Almost similar configurations

Abstract

Let $h(n)$ denote the maximum number of triangles with angles between $59^\circ$ and $61^\circ$ in any $n$-element planar set. Our main result is an exact formula for $h(n)$. We also prove $h(n)= n^3/24+ O(n \log n)$ as $n\to \infty$. However, there are triangles $T$ and $n$-point sets $P$ showing that the number of $\varepsilon$-similar copies of $T$ in $P$ can exceed $n^3/15$ for any $\varepsilon>0$.