smc

Suppose that $P$ is the intersection of $\overline{CM}$ and $\overline{BN}.$ Let $BN=x.$ By the properties of centroids, we have $BP=\frac{2}{3}x.$ Note that $△BPC\sim △BCN$ by AA. From the ratio of similitude $\frac{BP}{BC}=\frac{BC}{BN},$ we get $$\begin{array}{rl}BP\cdot BN& =B{C}^2\\ \frac{2}{3}x\cdot x& =s^2\\ x^2& =\frac{3}{2}{s}^2\\ x& =\boxed{\frac{s\sqrt{6}}{2}}.\end{array}$$