Are closed simple curves with this property necessarily circles? 
Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple curve and $\Gamma$ be the region enclosed by $\gamma$. Let $O$ be the center of mass of $\Gamma$.
Suppose that any line that goes through $O$ splits $\gamma$ into two regions with equal areas.
Is $\gamma$ a circle ?

I have no experience in differential geometry or complex analysis. I'd say the answer is yes, but I'm looking for a rigorous proof.

Any convex centrally symmetric region has this property.
(I include the assumption of convexity only to ensure that there's no ambiguity about what counts as "split into two regions".)
