If $\dfrac{x^2 + y^2}{xy} = \dfrac{y^2 + z^2}{yz}$ and $z \ne x$, then $x, y, z$ are in—

Clearing denominators gives $(x-z)(xz-y^2)=0$. Since $z\ne x$, we must have $y^2=xz$, which is exactly the condition for $x,y,z$ to be in continued proportion.