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arXiv · 2603.15657

A Vector Generalization of Euler's Quadrilateral Theorem

Mohammad Hassan Murad
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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.

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Mohammad Hassan Murad. 2026-03-08. A Vector Generalization of Euler's Quadrilateral Theorem. https://arxiv.org/abs/2603.15657

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