I've been trying to find ways to prove the circle inversion theorem without the use of algebra, and the last part requires the equality between a square and a rectangle. I've managed to find two proofs which are simply applications of Euclid's Book 1 Propositions 42 and Proposition 47 (this last one is Euclid's pythagorean theorem) (if you're interested, I answered my own question with the proofs: Multiplication in synthetic geometry (example using geometrical circle inversion theorem) )
I was wondering: does anyone know if there are synthetic-geometrical theorems, i.e. without algebra, which allows us to prove that the area of a rectangle is equal to the area of a square(other than Euclid's Bk 1 P41, 47)? Is there any book which deals with this, other than Euclid's 2300 year old stuff?
Thank you.
An old but useful idea is shown in diagram below. Place square $ABCD$ and rectangle $AEFG$ so that their sides with a vertex in common lie on the same ray. Then they have the same area if and only if lines $BG$ and $DE$ are parallel.