Question 45
If $P$ and $Q$ are two points in the plane, let $m(PQ)$ be the perpendicular bisector of $PQ$. $S$ is a finite set of $n \gt 1$ points such that:
(1) if $P$ and $Q$ belong to $S$, then some point of $m(PQ)$ belongs to $S$,
(2) if $PQ$, $P'Q'$, $P''Q''$ are three distinct segments, whose endpoints are all in $S$, then if there is a point in all of $m(PQ)$, $m(P'Q')$, $m(P''Q'')$ it does not belong to $S$.
What are the possible values of $n$?
(1) if $P$ and $Q$ belong to $S$, then some point of $m(PQ)$ belongs to $S$,
(2) if $PQ$, $P'Q'$, $P''Q''$ are three distinct segments, whose endpoints are all in $S$, then if there is a point in all of $m(PQ)$, $m(P'Q')$, $m(P''Q'')$ it does not belong to $S$.
What are the possible values of $n$?