Cross-ratios and 6-figures in some Moufang-Klingenberg planes

Abstract

This paper deals with Moufang-Klingenberg planes M(A) defined over a local alternative ring A of dual numbers. The definition of cross-ratio is extended to M(A). Also, some properties of cross-ratios and 6-figures that are well-known for Desarguesian planes are investigated in M(A); so we obtain relations between algebraic properties of A and geometric properties of M(A). In particular, we show that pairwise non-neighbour four points of the line g are in harmonic position if and only if they are harmonic, and that p is Menelaus or Ceva 6-figure if and only if r (mu) = - 1 or r (mu) = 1, respectively.