1993 OIM Problems/Problem 5

Problem

Let $P$ and $Q$ be two different points on the plane. Let us denote by $m(PQ)$ the bisector of the segment $PQ$ . Let $S$ be a finite subset of the plane, with more than one element satisfying the following properties:

a. If $P$ and $Q$ are points distinct from $S$ , then $m(PQ)$ intersects $S$ .

b. If $P_1Q_1$ , $P_2Q_2$ , and $P_3Q_3$ are three different segments whose ends are points of $S$ , then no point of $S$ belongs simultaneously to the three lines $m(P_1Q_1)$ , $m(P_2Q_2)$ , and $m(P_3Q_3)$ .