Distinct Triangle Areas in a Planar Point Set

Erdős, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the triangles determined by n noncollinear points in the plane is at least b 2 c, which is attained for dn/2e and respectively bn/2c equally spaced points lying on two parallel lines. We show that this number is at least 17 38 n − O(1) ≈ 0.4473n. The best previous bound, ( √ 2 − 1)n − O(1) ≈ 0.4142n, which dates back to 1982, follows from the combination of a result of Burton and Purdy [5] and Ungar’s theorem [23] on the number of distinct directions determined by n noncollinear points in the plane.