On the number of classes of triangles determined by $N$ points in $\R^2$

Abstract

Let $P$ be a set of $N$ points in the Euclidean plane, where a positive proportion of points lies off a single straight line. This note points out two facts concerning the number of equivalence classes of triangles that $P$ determines, namely that (i) $P$ determines $\Omega(N^2)$ different equivalence classes of congruent triangles, and (ii) $P$ determines $\Omega(\frac{N^2}{\log N})$ different equivalence classes of similar triangles. The first fact follows from the recent theorem by Guth-Katz on point-line incidences in $\R^3$. The second one, perhaps not so well known, is due to Solymosi and Tardos.