The **Altitude Theorem** or **Geometric Mean Theorem** is a result from
high-school geometry. In a right triangle, the *altitude* $h$ on the
hypotenuse divides the hypotenuse into two segments, $p$ and $q$. The
theorem now states that $h^2 = pq$ or, equivalently, $h = \sqrt{pq}$:
the altitude equals the geometric mean of the segments of the hypotenuse.

The content of this theorem is a bit surprising, because the altitude and
the hypotenuse segments seem geometrically somewhat “unrelated”: it’s
not clear how one could (geometrically) be transformed into the other.
And although the theorem can be proven in many different ways, many of
the proofs are at least partially algebraic, and therefore do not provide
an intuitive, geometric sense why it is true.
But it turns out that a very elegant, strictly geometric proof of this
proposition can be constructed.