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For any distinct points $P$ and $Q$ of the plane, we denote $m(P Q)$ the perpendicular bisector of the segment $P Q$. Let $S$ be a finite subset of the plane, with more than one element, which satisfies the following conditions:

(i) If $P$ and $Q$ are distinct points of $S$, then $m(P Q)$ meets $S$.

(ii) If $P_1 Q_1, P_2 Q_2$ and $P_3 Q_3$ are three distinct segments with endpoints in $S$, then no point of $S$ belongs simultaneously to the three lines $m\left(P_1 Q_1\right), m\left(P_2 Q_2\right), m\left(P_3 Q_3\right)$. (Mexico)
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