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From HandWiki
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 (p + q, r, s), (r, h, p) and (s, h, q),
[math]\displaystyle{ \begin{align} (p + q)^2 \quad &= \quad\;\, r^2 \quad + \quad s^2 \\ p^2 \!+ 2pq + q^2 &= \left(h^2 \!\!+\! p^2\right) \!+\! \left(h^2 \!\!+\! q^2\right) \\ 2pq \qquad\, &= 2h^2 \quad \therefore h = \sqrt{pq} \\ \end{align} }[/math]
[math]\displaystyle{ \begin{align} (p + q)^2 \quad &= \quad\;\, r^2 \quad + \quad s^2 \\ p^2 \!+ 2pq + q^2 &= \left(h^2 \!\!+\! p^2\right) \!+\! \left(h^2 \!\!+\! q^2\right) \\ 2pq \qquad\, &= 2h^2 \quad \therefore h = \sqrt{pq} \\ \end{align} }[/math]
