Degenerate Forms of Miquel Axiom and Dilatations in Benz Planes

Abstract

We study Benz planes with dilatations and show that for a Benz plane $\mathfrak B:=(\mathcal {P}, \mathcal {C}, \mathfrak G)$ the following statements are equivalent:

• $\mathfrak B$ satisfies a degenerate form of Miquel axiom, called $\mathbf{M}^{\mathbf{III}}_{\mathbf{6}}$ and for each point $W\in \mathcal {P}$ the affine derivation $\partial _W(\mathfrak B)$ is desarguesian.

• For each two nonparallel points $P,Q\in {\mathcal {P}}$ there exists an automorphism $\sigma _{_{P,Q}}$ of $\mathfrak B$ such that ${\mathrm{Fix}}~\sigma _{_{P,Q}}=\{P,Q\}$ and in the affine derivation $\partial _Q(\mathfrak B)$ the map $\sigma _{_{P,Q}}$ is the point reflection at P.

Hence by Monatsh Math 86:131–142, 1978, if $\mathfrak L$ is an ovoidal Laguerre plane where the associated skewfield has characteristic $\ne 2$ and satisfies $\mathbf{M}^{\mathbf{III}}_{\mathbf{6}}$ then it is a miquelian Laguerre plane, i.e. for these planes $\mathbf{M}^{\mathbf{III}}_{\mathbf{6}}$ and $\mathbf M_8$ are equivalent.(cf. Theorem 1.5).