ISL 2009 G5

Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P \subset R$ we have $$\frac{|R|}{|P|} \leq \sqrt{2}$$ where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.