The Minimum Number of Distinct Areas of Triangles Determined by a Set of n Points in the Plane

We prove a conjecture of Erdos, Purdy, and Straus on the number of distinct areas of triangles determined by a set of $n$ points in the plane. We show that if $P$ is a set of $n$ points in the plane, not all on one line, then $P$ determines at least $\lfloor\frac{n-1}{2}\rfloor$ triangles with pairwise distinct areas. Moreover, one can find such $\lfloor\frac{n-1}{2}\rfloor$ triangles all sharing a common edge.