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)