generalization for Ptolemy's theorem to all cyclic polygons, I've verified that the proof works with pentagons and hexagons so I'm relatively sure of its veracity. However, it seems that my formula is a bit simple, so I was wondering if there exist similar formulas for cyclic polygons which are corollary to my work. Also I was wondering if anyone could point me in the direction of other generalizations of Ptolemy's theorem to possibly other dimensions? I've attached a google doc with my proof, feel free to look over it.

link to proof: https://docs.google.com/document/d/1KrZu6aC-6YxY5Id9WqH5N3-7zhf968fpCiuHYrQPhSc/edit?usp=sharing

person
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There is a paper by R.J. Gregorac: "Feuerbach's Relation and Ptolemy's Theorem in R^n", Geometriae Dedicata 60: 65-88, 1996. https://link.springer.com/content/pdf/10.1007/BF00150868.pdf which may have what you are looking for.

Also there is a classical theorem by Fuhrmann for the cyclic hexagon:

https://pdfs.semanticscholar.org/3c38/c0f8b2e55447437b883975aac1f84669a057.pdf

msantopr
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