Given a set of $n$ points $\{A_1, \ldots , A_n\}$ of the plane and every possible triangle formed with $3$ points $A$, I would like to describe the intersections fo theses triangles. By intersection, I mean the part of the plane that is covered by the interiors of the triangles that intersect.
I am describing these triangles with convex hulls. For exemple, the triangle $A_1A_2A_3$ is represented by $\operatorname{Conv}(A_1, A_2, A_3)$.
I would like to define a set of convex such as every triangle (seen as a convex) is intersecting with at least of convex of this set. So basically, I would like to describe these intersections in order to count them or manipulate them.
I've heard of Helly's Theorem but I can't see a way to use it without knowing if the convex hulls are intersecting.
I am totally open to change my representation of triangles (for example by using something else than convex hulls) and I am not looking for algorithm to do it. If possible, my idea is to describe them 'mathematically'.