  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 $F(t) < 48(t+1)$ for all $t$. We also
completely characterize the optimal configurations for $t = 1, 2$.
