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Cross-ratios and 6-figures in some Moufang-Klingenberg planes. (English) Zbl 1138.51002

This paper deals with Moufang-Klingenberg planes over a local alternative ring of dual numbers. The cross-ratio of a quadruple of pairwise non-neighboring points is defined and some properties of the cross-ratio are investigated. Finally some special configurations, called 6-figures, are generalized from ordinary desarguesian and Moufang projective planes to Moufang-Klingenberg planes and the ratio of such a figure is defined (using cross-ratios). It is then proved that a 6-figure is a Menelaus or Ceva configuration iff the corresponding ratio equals \(-1\) or 1 respectively.

MSC:

51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
51A35 Non-Desarguesian affine and projective planes
17D05 Alternative rings
51A20 Configuration theorems in linear incidence geometry