### Zdenka Kolar-Begović

School of Applied Mathematics and Informatics

Josip Juraj Strossmayer University of Osijek

Josip Juraj Strossmayer University of Osijek

### Research Interests

- Nonassociative algebraic structures
- Geometry
- Mathematics Teaching

### Degrees

- PhD in Mathematics, Department of Mathematics, University of Zagreb, 2003
- Msc in Mathematics, Department of Mathematics, University of Zagreb, 1999
- Bsc in Mathematics and Physics, University of Osijek, 1993

### Publications

Journal Publications

- Z. Kolar-Begović, V. Volenec, Hamilton triangle of a triangle in the isotropic plane, Mathematica Pannonica New Series
**28**/1 (2022), 1-10In this paper we introduce the concept of the Hamilton triangle of a given triangle in an isotropic plane and investigate a number of important properties of this concept. We prove that the Hamilton triangle is homological with the observed triangle and with its contact and complementary triangles. We also consider some interesting statements about the relationships between the Hamilton triangle and some other significant elements of the triangle, like e.g. the Euler and the Feuerbach line, the Steiner ellipse and the tangential triangle. - V. Volenec, Z. Kolar-Begović, On some properties of Kiepert parabola in the isotropic plane, Rad HAZU, Matematičke znanosti. (2022), prihvaćen za objavljivanjeIn this paper we consider the curve which is an envelope of the axes of homology of a given triangle and the corresponding Kiepert triangles in the isotropic plane - the Kiepert parabola of the given triangle. We derive the equation of this parabola by using appropriate coordinate system. We give some new significant characterizations of this curve which are not valid in the Euclidean plane. We have also studied the relationships between Kiepert parabola and the Steiner point, the tangential triangle as well as the Jeřabek hyperbola of the given triangle.
- V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, On Some Properties of the First Brocard Triangle in the Isotropic Plane, Mathematics
**10**/9 (2022), 1-13In this paper we introduce the first Brocard triangle of an allowable triangle in the isotropic plane and derive the coordinates of its vertices in the case of a standard triangle. We prove that the first Brocard triangle is homologous to the given triangle and that these two triangles are parallelogic. We consider the relationships between the first Brocard triangle and the Steiner axis, the Steiner point, and the Kiepert parabola of the triangle. We also investigate some other interesting properties of this triangle and consider relationships between the Euclidean and the isotropic case. - A. Katalenić, Z. Kolar-Begović, Prospective Primary School Teachers’ Work in Continuous Online Assessments in the Course of Didactics of Mathematics, Mathematics Teaching-Research Journal
**14**/4 (2022), 80-105This study aimed to examine students’ work in online assessments to gain more understanding for designing continuous assessments in a blended learning environment for prospective primary school teachers. The study took place during emergency remote teaching due to the COVID-19 pandemic. Course work for prospective primary school teachers in the didactics of mathematics course included continuous, obligatory, non-graded, online assessments. We performed a qualitative content analysis of their answers. Students’ work was examined regarding the content knowledge and requirements in the questions, based on the categories of Subject Matter Knowledge and Mathematical Assessment Task Hierarchy taxonomy. The results showed that students’ study approach was strategic, relying heavily on peer support. Their work differed concerning the content and requirements of the questions. Students were more engaged in questions that required creating examples, discussing definitions and properties, and solving contextual problems. Questions related to horizon content knowledge were most challenging for students. We discussed how the results of our study could affect the design of continuous assessment in a blended learning environment for prospective primary school teachers. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Inflection Points in Cubic Structures, Mathematics
**9**/21 (2021)In this paper, we introduce and study new geometric concepts in a general cubic structure. We deﬁne the concept of the inﬂection point in a general cubic structure and investigate relationships between inﬂection points and associated and corresponding points in a general cubic structure. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Tangentials in cubic structures, Glasnik Matematički
**55**/2 (2020), 337-349In this paper we study geometric concepts in a general cubic structure. The well-known relationships on the cubic curve motivate us to introduce new concepts into a general cubic structure. We will define the concept of the tangential of a point in a general cubic structure and we will study tangentials of higher-order. The characterization of this concept will be also given by means of the associated totally symmetric quasigroup. We will introduce the concept of associated and corresponding points in a cubic structure, and discuss the number of mutually different corresponding points. The properties of the introduced geometric concepts will be investigated in a general cubic structure. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, A complete system of the shapes of triangles, Glasnik Matematički
**54**/2 (2019), 409-420In 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. - Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, Brocard circle of the triangle in an isotropic plane, Mathematica Pannonica
**26**/1 (2018), 103-113The concept of the Brocard circle of a triangle in an isotropic plane is deﬁned in this paper. 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. - Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, Jerabek hyperbola of a triangle in an isotropic plane, KoG (Scientific and Professional Journal of Croatian Society for Geometry and Graphics)
**22**/22 (2018), 12-19In this paper, we examine the Jerabek hyperbola of an allowable triangle in an isotropic plane. 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. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Kiepert hyperbola in an isotropic plane, Rad HAZU, Matematičke znanosti.
**22**/534 (2018), 129-143The 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. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Cubic structure, Glasnik Matematički
**52**/2 (2017), 247-256In 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. - R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, Steiner point of a triangle in an isotropic plane, Rad HAZU, Matematičke znanosti.
**20**/528 (2016), 83-95The 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. - Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, Equicevian points and equiangular lines of a triangle in an isotropic plane, Sarajevo Journal of Mathematics
**11**/23 (2015), 101-107The concepts of equicevian points and equiangular lines of a triangle in an isotropic plane are studied in this paper. A number of significant properties of the introduced concepts are considered. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Affine Fullerene C_60 in a GS-Quasigroup, Journal of Applied Mathematics
**2014**(2014), 1-8It 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 $C((1/2)(1+sqrt 5))$. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Reciprocity in an isotropic plane, Rad HAZU, Matematičke znanosti.
**519**/18 (2014), 171-181The concept of reciprocity with respect to a triangle is introduced in an isotropic plane. 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. - J. Beban-Brkić, V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Cosymmedian triangles in an isotropic plane, Rad HAZU, Matematičke znanosti.
**515**/2013 (2013), 33-42In 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. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Crelle-Brocard points of the triangle in an isotropic plane, Mathematica Pannonica
**24**/2 (2013), 167-181In this paper the concept of Crelle-Brocard points of the triangle in an isotropic plane is defined. 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. - J. Beban-Brkić, V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, On Gergonne point of the triangle in isotropic plane, Rad HAZU, Matematičke znanosti.
**515**/2013 (2013), 95-106Using the standard position of the allowable triangle in the isotropic plane relationships between this triangle and its contact and tangential triangle are studied. Thereby different properties of the symmedian center, the Gergonne point, the Lemoine line and the de Longchamps line of these triangles are obtained. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Affine regular icosahedron circumscribed around the affine regular octahedron in GS--quasigroup, Commentationes Mathematicae Universitatis Carolinae
**53**/3 (2012), 501-507The concept of the affine regular icosahedron and affine regular octahedron in a general GS- quasigroup will be introduced in this paper. The theorem of the 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. The geometrical representation of the introduced concepts and relations between them will be given in the GS- quasigroup $mathbb{ C} ((frac{1}{2}(1+sqrt 5))$. - Z. Kolar-Begović, A short direct characterization of GS-quasigroups, Czechoslovak Mathematical Journal
**61**/136 (2011), 3-6The 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. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Affine-regular hexagons in the parallelogram space, Quasigroups and Related Systems
**19**(2011), 353-358The 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. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, ARH-quasigroups, Mathematical Communications
**16**(2011), 539-550In this paper, the concept of an ARH-quasigroup is introduced and identities valid in that quasigroup are studied. The geometrical concept of an affine-regular heptagon is defined in a general ARH-quasigroup and geometrical representation in the quasigroup $C(2 cos pi/7)$ is given. Some statements about new points obtained from the vertices of an affine-regular heptagon are also studied. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Kiepert triangles in an isotropic plane, Sarajevo Journal of Mathematics
**7**/19 (2011), 81-90In this paper the concept of the Kiepert triangle of an allowable triangle in an isotropic plane is introduced. The relationships between the areas 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. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, ARO-quasigroups, Quasigroups and Related Systems
**18**(2010), 213-228In this paper the concept of ARO-quasigroup is introduced and some identities which are valid in a general ARO-quasigroup are proved. The "geometric" concepts of midpoint, parallelogram and affine-regular octagon is introduced in a general ARO-quasigroup. The geometric interpretation of some proved identities and introduced concepts is given in the quasigroup $C(1+sqrt2/2)$. - R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, Dual Feuerbach theorem in an isotropic plane, Sarajevo Journal of Mathematics
**18**(2010), 109-115The dual Feuerbach theorem for an allowable triangle in an isotropic plane is proved analytically by means of the so-called standard triangle. A number of statements about relationships between some concepts of the triangle and their dual concepts are also proved. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Steiner's ellipses of the triangle in an isotropic plane, Mathematica Pannonica
**21**/2 (2010), 229-238The concept of the Steiner's ellipse of the triangle in an isotropic plane is introduced. The connections of the introduced concept with some other elements of the triangle in an isotropic plane are also studied. - R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, Thebault circles of the triangle in an isotropic plane, Mathematical Communications
**15**(2010), 437-442In 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. - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Thebault's pencil of circles in an isotropic plane, Sarajevo Journal of Mathematics
**18**(2010), 237-239In the Euclidean plane Griffiths's and Thebault's pencil of the circles are generally different. 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. - Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, Brocard angle of the standard triangle in an isotropic plane, Rad HAZU, Matematičke znanosti.
**503**(2009), 55-60 - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Heptagonal triangle as the extreme triangle of Dixmier-Kahane-Nicolas inequality, Mathematical Inequalities and Applications
**12**/4 (2009), 773-779 - Z. Kolar-Begović, V. Volenec, LGS-quasigroups, Quasigroups and Related Systems
**17**(2009), 77-90 - V. Volenec, J. Beban-Brkić, R. Kolar-Šuper, Z. Kolar-Begović, Orthic axis, Lemoine line and Longchamp's line of the triangle in I_2., Rad HAZU, Matematičke znanosti.
**503**(2009), 13-19 - Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, The second Lemoine circle of the triangle in an isotropic plane, Mathematica Pannonica
**20**/1 (2009), 79-86 - V. Volenec, Z. Kolar-Begović, Affine regular decagons in GS-quasigroups, Commentationes Mathematicae Universitatis Carolinae
**49**/3 (2008), 383-395 - Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, Angle bisectors of a triangle in I_2, Mathematical Communications
**13**/1 (2008), 97-105 - R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, Apollonius circles of the triangle in an isotropic plane, Taiwanese journal of mathematics
**12**/5 (2008), 1239-1249The concept of Apollonius circle and Apollonius axes of an allowable triangle in an isotropic plane will be introduced. 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. - R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, J. Beban-Brkić, Isogonality and inversion in an isotropic plane, International Journal of Pure and Applied Mathematics
**44**/3 (2008), 339-346 - V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Two characterizations of the triangle with the angles $ frac{pi}{7}, frac{2 pi}{7}, frac{4 pi}{7}$, International Journal of Pure and Applied Mathematics
**44**/3 (2008), 335-338 - Z. Kolar-Begović, R. Kolar-Šuper, Six concyclic points, Mathematical Communications
**12**(2007), 255-256 - R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, The first Lemoine circle of the triangle in an isotropic plane, Mathematica Pannonica
**18**/2 (2007), 253-263 - Z. Kolar-Begović, V. Volenec, The meaning of computer search in the study of some classes of IM-quasigroups, Croatian Journal of Education
**53**(2007), 293-297 - J. Beban-Brkić, R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, On Feuerbach's theorem and a pencil of circles in I_2, Journal for Geometry and Graphics
**10**/2 (2006), 125-132

Others

- R. Kolar-Šuper, Z. Kolar-Begović, Dokaz bez riječi: kvadrat razlike, razlika kubova, kub razlike, Osječki matematički list
**19**/1 (2019), 45-48U radu je dan geometrijski dokaz bez riječi za kvadrat razlike, razliku kubova i kub razlike - Z. Kolar-Begović, V. Ždralović, Vivianijev teorem , Osječki matematički list
**19**/1 (2019), 31-41U ovom radu se razmatra tvrdnja poznata u literaturi kao Vivianijev teorem prema kojemu je suma udaljenosti bilo koje točke jednakostraničnog trokuta od stranica trokuta jednaka visini tog trokuta. Dano je nekoliko različitih dokaza teorema te navedena poopćenja i analogoni teorema. - M. Alilović, Z. Kolar-Begović, Lj. Primorac Gajčić, Wallace-Simsonov pravac, Osječki matematički list
**19**/2 (2019), 137-146U radu se razmatra pravac na kojem leže nožišta okomica, povuče- nih na stranice trokuta, iz točke koja leži na kružnici opisanoj tom tro- kutu. Taj pravac je u literaturi poznat pod imenom Wallace-Simsonov pravac. Navedeni su zanimljivi elementi povijesti otkrića ovog pravca. Promatrana su neka njegova zanimljiva geometrijska svojstva te veze s Eulerovom kružnicom trokuta.

Books

- Z. Kolar-Begović, R. Kolar-Šuper, A. Katalenić, Advances in Research on Teaching Mathematics, Odjel za matematiku, Fakultet za odgojne i obrazovne znanosti, Osijek, 2022.
- Z. Kolar-Begović, R. Kolar-Šuper, Lj. Jukić Matić, Towards new perspectives on mathematics education, Fakultet za odgojne i obrazovne znanosti i Odjel za matematiku, Sveučilište u Osijeku, Osijek, 2019.
- Z. Kolar-Begović, R. Kolar-Šuper, Lj. Jukić Matić, Mathematics Education as a Science and a Profession, Odjel za matematiku i Fakultet za odgojne i obrazovne znanosti, Osijek, 2017.
- Z. Kolar-Begović, R. Kolar-Šuper, I. Đurđević, Higher Goals in Mathematics Education , Odjel za matematiku, Fakultet za odgojne i obrazovne znanosti , Osijek, 2015.
- M. Pavleković, Z. Kolar-Begović, R. Kolar-Šuper, Mathematics teaching for the future, Odjel za matematiku, Fakultet za odgojne i obrazovne znanosti, Osijek, 2013.

### Projects

**Participation (as researcher) in work of the following projects:**

- Regionalni znanstveni centar Panonske Hrvatske, stručni projekt, nositelj Osječko-baranjska županija (2022.-2024.)
- Non associative algebraic structures and their application (Neasocijativne algebarske strukture i njihove primjene), Ministry of Science and Education of the Republic Croatia, Department of Mathematics, University Of Zagreb, Principal investigator: Vladimir Volenec
- Geometric and algebraic geometric structures (Geometrije i algebarsko geometrijske strukture), Ministry of Science and Education of the Republic Croatia, Department of Mathematics, University Of Zagreb, Principal investigator: Vladimir Volenec

### Professional Activities

**Editorial Board**

Editor in Chief of the Journal Osječki matematički list (since 2012)

**Refereeing/Reviewing**

Mathematical Reviews

### Teaching

2023/2024

Elementary Geometry

Analytic and Constructive Geometry

### Personal

- Birthdate: March 24, 1969
- Birthplace: Sremska Mitrovica
- Citizenship: Croatian
- Family: Married with Ivica, daughter Dolores, son Alojzije