Pick n points independently at random in R^2, according to a prescribed probability
measure mu, and let D^n_1 <= D^n_2 <= ... be the areas of the binomial n choose 3
triangles thus formed, in non-decreasing order. If mu is absolutely continuous with respect
to Lebesgue measure, then, under weak conditions, the set {n^3 D^n_i : i >= 1} converges
as n --> infinity to a Poisson process with a constant intensity c(mu). This result, and
related conclusions, are proved using standard arguments of Poisson approximation, and may
be extended to functionals more general than the area of a triangle. It is proved in
addition that, if mu is the uniform probability measure on the region S, then c(mu) <=
2/|S|, where |S| denotes the area of S. Equality holds in that c(mu) = 2/|S| if S is
convex, and essentially only then. This work generalizes and extends considerably the
conclusions of a recent paper of Jiang, Li, and Vitanyi.