19.2.1.5 Properties of normed spaces

Problem: Prove that the Apollonius identity holds in Euclidean space: \[ 2\|z-x\|^{2}+2\|z-y\|^{2}=\|x-y\|^{2}+4\left\|z-\frac{x+y}{2}\right\|^{2} . \]