On Miquel's theorem and inversions in normed planes
A normed plane \(\mathbb M\) is a 2-dimensional real vector space equipped with a norm. If the unit circle \(\mathcal C\) of all vectors of norm 1 in \(\mathbb M\) is strictly convex and smooth, then \textit{G. Ewald} [Abh. Math. Semin. Univ. Hamb. 30, 179--187 (1967; Zbl 0147.19802)] showed geometrically that the collection of all homothethic images of \(\mathcal C\) plus all Euclidean lines extended by a new point \(\infty\) form the circles of a spherical Möbius plane. The case where \(\mathcal C\) is an ellipse is characterized by the validity of Miquel's theorem, a well-known result which can be traced back to \textit{B. Hesselbach} [Abh. Math. Semin. Hamb. Univ. 9, 265--271 (1933; Zbl 0007.07303)]. In view of the correspondence between strictly convex, smooth normed planes and Möbius planes it is natural to consider Miquel's configuration in normed planes. \textit{H. Martini} and the author [Discrete Math. 309, 5158--5168 (2009; Zbl 1186.52007)] showed that Miquel's configuration closes in a strictly convex normed plane if the considered circles have equal radii. In the paper under review, the author extends a result of this kind to all normed planes without the assumption of strict convexity. Since circles can now intersect in segments, the formulation of an analogue to Miquel's configuration is rather involved. To this end the author introduces the notion of conjugate points with respect to two circles of equal radii. Four points \(p_i\), \(i=1,2,3,4\), in a cyclic order on a circle \(x+\lambda\mathcal{C}\) determine four further circles of radius \(\lambda\), whose centres \(y_i\) are obtained as conjugate points of \(x\) with respect to the consecutive circles \(p_i+\lambda\mathcal{C}\) and \(p_{i+1}+\lambda\mathcal{C}\). It is then shown that the four points that are the conjugates to the \(p_i\) with respect to consecutive pairs \(y_i+\lambda\mathcal{C}\) and \(y_{i+1}+\lambda\mathcal{C}\) of these four circles are on a circle with the same radius \(\lambda\). Further properties of this special configuration of points and circles are obtained. In the last section the author investigates inversions of a strictly convex, smooth normed plane \(\mathbb M\), which are defined as involutions of the associated Möbius plane that fix precisely the points of one circle. She shows that the existence of one inversion at a circle of \(\mathbb M\) implies that \(\mathbb M\) is Euclidean (ie, \(\mathcal C\) is an ellipse). This result makes slightly weaker assumptions than a characterization by \textit{K. Strambach} [Math. Z. 113, 266--292 (1970; Zbl 0192.57503)] of the spherical Miquelian Möbius plane as a symmetric spherical Möbius plane (one that admits an inversion at each of its circles). As a consequence, a new proof of the fact that Miquelian in the associated spherical Möbius plane implies Euclidean in the strictly convex, smooth normed plane is obtained.
scientific article; zbMATH DE number 5806775
scientific article