Ptolemy's theorem connects the lengths of sides and diagonals in a cyclic quadrilateral on Euclidean plane. This theorem can be considered as a generalisation of Pythagorean theorem. In the recent decades, identities similar to the one in the Ptolemy's theorem started to pop up in many fields of mathematics in connection to the notion of cluster algebras introduced and studied since 2000 by Fomin and Zelevinsky. The list of contexts where the identities appear include (but are not limited to):

• triangulated polygons
• Grassmannians;
• Conway-Coxeter Frieze Patterns;
• hyperbolic geometry;
• hyperbolic surfaces;
• cluster algebra;
• CAT(0) metric spaces.

Prerequisites: Geometry III.

Resources:

One can with the brief review (7 pages) and the references therein. One can find more references here.