  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 $\varepsilon>0$.
