Why does absolute geometry with the elementary continuity axiom have only two models?

Wanda Szmielew showed in 1959 that A−−, plane absolute geometry with the elementary continuity axiom schema, has precisely two models: Euclidean planes and hyperbolic planes over real-closed fields. In this note we determine the reason why this surprising result holds. We find that two axioms are responsible for it, the circle axiom and Aristotle’s axiom; both hold in A−−.