Assume that $A$ is compact and convex. If there is a point $P$ such that any line through it is a bisector of $A$ then $A$ has to be centrally symmetric. In fact a stronger result is known (see the [paper][1] by V. Menon):

> **Theorem.** Let $K$ be a compact convex figure. The following
four statement are equivalent:

> -  the point $P$ through which three bisectors of $K$ pass is unique,
> - all bisectors of $K$ are concurrent in $P$;
> - there exists a point $P$ such that any line through it is a bisector of $K$;
> - $K$ is a centrally symmetric figure with $P$ as its centre.




  [1]: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102995091