An invariance property of the tridens curve in the isotropic plane

Jürgen Tölke

Abstract: The tridens curves of third order and their generalizations in the isotropic plane over $\mathbb R$ were studied by D.Palman [P] and H.Sachs [S1,S2]. For additional properties see [T1,T2]. In this paper we prove that for every such tridens curve $T$ of third order there exists an inscribed triangle $\Delta$ with the property: $T$ remains invariant under the correspondence of opposite angle points of $\Delta$.

[P] Palman, D.: Über zirkuläre Kurven 3.Ordnung der isotropen Ebene. Rad. JAZU {\bf 444} (1989), 37--46. %

[S1] Sachs, H.: Ebene isotrope Geometrie. Vieweg-Verlag, Wiesbaden 1987. %

[S2] Sachs, H.: Vollständige zirkuläre Kurven n-ter Ordnung der isotropen Ebene. Stud. Sci. Math. Hungarica {\bf 24} (1989), 377--383. %

[T1] Tölke, J.: Zur Geometrie der Tridens-Kurven der Isotropen Ebene. Stud. Sci. Math. Hungarica {\bf 33} (1997), 433--440. %

[T2] Tölke, J.: A noteworthy property of the tridens curve in the isotropic plane. Beiträge Algebra Geom. {\bf 40} (1999), 275--282.

Keywords: isotropic plane, tridens curve of third order, opposite angle points

Classification (MSC2000): 51N25

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