it is not too daunting to prove (or "provide a logical explanation") that the length of the altitude of a right angled triangle is the geometric mean of the two segments at the foot of the altitude (where it divides the hypotenuse)
this is an important result in the Russian geometry of Kiselev (translated by Alexander Givental into English)
it is simply established with similar triangles
or you could use Pythagoras
or you could even use the intersecting chord theorem (if you are fortunate enough to know it)
how could this be used to create a square equal in area to a rectangle?
how could this be used to prove Pythagoras' theorem?