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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?

Likk
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  • In your restricted situation, how do you define the area of a circle? – Hagen von Eitzen Jan 17 '19 at 20:40
  • @HagenvonEitzen: that's an interesting question, that I also asked myself. I would say "however Euclid would have defined it", but I think he never did, although the problem of squaring the circle was well known at the time – Likk Jan 17 '19 at 20:42
  • Let $\Delta ABC$ have the right angle in $A$, construct $D$ on $BC$ with $AD\perp BC$. Now arrange that $BD,DC$ are $1,2$. What is $AD$? – dan_fulea Jan 17 '19 at 20:47
  • @dan_fulea: this requires the knowledge that $2 \text{Area}(r) = \text{Area}(\sqrt{2} r)$. How to prove that? – Likk Jan 17 '19 at 21:05
  • Are you willing to accept that the ratio of Area of a circle to that of a circumscribed square is constant? – whoisit Oct 03 '22 at 22:28

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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.

Hagen von Eitzen
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    I wasn't aware that Euclid used certain arguments involving "limits"! Do you have any examples of a similar argument from those times, maybe from the Elements? – Likk Jan 17 '19 at 21:03
  • Finally found something: Archimedes, who indeed used the method of inscribed polygons to approximate π. I was so much focused on Euclid's Elements that I completely missed him. Thanks for your answer! You saved me a lot of time! – Likk Jan 17 '19 at 21:32
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The centre of the circle is known; it is the basis from which a circle is drawn. Draw a straight line from the centre to the circumference of the circle; a radius. Draw a second radius at a right angle to the first. Draw a straight line between the points where the two radii meet the circumference. Use this line as the radius for the new circle.

The ancients knew how to double the area of a square using this method. Since the area of a circle must be proportional to its radius, using the same method could be expected to give the same result for a circle.

Lawrence White
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