Generalizations of the Theorems of Apollonius and Euler

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 $n\ge 4$ 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.