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.