Let $x,y$ and $z$ be points in an inner product space. Prove that $$\lvert\lvert x\rvert\rvert\times\lvert\lvert y - z\rvert\rvert \leq \lvert\lvert y\rvert\rvert\times \lvert\lvert z - x\rvert\rvert + \lvert\lvert z\rvert\rvert\times \lvert\lvert x - y\rvert\rvert.$$
My current approach has been to square both terms, then collect them to one side and apply the parallelogram law but this just leaves me where I started with some signs changed. I also tried to expand out terms after squaring and apply cauchy-schwarz but did not seem very wise. Could do with a hint/solution.