generalization of the parallelogram law

Theorem.

In an inner product space (http://planetmath.org/InnerProductSpace), let $x\mathrm{,}y\mathrm{,}z$ be vectors. Then

$${\parallel x+y\parallel }^2+{\parallel y+z\parallel }^2+{\parallel z+x\parallel }^2={\parallel x\parallel }^2+{\parallel y\parallel }^2+{\parallel z\parallel }^2+{\parallel x+y+z\parallel }^2.$$

Taking $x+z=0$ we have the usual parallelogram law.