Characterization of the center of a polygon

Question

Let $O$ be the circumcenter of the regular polygon $P_n$. Then for any $A\in P_n$ one has $d(A,O)\leq r$, where $d$ is the usual Euclidean distance and $r$ is the polygon's circumradius.

Prove that this property uniquely determines the point $O$.

Answer 1

Let $X$ be a point in the plane. We say that $X$ is a central point of $P_n$ if $d(A,X)\leq r$ for all $A \in P_n$.

If $\sigma$ is a symmetry of $P_n$ and $X$ is a central point, then so is $\sigma(X)$.

Thus, set of central points is fixed by all symmetries of $P_n$. But the only point fixed by all symmetries of $P_n$ is its circumcenter $O$.