## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Cross-Ratios and 6-Figures in some Moufang-Klingenberg Planes

#### Abstract

This paper deals with Moufang-Klingenberg planes $\boldsymbol{M}(\mathcal{A})$ defined over a local\ alternative ring $\mathcal{A}$\ of dual numbers. The definition of cross-ratio is extended to $\boldsymbol{M}(\mathcal{A})$. Also, some properties of cross-ratios and 6-figures that arewell-known for Desarguesian planes are investigated in $\boldsymbol{M}(\mathcal{A})$; so we obtain relations between algebraic properties of $\mathcal{A}$ and geometric properties of $\boldsymbol{M}(\mathcal{A})$. 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 $\mu$ is Menelaus or Ceva 6-figure if and only if $r\left( \mu \right) =-1$ or $r\left( \mu \right) =1,$ respectively.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 1 (2008), 49-64.

Dates
First available in Project Euclid: 22 February 2008

https://projecteuclid.org/euclid.bbms/1203692446

Digital Object Identifier
doi:10.36045/bbms/1203692446

Mathematical Reviews number (MathSciNet)
MR2406086

Zentralblatt MATH identifier
1138.51002

#### Citation

Akpinar, Atilla; Celik, Basri; Ciftci, Süleyman. Cross-Ratios and 6-Figures in some Moufang-Klingenberg Planes. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 1, 49--64. doi:10.36045/bbms/1203692446. https://projecteuclid.org/euclid.bbms/1203692446