Rad HAZU, Matematičke znanosti, Vol. 27 (2023), 189-202.

ON SOME PROPERTIES OF KIEPERT PARABOLA IN THE ISOTROPIC PLANE

Vladimir Volenec and Zdenka Kolar-Begović

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: volenec@math.hr

Department of Mathematics and Faculty of Education, University of Osijek, 31 000 Osijek, Croatia
e-mail: zkolar@mathos.hr

Abstract.   In this paper we consider the curve which is an envelope of the axes of homology of a given triangle and the corresponding Kiepert triangles in the isotropic plane - the Kiepert parabola of the given triangle. We derive the equation of this parabola by using appropriate coordinate system. We give some new significant characterizations of this curve which are not valid in the Euclidean plane. We have also studied the relationships between Kiepert parabola and the Steiner point, the tangential triangle as well as the Jerabek hyperbola of the given triangle.

2020 Mathematics Subject Classification.   51N25.

Key words and phrases.   Isotropic plane, Kiepert parabola, Steiner point.

Full text (PDF) (free access)

DOI: https://doi.org/10.21857/yl4okf5qp9

References:

  1. Z. Kolar-Begović, R. Kolar-Šuper and V. Volenec, Jeřabek hyperbola of a triangle in an isotropic plane, KOG 22 (2018), 12-19.
    MathSciNet     CrossRef

  2. R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec and J. Beban-Brkić, Metrical relationships in a standard triangle in an isotropic plane, Math. Commun. 10 (2005), 149-157.
    MathSciNet

  3. R. Kolar-Šuper, Z. Kolar-Begović and V. Volenec, Steiner point of a triangle in an isotropic plane, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 20 (2016), 83-95.
    MathSciNet

  4. H. Sachs, Ebene isotrope Geometrie, Vieweg–Verlag, Braunschweig, 1987.
    MathSciNet     CrossRef

  5. K. Strubecker, Geometrie in einer isotropen Ebene, Math. Naturwiss. Unterricht 15 (1962/1963), 297-306, 343-351, 385-394.
    MathSciNet

  6. V. Volenec, Z. Kolar-Begović and R. Kolar-Šuper, Kiepert triangles in an isotropic plane, Sarajevo J. Math. 19 (2011), 81-90.
    MathSciNet

  7. V. Volenec, Z. Kolar-Begović and R. Kolar-Šuper, Kiepert hyperbola in an isotropic plane, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 22 (2018), 129-143.
    MathSciNet     CrossRef

  8. V. Volenec, Z. Kolar-Begović and R. Kolar-Šuper, Steiner's ellipses of the triangle in an isotropic plane, Math. Pannon. 21 (2010), 229-238.
    MathSciNet

Rad HAZU Home Page