The resistance across M and N in the given fig. is 

The resistance of each side of the triangle would be r / 2 as the two resistances are connected in parallel. The two sides are in series and hence their equivalent resistance would be (r / 2) + (r / 2) = r. This equivalent resistance is connected in parallel with third one. Hence
$\frac{1}{\mathrm{R}}=\frac{1}{\mathrm{r}}+\frac{1}{(\mathrm{r}/2)}=\frac{1}{\mathrm{r}}+\frac{2}{\mathrm{r}}=\frac{3}{\mathrm{r}}\text{ or }\mathrm{R}=\frac{\mathrm{r}}{3}$