g.m. in a right angled triangle

it is not too daunting to prove (or "provide a logical explanation") why the length of the altitude of a right angled triangle is the geometric mean of the two segments either side of the foot of the altitude (where it splits the hypotenuse)

this result in the Russian geometry texts of Kiselev (born 1852) is introduced after proportionality
(translated by Alexander Givental into English)
pdf version of the first book, 'planimetry'
and forms his proof of the 'pythagoras' theorem (in chapter 6, theorem 188)

it is simply established using two similar triangles

or you could use the intersecting chord theorem (if you are fortunate enough to know it) - usually established from similar triangles

how was this used as a method to create a square equal in area to a rectangle?

you could use pythagoras to establish the result
but if you used similarity, how could this result be used to prove the pythagoras theorem?