construction of contraharmonic mean of two segments

Let $a$ and $b$ two line segments (and their ).  The contraharmonic mean

$$x:=\frac{a^2+b^2}{a+b}=\frac{{\left(\sqrt{a^2+b^2}\right)}^2}{a+b},$$

satisfying the proportion equation

$$\frac{a+b}{\sqrt{a^2+b^2}}=\frac{\sqrt{a^2+b^2}}{x},$$

can be constructed geometrically (http://planetmath.org/GeometricConstruction) as the third proportional of the segments$a+b$ and $\sqrt{a^2+b^2}$, the latter of which is gotten as the hypotenuse of the right triangle with catheti$a$ and $b$.  See the construction of fourth proportional.

(The dotted lines are parallel, with declivity ${45}^{\circ }$.)