Ninth STARS OF MATHEMATICS Competition, 2015

Given a point $x$ in $S$ and a real number $r$, let $S(x, r) = \{y : y \in S, \text{dist}(x, y) = r\}$, and notice that the $S(x, r)$, $r \ge 0$, partition $S$.
The number of non-degenerate isosceles triangles with vertices in $S$ and apex at $x$
is $\sum_{r>0} (|S(x,r)|)^2$, so the total number of non-degenerate isosceles triangles with vertices
in $S$ is $N = \sum_{x \in S} \sum_{r > 0} \binom{|S(x,r)|}{2}$, equilateral triangles with vertices in $S$ being counted three times each. Now,
$$\begin{align*} N &= \sum_{x \in S} \sum_{r > 0} \binom{|S(x, r)|}{2} = \sum_{r > 0} \sum_{x \in S} \binom{|S(x, r)|}{2} \\ &\geq \sum_{r > 0} |S| \binom{\frac{1}{|S|} \sum_{x \in S} |S(x, r)|}{2} = \sum_{r > 0} |S| \binom{\frac{2|D(S,r)|}{|S|}}{2} \\ &= \frac{2}{|S|} \sum_{r > 0} |D(S, r)|^2 - \sum_{r > 0} |D(S, r)| = \frac{2}{|S|} \sum_{r > 0} |D(S, r)|^2 - \binom{|S|}{2}, \end{align*}$$