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.
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References
Böhm, J. u.a.: Geometrie I, Deutscher Verlag der Wissenschaften, 5. Aufl., Berlin, 1988
Degen, W. und Profke, L.: Grundlagen der affinen und euklidischen Geometrie. B.G.Teubner, Stuttgart, 1976.
Karzel, H. und Kroll, H.-J.: Geschichte der Geometrie seit Hilbert.Wissensch. Buchgesellschaft, Darmstadt, 1988.
Karzel, H., Sörensen, K. und Windelberg, D.: Einf ührung in die Geometrie. Vandenhoeck & Ruprecht, Göttingen, 1973.
Sperner, E.: Die Ordnungsfunktionen einer Geometrie, Math. Ann. 121 (1949), 107–130.
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Jaritz, R. Eine Variante des Fano-Axioms fü angeordnete Ebenen. Geometriae Dedicata 64, 365–372 (1997). https://doi.org/10.1023/A:1004987026217
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DOI: https://doi.org/10.1023/A:1004987026217