The Number of Unit-Area Triangles in the Plane: Theme and Variations∗

We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n), improving the earlier bound O(n) of Apfelbaum and Sharir [2]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans, with one vertex on each of the lines, can be Ω(n), for any triple of lines (this number is always O(n) in this case). (ii) We show that if S is a convex grid of the form A × B, where A, B are convex sets of n real numbers each (i.e., the sequences of differences of consecutive elements of A and of B are both strictly increasing), then S determines O(n) unit-area triangles.