Circle configurations in strictly convex normed planes

Abstract

Abstract We present extensions of results of P. J. Kelly [Amer. Math. Monthly 57: 677–678, 1950] on the so-called re-entrant property of circles in strictly convex normed (or Minkowski) planes, and also further properties of circles, well known for the Euclidean plane, are generalized for all strictly convex Minkowski planes. More precisely, we present “Minkowskian analogues” of the philosophical symbol Yin-Yang, of the Arbelos, a special case of the famous Apollonius problem on circles touching each other, and one of the Sangaku-circles problems (coming from the Japanese Temple Geometry). The latter is remarkable since the consideration of this type of problems in the Euclidean plane requires the use of inversion or of the Pythagorean Theorem (i.e., of tools having no analogues in normed planes). Finally we observe that a strictly convex normed plane which, in addition, is smooth can be considered as a flat Möbius plane where, for example, Apollonius' problem is solved.