  Triangles of groups have been introduced by Gersten and Stallings. They are,
roughly speaking, a generalisation of the amalgamated free product of two
groups and occur in the framework of Corson diagrams. First, we prove an
intersection theorem for Corson diagrams. Then, we focus on triangles of
groups. It has been shown by Howie and Kopteva that the colimit of a hyperbolic
triangle of groups contains a non-abelian free subgroup. We give two natural
conditions, each of which ensures that the colimit of a non-spherical triangle
of groups either contains a non-abelian free subgroup or is virtually solvable.
