Find the fourth proportional to 2xy, x² and y²

Let fourth proportional to 2xy, x² and y² be n.

$\therefore \dfrac{2xy}{x^2} = \dfrac{y^2}{n} \\[1em] \Rightarrow n = \dfrac{x^2y^2}{2xy}\\[1em] \Rightarrow n = \dfrac{xy}{2}.$

Hence, fourth proportional = $\dfrac{xy}{2}$.