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.
U radu proučavamo potpuni četverostran u euklidskoj ravnini. Poput trokuta i potpuni četverostran ima mnogo zanimljivih svojstava te pridruženih točaka, pravaca i konika. Ovdje je proučavanje provedeno korištenjem pravokutnih koordinata, simetrično po sve četiri stranice četverostrana s četiri parametra
a, b, c, d.
Proučavamo svojstva točaka, pravaca i kružnica pridruženih četverostranu. Gotovo sve tvrdnje prikazane u ovom radu su dobro poznate, ali su se ipak ponegdje usput pojavili i neki novi rezultati.
Geometrija netetivnog četverovrha u izotropnoj ravnini uvedena je u člancima 2 and 6. Ovdje se proučava dijagonalni trokut i daju se neka njegova lijepa svojstva.
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.
U radu se uvodi pojam covertex trokuta upisanog paraboli u izotropnoj ravnini. To je trokut upisan paraboli čije težište leži na osi parabole, tj. čija opisana kružnica prolazi tjemenom parabole. Određuju se koordinate točaka te jednadžbe pravaca, kružnica i konika povezanih s tim trokutom.
U radu se uvodi pojam covertex trokuta upisanog paraboli u izotropnoj ravnini. To je trokut upisan paraboli čije težište leži na osi parabole, tj. čija opisana kružnica prolazi tjemenom parabole. ...Određuju se koordinate točaka te jednadžbe pravaca, kružnica i konika povezanih s tim trokutom.
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.
This paper gives a complete classification of conics in $PE_2(\mathbb{R})$.
The classification has been made earlier (Reveruk 5), but it showed to be
incomplete and not possible to cite and use in ...further studies of properties of
conics, pencil of conics, and of quadratic forms in pseudo-Euclidean spaces.
This paper provides that. A pseudo-orthogonal matrix, pseudo-Euclidean values
of a matrix, diagonalization of a matrix in a pseudo-Euclidean way are
introduced. Conics are divided in families and by types, giving both of them
geometrical meaning. The invariants of a conic with respect to the group of
motions in $PE_2(\mathbb{R})$ are determined, making it possible to determine a
conic without reducing its equation to canonical form. An overview table is
given.
This paper gives a complete classification of conics in
PE_2(R). The classification has been made earlier (Reveruk 5), but it
showed to be incomplete and not possible to cite and use in further ...studies of properties of conics, pencil of conics, and of quadratic forms in pseudo-Euclidean spaces. This paper provides that. A pseudo-orthogonal matrix, pseudo-Euclidean values of a matrix, diagonalization of a matrix in a pseudo-Euclidean way are introduced. Conics are divided in families and by types, giving both of them geometrical meaning. The invariants of a conic with respect to the group of motions in PE_2(R) are determined, making it possible to determine a conic without reducing its equation to canonical form. An overview table is given.
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.
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