Generalizations of the Theorems of Apollonius and Euler

TL;DR

This paper generalizes classical geometric theorems of Apollonius and Euler using algebraic and combinatorial methods within inner product spaces, extending their applicability to n-vectors.

Contribution

It introduces a unified algebraic framework that generalizes Euler's theorem for quadrilaterals to arbitrary n-vectors in inner product spaces.

Findings

Derived a unified vector framework for Apollonius' and Euler's identities.

Established a general algebraic relation for n vectors in inner product spaces.

Showed Euler's relation as a special case of a broader identity.

Abstract

We present an algebraic generalization of Euler's theorem for quadrilaterals. Starting from the parallelogram identity in an inner product space, we derive Apollonius' identity and obtain Euler's quadrilateral identity in a unified vector framework. Using combinatorial approach, we establish a general algebraic relation for $n\geq 4$ vectors in an arbitrary real or complex inner product space. This result shows that Euler's classical relation is a special case of a general identity involving sums of squared norms of vectors.