Quaternions, Ptolemy's inequality and the inversion process.

Abstract

In this work, we introduce quaternions. We discuss how their operations can be performed and their connections to some results in analytic geometry. We also verify some properties of these operations and how they relate to some little-known results such as Ptolemy's inequality. This inequality involves the norms of the differences of four vectors in a real vector space with an inner product. We present a proof using quaternions and also discuss when equality occurs. Finally, we show how this result can be extended using the inversion process, which generalizes the inversion of elements within quaternions. For this, we use some results on Gram matrices and matrices involving distances between elements.