Geometric mean / altitude-on-hypotenuse theorem

Dependencies: AA similarity.

Statement

Suppose $\triangle ABC$ has a right angle at $C$, i.e. $\angle C = 90^\circ$. Denote the three sides by $a = BC$, $b = CA$, $c = AB$. Drop a perpendicular from the vertex $C$ to the hypotenuse $AB$, with foot $H$, i.e. $CH \perp AB$ at $H$.

Then the altitude $CH$ to the hypotenuse splits the right triangle $\triangle ABC$ into two small triangles similar to the original:

From this triple similarity one immediately reads off three length identities — together they are called the altitude-on-hypotenuse theorem:

The first two say "the square of each leg equals its projection onto the hypotenuse times the whole hypotenuse"; the third says "the altitude to the hypotenuse is the geometric mean of the two segments into which the hypotenuse is divided".