Given a set A composed by six non collinear distinct points on the plane, let us consider only the partitions whose elements are pairs of points in A. Then, we call the set of such partitions by P(A). So, fix a partition in P(A). Then, take each pair in this partition and consider the 3 segments joining the two points in A belonging to each pair. For some partitions those 3 segments intersect in one single point. Let S(P(A)) be the set of such partitions. Then can you provide an example of 6 not collinear points on the plane such that there are two partitions P, P^’ in S(P(A))=\nonempty, whose associated intersection points are different.

I am looking for an example where six non collinear distinct points on the plane are located in such a way that the above configuration holds. I do not know if this configuration exists and I guess it doesn't.

Thanks.