### Marija Šimić Horvath, Vladimir Volenec and Jelena Beban-Brkić

Faculty of Architecture, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: marija.simic@arhitekt.hr

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

Faculty of Geodesy, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: jbeban@geof.hr

Abstract.   The geometry of the cyclic quadrangle in the isotropic plane has been discussed in [11]. Therein, its diagonal triangle and diagonal points were introduced. Hereby, we turn our attention to parabolas inscribed to non tangential quadrilaterals of the cyclic quadrangle. Non tangential quadrilaterals of the cyclic quadrangle are formed by taking its four sides out of six. Several properties of these parabolas related to diagonal points of the cyclic quadrangle are studied.

2010 Mathematics Subject Classification.   51N25.

Key words and phrases.   Isotropic plane, cyclic quadrangle, parabolas related to the cyclic quadrangle.

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References:

1. J. Beban-Brkić, Isometric invariants of conics in the isotropic plane-classification of conics, J. Geom. Graph. 6(1) (2002), 17-26.
MathSciNet

2. J. Beban-Brkić, M. Šimić and V. Volenec, On some properties of non cyclic quadrangle in isotropic plane, Proc. 13th Internat. Conf. Geom. Graphics, Dresden, 2008.

3. J. Beban-Brkić, M. Šimić and V. Volenec, On foci and asymptotes of conics in isotropic plane, Sarajevo J. Math. 3(16) (2007), 257-266.
MathSciNet

4. J. Coissard, Sur les quadrilateres inscrits dans un cercle, et circonscrits a une parabole, Rev. Math. Spec. 38 (1927-28), 217-219.

5. R. Goormaghtigh and R. Deaux, Sur le quadrilatere inscriptible, Mathesis 62 (1953), 155-157.
MathSciNet

6. R. Kolar-Šuper, Z. Kolar-Begović, J. Beban-Brkić and V. Volenec, Metrical relationships in standard triangle in an isotropic plane, Math. Commun. 10 (2005), 159-167.
MathSciNet

7. P. Lepiney, Question 2004, Mathesis 35 (1915), 31.

8. H. Sachs, Ebene isotrope Geometrie, Vieweg-Verlag, Braunschweig-Wiesbaden, 1987.
MathSciNet     CrossRef

9. K. Strubecker, Geometrie in einer isotropen Ebene, Math. Naturwiss. Unterr. 15 (1962–63), 297–306, 343–351, 385–394.
MathSciNet

10. V. Volenec, J. Beban-Brkić and M. Šimić, The focus and the median of a non-tangential quadrilateral in the isotropic plane, Math. Commun. 15 (2010), 117-127.
MathSciNet

11. V. Volenec, J. Beban-Brkić and M. Šimić, Cyclic quadrangle in the isotropic plane, Sarajevo J. Math. 7(20) (2011), 265-275.
MathSciNet