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On orthocentric systems in Minkowski planes

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Abstract

By studying some properties of suitable point quadruples in a normed plane, a natural way to define the notion of \(\mathcal {C}\)-orthocenter is presented. This allows to extend in a new form the notions or concepts of orthocentric systems, Euler’s line, six-point circles and the three-circles theorem to normed planes. In addition, new characterizations of the Euclidean plane are obtained.

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References

  • Amir, D.: Characterizations of Inner Product Spaces. Birkhäuser, Basel (1986)

    Book  MATH  Google Scholar 

  • Alonso, J.: Uniqueness properties of isosceles orthogonality in normed linear spaces. Ann. Sci. Math. Que. 18(1), 25–38 (1994)

    MATH  Google Scholar 

  • Alonso, J., Martini, H., Spirova, M.: Minimal enclosing discs, circumcircles, and circumcenters in normed planes (Part I). Comput. Geom. 45, 258–274 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Alonso, J., Martini, H., Wu, S.: On Birkhoff ortghogonality and isosceles orthogonality in normed linear spaces. Aequ. Math. 83, 153–189 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Asplund, E., Grünbaum, B.: On the geometry of Minkowski planes. Enseign. Math. 6, 299–306 (1960)

    Google Scholar 

  • Busemann, H.: Planes with analogues to Euclidean angular bisectors. Math. Scand. 36, 5–11 (1975)

    MATH  MathSciNet  Google Scholar 

  • Düvelmeyer, N.: Angle measures and bisectors in Minkowski planes. Can. Math. Bull. 48, 523–534 (2004)

    Article  Google Scholar 

  • Johnson, R.A.: Advanced Euclidean Geometry. Dover Publications Inc., Mineola (2007)

    Google Scholar 

  • Martini, H., Swanepoel, K.J., Weiß, G.: The geometry of Minkowski spaces—a survey. Part I. Expos. Math. 19, 97–142 (2001)

    Article  MATH  Google Scholar 

  • Martini, H., Spirova, M.: The Feuerbach circle and orthocentricity in normed planes. L’Enseign. Math. 53(2), 237–258 (2007)

    MATH  MathSciNet  Google Scholar 

  • Martini, H., Swanepoel, K.J.: The geometry of Minkowski spaces—a survey. Part II. Expos. Math. 22, 93–144 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Martini, H., Wu, S.: On orthocentric systems in strictly convex normed planes. Extr. Math. 24(1), 31–45 (2009)

  • Thompson, A.C.: Minkowski Geometry. Encyclopedia of Mathematics and Its Applications, vol. 63. Cambridge University Press,Cambridge (1996)

  • Wu, S.: Geometry of Minkowski Planes and Spaces—Selected Topics. Chemnitz University of Technology. PhD Thesis (2009)

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Authors and Affiliations

  1. Departamento de Matemática, Facultad Experimental de Ciencias, Universidad del Zulia, Maracaibo, Venezuela

    Tobías Rosas Soto & Wilson Pacheco Redondo

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  1. Tobías Rosas Soto
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  2. Wilson Pacheco Redondo
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Correspondence to Wilson Pacheco Redondo.

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Rosas Soto, T., Pacheco Redondo, W. On orthocentric systems in Minkowski planes. Beitr Algebra Geom 56, 249–262 (2015). https://doi.org/10.1007/s13366-014-0214-6

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  • DOI: https://doi.org/10.1007/s13366-014-0214-6

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