  A convex geometric hypergraph or cgh consists of a family of subsets of a
strictly convex set of points in the plane. There are eight pairwise
nonisomorphic cgh's consisting of two disjoint triples. These were studied at
length by Bra{\ss} (2004) and by Aronov, Dujmovi\'c, Morin, Ooms, and da
Silveira (2019). We determine the extremal functions exactly for seven of the
eight configurations.
  The above results are about cyclically ordered hypergraphs. We extend some of
them for triangle systems with vertices from a non-convex set. We also solve
problems posed by P. Frankl, Holmsen and Kupavskii (2020), in particular, we
determine the exact maximum size of an intersecting family of triangles whose
vertices come from a set of $n$ points in the plane.
