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 Braß (2004) and by Aronov, Dujmović, 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.