Intersection of segments on a plane 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.
 A: For every set $A$ of six points in general position (no three on a line) there is at most one such partition, which can be seen as follows. 
First, if three segments determined by $A$ cross at the same point, then each of the segments is a halving segment, which means that the line determined by the segment splits the remaining four points evenly in two halfplanes. 
Second, the convex hull of $A$ has at least three extremal points (vertices of the polygon). Each extremal point is incident to exactly one halving segment, so this determines at least two pairs in the partition. Since $A$ only has six points, the whole partition is determined uniquely.
If $A$ is not in general position, a similar argument shows that for each extremal point $x$ of $A$, the line extending the segment with endpoint $x$ is determined uniquely. There are at least two extremal points for which the lines are different, and this determines the intersection point of the three segments. 
There are examples where two different partitions are possible but with the same intersection point. For example $A=\{(-2,0),(-1,0),(1,0),(2,0),(0,1),(0,-1)\}$.
A: Let $Q=(-6,0)$, $R=(6,0)$, $S=(0,2)$, $T=(-3,3)$, $U=(3,3)$, $V=(0,6)$. The partition $QU,RT,SV$ gives the point $S$, the partition $QT,RU,SV$ gives the point $V$. 
EDIT: I just noticed that the question specifies line segments, not lines. If that's what's wanted, the this example fails, as $QT$ and $RU$ don't intersect, unless extended. 
