Summer of Math Exposition

If I tell you that I see a finite set of points, which are characterized by a special propriety; each line that joins two points of this set will inevitably reach another point of this set. What do I see? Are these points co-linear? well, if they WERE co-linear, this propriety WOULD be satisfied. but how could you be certain that this is the only positioning that would assemble such propriety? What I learned solving this is how to turn intuition and what seems "obvious" into a rigorous argument.