# (Synthetic-)Geometrical theorems on the quadrature of a rectangle

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.

## 1 Answer

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.

• very neat, thank you! Apr 1, 2021 at 17:42