The impossibility of certain constructions in Euclidean geometry, such as squaring the circle with straight-edge and compass is usually shown by using algebraic methods. I am wondering if there are "purely" geometric proofs of the impossibility of these constructions.
I searched the internet and could only find this blog entry terrytao's geometrical proof where a "more geometrical" proof of the impossibility of trisecting an angle by straight-edge and compass is presented. As far as my understanding goes, algebra is however still needed ("Lemma Two" in the blog entry).
I am asking this question also as there were some eminent early modern mathematicians, for example Johannes Kepler, that rejected on philosophical grounds altogether algebraic methods in geometry. So I wonder if today there would be a kind of impossibility proof that would be accepted by someone like Kepler.
I also remember vaguely but cannot find the reference right now that Christiaan Huygens presented a purely geometrical argument against the claims of Thomas Hobbes having squared the circle by straight-edge and compass. But I do not know if that would amount to a kind of geometrical proof of the impossibility of squaring the circle by straight-edge and compass.
Any comment is appreciated.
"The impossibility of the quadrature of the circle wold follow if it were shown that the square root of $\pi$ is not algebraic. This was done by the German mathematician C.L.F. Lindemann, in 1882. The proof requires methods that are not algebraic."
Source: Modern Algebra: An Introduction 6th edition by John R. Durbin
You should reference the Gelfond-Schneider Theorem as well. As far as I know, there are no "purely geometric proofs" available.