An algebraic approach uses the isomorphism between the Euclidean plane and the real coordinate space \mathbb{R}^2. In this way, a stronger version of the theorem was proven in 1990.Arnon Avron, "On strict strong constructibility with a compass alone", Journal of Geometry (1990) 38: 12. It also shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language).
