Two remarks on ordering in absolute planes

As well known, the underlying field K of an Euclidean plane E is Euclidean if and only if E fulfills the Circle Axiom. In this paper we consider an apparently weaker form of the Circle Axiom which leads to weaker properties of K. It will be shown that these weaker properties still characterize Euclidean fields. In the second remark we complete the construction of a certain kind of Hilbert plane considered in Pambuccian and Schacht (Beitr Algebra Geom, 2019. https://doi.org/10.1007/s13366-019-00445-y).