There's an interesting proof of the impossibility of trisecting an arbitrary angle with only straight-edge and compass on Terry Tao's blog [1]. It's interesting because it is mostly a geometric proof rather than the usual Galois theory approach. The only major non-geometric part is that it depends on the fact that 3 does not divide any integral power of 2.

Here are formal proofs of the impossibility of trisection and of doubling the cube, using the Isabelle proof assistant [2].

[1] https://terrytao.wordpress.com/2011/08/10/a-geometric-proof-...

[2] https://www.isa-afp.org/entries/Impossible_Geometry.shtml