Peter Braß, Günter Rote, and Konrad J. Swanepoel:

Triangles of extremal area or perimeter in a finite planar point set

Abstract

We show the following two results on a set of n points in the plane, thus answering questions posed by Erdös and Purdy (1971): 1. The maximum number of triangles of maximum area (or of maximum perimeter) in a set of n points in the plane is exactly n. 2. The maximum possible number of triangles of minimum positive area in a set of n points in the plane is Θ(n²).