Geometry where direction matters -- or does it?
In the present survey and its predecessors [East-West J. Math., Spec. Vol., 59--101 (2007; Zbl 1186.52004) or Bangkok: Mahidol University. 45--83 (2007; Zbl 1146.52004), Expo. Math. 19, No. 2, 97--142 (2001; Zbl 0984.52004), Expo. Math. 22, No. 2, 93--144 (2004; Zbl 1080.52005)] the authors order and summarize the hundreds of widespread papers on Minkowski geometry (=geometry of finite dimensional normed spaces) and aim to demonstrate the power of geometry as method in this discipline. Results concerning the following topics of elementary triangle geometry are exhibited: the re-entrant property (with the Flower of Life); the three-circle theorem; the orthocenter, the centroid, the circumcenter, and the Euler line of a triangle; the Miquel theorem; Clifford's chain theorem; a theorem on a regular \(4\)-covering; the Apollonius theorem; Glogovskii's definition of an angle-bisector; incircle of a triangle; the nine-point (or Feuerbach) circle of Euclidean geometry is generalized by a six-point circle; the Feuerbach circle of a quadrangle. Furthermore, the authors provide results regarding Kusner's problem and the lower and upper bounds for the equilateral number (cf. [\textit{K. J. Swanepoel}, Arch. Math. 83, No. 2, 164--170 (2004; Zbl 1062.52017); \textit{K. J. Swanepoel} and \textit{R. Villa}, Proc. Am. Math. Soc. 136, No. 1, 127--131 (2008; Zbl 1142.46006)]). The final chapter is devoted to the Fermat-Torricelli problem in Minkowski spaces (cf. [\textit{H. Martini}, \textit{K. J. Swanepoel} and \textit{G. Weiss}, J. Optimization Theory Appl. 115, No. 2, 283--314 (2002; Zbl 1047.90032)]). The article is accompanied by 11 very illustrative and helpful figures and a bibliography with 57 items.
floating Fermat-Torricelli configuration
absorbing Fermat-Torricelli configuration
scientific article; zbMATH DE number 5971478
scientific article