Using only an unmarked straightedge and compass: (1) trisect an arbitrary angle; (2) construct a cube of volume twice a given cube (Delian problem); (3) construct a square equal in area to a given circle. All three require constructible quantities; the targets — a generic third-angle cosines, ∛2, and √π respectively — live outside the constructible (tower-of-quadratic) field extensions.
Greek geometers invented conics, neusis and mechanical curves (Archimedes' spiral, quadratrices) attacking these — the failures themselves generated mathematics. Pierre Wantzel (1837) proved angle trisection and cube duplication impossible via cubic irreducibility over ℚ (his 1844 paper also gave non-constructibility criteria); Ferdinand von Lindemann (1882) proved π transcendental, killing circle-squaring — building on Hermite's e-transcendence (1873).
All three proven impossible: Wantzel 1837 (trisection, Delian), Lindemann 1882 (squaring). Impossibility proofs of the 19th century did for geometry what incompleteness would do for logic: drew the exact boundary of a method's power. Galois theory is their direct descendant; every 'crank proof' of trisection since is a refresher course in why the boundary is real.