2015 Ninth STARS OF MATHEMATICS Competition

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 $N={\sum }_{x\in S}{\sum }_{r>0}\left(\genfrac{}{}{0ex}{}{|S(x,r)|}{2}\right)$, equilateral triangles with vertices in $S$ being counted three times each. Now, $$\begin{array}{rl}N& =\sum _{x\in S}\sum _{r>0}\left(\genfrac{}{}{0ex}{}{|S(x,r)|}{2}\right)=\sum _{r>0}\sum _{x\in S}\left(\genfrac{}{}{0ex}{}{|S(x,r)|}{2}\right)\\ & \ge \sum _{r>0}|S|\left(\genfrac{}{}{0ex}{}{\frac{1}{|S|}{\sum }_{x\in S}|S(x,r)|}{2}\right)=\sum _{r>0}|S|\left(\genfrac{}{}{0ex}{}{\frac{2|D(S,r)|}{|S|}}{2}\right)\\ & =\frac{2}{|S|}\sum _{r>0}|D(S,r){|}^2-\sum _{r>0}|D(S,r)|=\frac{2}{|S|}\sum _{r>0}|D(S,r){|}^2-\left(\genfrac{}{}{0ex}{}{|S|}{2}\right),\end{array}$$