the altitude-on-hypotenuse relations
Take a right triangle and drop a perpendicular from the right-angle corner straight down to the hypotenuse. This altitude carves the big right triangle into two smaller right triangles. The beautiful surprise is that all three triangles — the original and the two pieces — are similar to each other, and that single fact produces three clean geometric-mean relationships.
Label the right angle at C, the hypotenuse AB, and the foot of the altitude D, which splits AB into segments AD (next to leg CA, call its length p) and DB (next to leg CB, call its length q). Writing h for the altitude CD: (1) the altitude is the geometric mean of the two hypotenuse pieces, h = sqrt(p·q), equivalently h^2 = p·q; (2) each leg is the geometric mean of the whole hypotenuse and the piece adjacent to it, so CA^2 = p·(p + q) and CB^2 = q·(p + q). These all flow from the three similar triangles set up by the AA criterion (each small triangle shares an acute angle with the big one).
These relations are a workhorse for finding missing lengths in right triangles without any trigonometry, and they give one of the slickest proofs of the Pythagorean theorem: add the two leg relations, CA^2 + CB^2 = p(p + q) + q(p + q) = (p + q)^2 = AB^2. So the geometric-mean picture is not a curiosity — it is Pythagoras hiding in similar triangles. A common slip is to pair the wrong segment with the wrong leg, so always match each leg to the hypotenuse piece that touches it.
A right triangle has its altitude meeting the hypotenuse, splitting it into pieces 4 and 9. The altitude is sqrt(4·9) = 6. One leg is sqrt(4·13) = 2·sqrt(13); the other is sqrt(9·13) = 3·sqrt(13).
Altitude h = sqrt(pq); each leg is the geometric mean of the whole hypotenuse and its adjacent piece.
These relations hold only when the altitude is dropped from the right angle in a right triangle. In a non-right triangle the altitude foot has no such geometric-mean property — the whole structure depends on that 90-degree corner.