Almost similar configurations

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 $h(n)=n^3\mathrm{/}24+O(n\log n)$ as $n\to \infty$\mathrm{.}However,therearetriangles$T$and$n$-pointsets$P$ showing that the number of $\epsilon$-similar copies of $T$ in $P$ can exceed $n^3\mathrm{/}15$ for any $\epsilon >0$.