Covertex Inscribed Triangles of Parabola in Isotropic Plane

Abstract

In the paper the concept of a covertex inscribed triangle of a parabola in an isotropic plane is introduced. It is a triangle inscribed to the parabola that has the centroid on the axis of parabola, i.e. whose circumcircle passes through the vertex of the parabola. We determine the coordinates of the triangle centers, and the equations of the lines, circles and conics related to the triangle.Covertex trokuti upisani paraboli u izotropnoj ravnini SAŽETAK U radu se uvodi pojam covertex trokuta upisanog paraboli u izotropnoj ravnini. To je trokut upisan paraboličije težište leži na osi parabole, tj.čija opisana kružnica prolazi tjemenom parabole. Odreduju se koordinate točaka te jednadžbe pravaca, kružnica i konika povezanih s tim trokutom.Ključne riječi: izotropna ravnina, trokut, parabola MotivationThe following theorem, which can be found in [10], is a well-known fact from the geometry of Euclidean plane:Let A, B,C be three points on a parabola P different from its vertex and different mutually. These are the equivalent statements:1 0 The normal lines to P at A, B,C are concurrent. 2 0 The centroid of the triangle ABC lies on the axis of parabola P .3 0 The circumcircle of the triangle ABC passes through the vertex of parabola P .The perpendicularity is not defined in the isotropic plane, and often an isotropic line plays a role of a line perpendicular to a given one. Therefore, every normal to P passes through the absolute point. From that point of view, the property 1 0 is fulfilled for any three points on the parabola, and it is interesting to study the triangles having properties 2 0 and 3 0 .The result above together with other results stated in [10] inspired the authors to write this paper.