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

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 ${\mathbb{R}}^3$. The second one, perhaps not so well known, is due to Solymosi and Tardos.

6pp

References in corpus (1)

• Simple Proofs of Classical Theorems in Discrete Geometry via the Guth--Katz Polynomial Partitioning Technique

Cited by in corpus (1)

• On Optimal Point Sets Determining Distinct Triangles