Ptolemy Relation and Friends

In recent decades, identities similar to the one in the Ptolemy’s theorem started to pop up in many fields in connection to the notion of cluster algebras introduced and studied since 2000 by Fomin and Zelevinsky. In this brief note we will try to describe several animals from this big and rich zoo. The note was written for the Association for Mathematical Research (AMR) “Reviews of Classic Results” series.