We generalize work of Erdos and Fishburn to study the structure of finite point sets that determine few distinct triangles. Specifically, we ask for a given t, what is the maximum number of points that can be placed in the plane to determine exactly t distinct triangles? Denoting this quantity by F(t), we show that F(1) = 4, F(2) = 5, and we completely characterize the optimal configurations for t = 1, 2. We also discuss the general structure of optimal configurations and conjecture that regular polygons are always optimal. This differs from the structure of optimal configurations for distances, where it is conjectured that optimal configurations always exist in the triangular lattice. We also prove that the number of distinct triangles determined by a regular n-gon is asymptotic to n2/12; so if the conjecture about regular n-gons being optimal is true, we identify the constant for the lower bound of distinct triangles determined by any point configuration.
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