The concept of the harmonic quadrangle and the associated Brocard points are introduced and investigated in the isotropic plane by employing suitable analytic methods.

U radu se prikazuju neki novi rezultati o Brocardovim točkama harmoničnog četverokuta u izotropnoj ravnini. Konstruiraju se novi harmonični četverokuti pridruženi danom četverokutu, te se proučavaju njihova svojstva vezana uz Brocardove točke.

A complete quadrilateral in the Euclidean plane is studied. The geometry of such quadrilateral is almost as rich as the geometry of a triangle, so there are lot of associated points, lines and conics. Hereby, the study was performed in the rectangular coordinates, symmetrically on all four sides of the quadrilateral with four parameters a, b, c, d. In this paper we will study the properties of some points, lines and circles associated to the quadrilateral. All these properties are well known, but here they are all proved by the same method. During this process, still some new results have appeared.

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.

The notion of the Gergonne point of a triangle in the Euclidean plane is very well known, and the study of them in the isotropic setting has already appeared earlier. In this paper, we give two generalizations of the Gergonne point of a triangle in the isotropic plane, and we study several curves related to them. The first generalization is based on the fact that for the triangle ABC and its contact triangle AiBiCi, there is a pencil of circles such that each circle km from the pencil the lines AAm, BBm, CCm is concurrent at a point Gm, where Am, Bm, Cm are points on km parallel to Ai,Bi,Ci, respectively. To introduce the second generalization of the Gergonne point, we prove that for the triangle ABC, point I and three lines q1,q2,q3 through I there are two points G1,2 such that for the points Q1,Q2,Q3 on q1,q2,q3 with d(I,Q1)=d(I,Q2)=d(I,Q3), the lines AQ1,BQ2 and CQ3 are concurrent at G1,2. We achieve these results by using the standardization of the triangle in the isotropic plane and simple analytical method.