The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments into which the hypotenuse is split.

${\displaystyle {\begin{aligned}r^{2}&=p^{2}+h^{2}\qquad \cdots (1)\\s^{2}&=q^{2}+h^{2}\qquad \cdots (2)\\(p+q)^{2}&=r^{2}+s^{2}\\p^{2}+2pq+q^{2}&=(p^{2}+h^{2})+(q^{2}+h^{2})\\2pq&=2h^{2}\\\therefore h&={\sqrt {pq}}\\\end{aligned}}}$