If $\ell x = my = nz$, then $\dfrac{x^2}{yz} + \dfrac{y^2}{zx} + \dfrac{z^2}{xy}$ equals—

Let $\ell x = my = nz = k$, so $x = k/\ell$, $y = k/m$, $z = k/n$. Then $\dfrac{x^2}{yz} = \dfrac{mn}{\ell^2}$, and summing the three symmetric terms gives $\dfrac{mn}{\ell^2} + \dfrac{n\ell}{m^2} + \dfrac{\ell m}{n^2}$.