Generalizations of the Theorems of Apollonius and Euler

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.