  We study the following family of problems: Given a set of $n$ points in
convex position, what is the maximum number triangles one can create having
these points as vertices while avoiding certain sets of forbidden
configurations. As forbidden configurations we consider all 8 ways in which a
pair of triangles in such a point set can interact. This leads to 256 extremal
Tur\'an-type questions. We give nearly tight (within a $\log n$ factor) bounds
for 248 of these questions and show that the remaining 8 questions are all
asymptotically equivalent to Stein's longstanding tripod packing problem.
