Template:Right angle altitude.svg

The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using Pythagoras' theorem on the 3 triangles of sides Template:Nowrap, Template:Nowrap and Template:Nowrap,
$\begin{array}{cc}(p+q{)}^2& =r^2+s^2\\ p^2+2pq+q^2& =\overbrace{p^2+h^2}+\overbrace{h^2+q^2}\\ 2pq& =2h^2\therefore h=\sqrt{pq}\\ \end{array}$