Functional Analysis_15. Inner Products and Orthogonality_15.2. Orthogonality via the norm

Let $X$ be a Hilbert space and $x, y \in X$. Show that $\langle x, y \rangle = 0$ if and only if $$\|x + \alpha y\| = \|x - \alpha y\|\quad \text{for all } \alpha \in \mathbb{F}.$$