In Euclid's Elements, Book 2, Proposition 14, We are shown how to construct a square from a given rectilinear figure. This allows us to square a rectangle.
Is it possible to do the inverse, creating a rectangle with a given side, with the same area as a given square?
I have looked through the rest of Euclid's Elements, and I can't see a proposition doing this. Did I miss one? I also saw references online to the Bolyai-Gerwien Theorem, but I couldn't figure out how to apply it using only a compass and straightedge. I have also tried using the Pythagorean theorem and a 3-4-5 triangle, but it doesn't apply to all squares, so I dropped that.
For a bit of context: I have been plotting different equations on paper, using only a compass and straightedge. With the above mentioned proof, we can plot variations of $\sqrt{x}$, however, we would need to turn a square into a rectangle of width 1 to be able to graph any equations using exponents, like $x^2$.
Note: The construction must be possible with an unmarked straightedge and a compass as the only tools. Preferably something that can be proven to be always true by euclidian methods.
Best Answer
Let $PC$ be the side length of the given square and $PA$, with $\angle CPA=90^\circ$, the given side length of the desired rectangle. Let $O$ be the intersection of the bisector of $AC$ with $\overleftrightarrow{AP}$. The circle around $O$ through $A$ (and $C$) intersects $AP$ in a second point $B$. Then $PB$ is the other side length of the desired rectangle.