A Vector Generalization of Euler's Quadrilateral Theorem

Abstract

In this paper, we develop a unified algebraic framework for Euler-type identities in real and complex inner product spaces. Starting from the parallelogram identity, we derive Apollonius' identity and recover Euler's classical theorem. We then establish a general Euler-type identity valid for every finite collection of $n\geq 4$ vectors. The proof is based on a combinatorial analysis of pairwise distances. The resulting identity recovers Euler's theorem when $n=4$. Several previously known identities thus arise naturally within a single algebraic framework.