A constructive theory of ordered affine geometry. Summary
We give a constructive axiomatization of ordered geometry, based on an ordering with directed lines and using constructions instead of existential axioms. A new duality is found such that, classically, equally and oppositely directed lines turn out dual to parallel and orthogonal lines. Principles such as the axiom of Pasch and ordered versions of the triangle axioms are shown to follow naturally from our approach. Then combinatorial properties of the geometrical plane are studied, and the relation to the usual axiomatization in terms of betweenness is established.