Q:

How to double the circle?

I'm looking for a compass-and-straightedge method to construct a circle that has area twice of the area of another circle, with no prior knowledge of π, without knowledge of the formula for the area of the circle, without calculus and without any modern (post-1800) method.
I proved constructions like: doubling the inscribed square, but this requires the assumption that the are of the circle and the inscribed square are proportional, which I failed to prove.
In general, for all my constructions, the main problem that I'm facing is that I'm unable to prove that there's a linear relationship between the area of the circle and the square of its radius. I think that even without knowing what the constant of proportionality is, I should still be able to prove that such a constant exists.
I believe this problem is solvable, because it involves the construction of $\sqrt{2}$, and because the ratio of the areas/radii are all constructible numbers. Any hints?

A:

Scaling by $\sqrt 2$ doubles the area of a square (by assuming that congruent triangles have equal area and areas of shapes having at most edges in common can be added).
Then the same is true for scaling isosceles right triangles (=half-squares).
By combing this with shearing, we see that scaling any triangle by $\sqrt 2$ doubles area.
As a consequence, arbitrary polygons (=unions of triangles) double their area when scaled by $\sqrt 2$.
Now whatever the area of a circular disk is, we can inscribe and circumscribe high-order polygons to it and these differ by an arbitrarily small area (this sounds like modern epsilontics, but that's how Euclid would also tackle this). Scaling by $\sqrt 2$ doubles the inscribed and circumscribed polygon's area (while preserving the inscribed/circumscribed property). It follows that the area of the scaled disk cannot be larger nor smaller than twice the original area.

