The Minimum Number of Distinct Areas of Triangles Determined by a Set of n Points in the Plane
Rom Pinchasi · SIAM Journal on Discrete Mathematics · 2008
We prove a conjecture of Erdős, 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.