Equisegmentary lines of a triangle in the isotropic plane

In this paper we introduce the concept of equisegmentary lines in the isotropic plane. We derive the equations of equisegmentary lines for a standard triangle and prove that the angle between them is equal to the Brocard angle of a standard triangle. We study the dual Brocard circle, the circle whose tangents are equisegmentary lines, as well as the inertial axis and the Steiner axis. Some interesting properties of this circle are also investigated

Bouvaistova kubika trokuta u izotropnoj ravnini

The cubic in an isotropic plane which passes through the intersections of the sides of an orthic triangle with the sides of a complementary triangle of a given triangle, and through the point which is complementary to the Steiner point of triangle is studied in this paper. It is proved that its non-isotropic asymptote is parallel to Lemoine line of a given triangle.U članku se proučava kubika koja prolazi kroz sjecišta stranica ortotrokuta i komplementarnog trokuta danog trokuta i kroz točku komplementarnu Steinerovoj točki tog trokuta. Dokazuje se da je neizotropna asimptota kubike paralelna s Lemoineovim pravcem danog trokuta

Kiepert hyperbola of a triangle in an isotropic plane

The concept of the Kiepert hyperbola of an allowable triangle in an isotropic plane is introduced in this paper. Important properties of the Kiepert hyperbola will be investigated in the case of the standard triangle. The relationships between the introduced concepts and some well known elements of a triangle will also be studied

Thebault circles of the triangle in an isotropic plane

In this paper the existence of three circles, which touch the circumscribed circle and Euler circle of an allowable triangle in an isotropic plane, is proved. Some relations between these three circles and elements of a triangle are investigated. Formulae for their radii are also given

Cubic structure

In this paper we examine the relationships between cubic structures, totally symmetric medial quasigroups, and commutative groups. We prove that the existence of a cubic structure on the given set is equivalent to the existence of a totally symmetric medial quasigroup on this set, and it is equivalent to the existence of a commutative group on this set. We give also some interesting geometric examples of cubic structures. By means of these examples, each theorem that can be proved for an abstract cubic structure has a number of geometric consequences. In the final part of the paper, we prove also some simple properties of abstract cubic structures

Affine Fullerene C 60

It will be shown that the affine fullerene C60, which is defined as an affine image of buckminsterfullerene C60, can be obtained only by means of the golden section. The concept of the affine fullerene C60 will be constructed in a general GS-quasigroup using the statements about the relationships between affine regular pentagons and affine regular hexagons. The geometrical interpretation of all discovered relations in a general GS-quasigroup will be given in the GS-quasigroup C(1/2(1+5))

Reciprocity in an isotropic plane

The concept of reciprocity with respect to a triangle is introduced in an isotropic plane. A number of statements about the prop erties of this mapping is proved. The images of some well known elements of a triangle with respect to this mapping will be studied

Cosymmedian triangles in isotropic plane

In this paper the concept of cosymmedian triangles in an isotropic plane is defined. A number of statements about some important properties of these triangles will be proved. Some analogies with the Euclidean case will also be considered

A complete system of the shapes of triangles

In this paper we examine the shape of a triangle by means of a ternary operation which satisfies some properties. We prove that each system of the shapes of triangles can be obtained by means of the field with defined ternary operation. We give a geometric model of the shapes of triangles on the set of complex numbers which motivate us to introduce some geometric concepts. The concept of transfer is defined and some interesting properties are explored. By means of transfer the concept of a parallelogram is introduced

Steiner point of a triangle in an isotropic plane

The concept of the Steiner point of a triangle in an isotropic plane is defined in this paper. Some different concepts connected with the introduced concepts such as the harmonic polar line, Ceva’s triangle, the complementary point of the Steiner point of an allowable triangle are studied. Some other statements about the Steiner point and the connection with the concept of the complementary triangle, the anticomplementary triangle, the tangential triangle of an allowable triangle as well as the Brocard diameter and the Euler circle are also proved