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 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.

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 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.