JBMO-SL 2008 G8

The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$, Knowing that $a b=\frac{x y}{2}$, show that

$$a=\frac{x}{\sqrt{2}}, b=\frac{y}{\sqrt{2}} \text { or } a=\frac{y}{\sqrt{2}}, b=\frac{x}{\sqrt{2}}$$