Eine Variante des Fano-Axioms fü angeordnete Ebenen

Abstract

An ordered plane is an incidence structure (\((E,\mathcal{G})\)) with an order function ω, which satisfies the axioms (G), (V) and (S), but no continuation--axiom is required. Points a, b ∈ E are said to be in distinct sides of a line \(G \in \mathcal{G}\) iff \(\omega (G,a,b) = 1\) and in the same side if \(\omega (G,a,b) = 1\) , respectively. For any lines \(A = \overline {pa}\) , \(B = \overline {pb}\) and \(C = \overline {pc}\) we prove that if b,c are in the same side of line A and a,c are in the same side of B , then a and b are in distinct sides of C. As conclusions we deduce that ω is harmonic and that in each complete quadrangle the intersection points of the diagonals are never collinear, which is known as the axiom of Fano. So the Fano-axiom holds in each ordered plane, and also in those with boundary points.