Konstruktivna algebra – algebarske strukture i prsten endomorfizma: (doktorska disertacija). Front Cover. Daniel A. Romano. D. A. Romano, QR code for Osnove algebarske strukture. Title, Osnove algebarske strukture. Author, Vladimir Benčić. Publisher, Znanje, Export Citation, BiBTeX EndNote. Title: Konstruktivna algebra – algebarske strukture i prsten endomorfizama. Author: Romano, Daniel. URI: Date:

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Some other statements about the introduced concepts and the connection with the concept of complementarity, isogonality, reciprocity, as well as the Brocard diameter, the Euler line, and the Steiner point of an allowable triangle are also considered. We investigate different ways of generating this special hyperbola and derive its equation in the case of a standard triangle in an isotropic plane. We prove that some remarkable points of a triangle in an isotropic plane lie on that hyperbola whose center is at the Feuerbach point of a triangle.

We also explore some other interesting properties of this hyperbola and its connection with some other significant elements of a triangle in an isotropic plane. 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. 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.

## Konstruktivna algebra – algebarske strukture i prsten endomorfizama

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. 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.

A number of significant algebarsks of the introduced concepts are considered. 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.

A number of statements about the properties of this mapping is proved.

The images of some well known elements of a triangle with respect to this mapping will be studied. 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 number of statements about the relationship between Crelle-Brocard points and some other significant elements of a triangle in an isotropic plane are also proved. Some analogies with the Euclidean case are considered as well.

Thereby different properties of the symmedian center, the Gergonne point, the Lemoine line and the de Longchamps line of these triangles are obtained. The theorem of struktrue unique determination of the affine regular icosahedron by means of its four vertices which satisfy certain conditions will be proved. The connection between affine regular icosahedron and affine regular octahedron in a general GS- quasigroup will be researched.

Some statements about new points obtained strujture the vertices of an affine-regular heptagon are also studied. The relationships between the shrukture and the Brocard angles of the standard triangle and its Kiepert triangle are studied. It is also proved that an allowable triangle and any of its Kiepert triangles are homologic.

In the case of a standard triangle the expressions for the center and the axis of this homology are given. The “geometric” concepts of midpoint, parallelogram and affine-regular octagon is introduced in a general ARO-quasigroup. Volenec, Dual Feuerbach theorem in an isotropic plane, Sarajevo Journal of Mathematics 18Abstract The dual Feuerbach theorem for an allowable triangle in an isotropic plane is proved analytically by means of the so-called standard triangle.

## Literatura

A number of statements about relationships between some concepts of the triangle and their dual concepts are also proved.

The connections of the introduced concept with some other elements of the triangle in an isotropic plane are also studied. Volenec, Thebault circles of the triangle in an isotropic plane, Mathematical Communications 15Abstract 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.

In this paper it is shown that in an isotropic plane the pencils of circles, corresponding to the Griffiths’s and Thebault’s pencil of circles in the Euclidean plane, coincide. Some statements about relationships between introduced concepts and some other previously studied geometric concepts about triangle will be investigated in an isotropic plane and some analogies with the Euclidean case will be also considered.

Volenec, The meaning of computer search in the study of some classes of IM-quasigroups, Croatian Journal of Education 53J. Non associative algebraic structures and their application Neasocijativne algebarske strukture i njihove primjeneMinistry of Science, Education and Sports of the Republic Croatia, Department of Mathematics, University Of Zagreb, Principal investigator: Prijedlog tema diplomskih radova.

In this paper, we examine the Jerabek hyperbola of an allowable triangle in an isotropic plane. The concept of the Kiepert hyperbola of an allowable triangle in an isotropic plane is introduced in this paper.

In this paper we examine the relationships between cubic structures, totally symmetric medial quasigroups, and commutative groups.

The concept of the Steiner point of a triangle in an algebarsoe plane is defined in this paper. The concepts of equicevian points and equiangular lines of a triangle in an isotropic plane are studied in this paper. 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 reciprocity with respect to a triangle is introduced in an isotropic plane. In this paper the concept of cosymmedian triangles in an isotropic plane is defined.

In this paper the concept of Crelle-Brocard points of the triangle in an isotropic plane is defined. Using the standard position of the allowable triangle in the isotropic plane relationships between this triangle and its contact and tangential triangle are studied. The concept of the affine regular icosahedron and affine regular octahedron in a general GS- quasigroup will be introduced in this paper.

The theorem about the characterization of a GS- quasigroup by means of a commutative group in which there is an automorphism which satisfies certain conditions, is proved directly. The concept of the affine-regular hexagon, by means of six parallelograms, is defined and investigated in any parallelogram space and geometrical interpretation in the affine plane is also given. In this paper, the concept of an ARH-quasigroup is introduced and identities valid in that quasigroup are studied.

In this paper the concept of the Kiepert triangle of an allowable triangle in an isotropic plane is introduced. In this paper the concept of ARO-quasigroup is introduced and some identities which are valid in a general ARO-quasigroup are proved.

### Faculty of Science and Education – Matematika

The slgebarske Feuerbach theorem for an allowable triangle in an isotropic plane is proved analytically by means of the so-called standard triangle. The concept of the Steiner’s ellipse of the triangle in an isotropic plane is introduced.

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.

In the Euclidean plane Griffiths’s and Thebault’s pencil of the circles are generally different. The concept of Apollonius circle and Apollonius axes of an allowable triangle in an isotropic plane will be introduced.