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Didactic materials can be used to introduce abstract mathematical concepts. Various studies found positive effects of using manipulatives in mathematics teaching and learning. Teachers’ knowledge and attitudes towards manipulatives influence the profound and helpful implementation of manipulatives in the didactic process. There is a variety of ready-made and self-made manipulatives that can be utilized in mathematics education. Following that perspective, we questioned what role manipulatives play in Croatian primary education. For that purpose, we conducted research with teachers on the use of didactic materials in mathematics classroom teaching. Results showed that teachers use all sorts of manipulatives. They acknowledged the benefits of using manipulatives and most of them claimed they used manipulatives often in the didactic process. However, some answers indicate the lack of knowledge about the characteristics of manipulatives and the implementation of manipulatives in the didactic process.

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.

U radu je dan geometrijski dokaz bez riječi za kvadrat razlike, razliku kubova i kub razlike.

The 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.

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.

Some properties of parallelogram spaces will be examined. Some concrete examples of the quaternary relation on special sets which satisfy the required properties of a parallelogram space will be mentioned. The concepts of a vector and translation in a parallelogram space will be introduced. The concept of symmetry with respect to the pair of points will also be defined. A geometrical representation of the introduced concepts and relations between them will be given.

An idempotent medial quasigoup is a quasigroup which satisfies the identities of idempotency and mediality. Some geometric concepts can be introduced in each idempotent medial quasigroup. However, the definition of these concepts is given only in an implicit way. In some idempotent medial quasigroup with additional identities, these concepts could be defined in an explicit way. A number of geometric concepts in some special classes of idempotent medial quasigroups will be considered in this presentation. Specially, we are going to consider geometric concepts in a general GS–quasigroup and a quadratical quasigroup. We will study some simple geometric concepts whose existence will allow us to construct some more complicated geometric concepts. We will present the construction of an affine fullerene $C_{; ; 60}; ; $ in a general GS– quasigroup.

U ovom radu je prezentirana metoda rješavanja problemskih zadataka promjenom fokusa. Glavna karakteristika ove metode je usmjeravanje pozornosti rješavača na elemente koji nisu direktno istaknuti u postavci zadatka, a čije promatranje će omogućiti uspješno, često i vrlo elegantno i brzo, rješavanje zadatka. Primjena i korisnost ove metode je ilustrirana na zadacima i tvrdnjama različitog karaktera.

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.

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.

In this paper the concept of Crelle-Brocard points of the triangle in an isotropic plane is de ned. A number of statements about the relationship between Crelle-Brocard points and some other signi cant elements of a triangle in an isotropic plane are also proved. Some analogies with the Euclidean case are considered as well.

In this presentation a new class of idempotent medial quasigroups will be introduced, the so– called ARO–quasigroups. A quasigroup will be called ARO–quasigroup if it satisfies the identities of idempotency and mediality, i.e. we have the identities aa = a, ab · cd = ac · bd, and besides that if the identity ab · b = ba · a is also valid. Some examples of ARO–quasigroups will be given as well. These quasigroups are interesting because of the possibility of defining affine–regular octagons and to study them by means of formal calculations in a quasigroup. The “geometrical” concepts of a parallelogram and midpoint will be introduced in a general ARO– quasigroup. Some results about the introduced geometric concepts will be proved and a number of statements about new points obtained from the vertices of an affine–regular octagon will also be 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 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 $\mathbb{; ; ; C}; ; ; ((\frac{; ; ; 1}; ; ; {; ; ; 2}; ; ; (1+\sqrt 5))$.

A golden section quasigroup (shortly GS-quasigroup) is defned as an idempotent quasigroup which satis es the mutually equivalent identities a(ab c) c = b, a (a bc)c = b. In a general GS-quasigroup the geometrical concept of an a ne{;regular octahedron will be introduced. A number of statements about the relationships between an a ne{;regular octahedron and some other geometric concepts in a general GS-quasigroup will be proved. The geometrical representation of all proved statements will be given in the GS-quasigroup $C( 1\2(1 +\sqrt5))$.

In this lecture the concept of the Kiepert triangle of an allowable triangle in an isotropic plane will be studied. We are going to investigate the relationships between the areas and the Brocard angles of the standard triangle and its Kiepert triangle. It will also be 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 will be given. A number of statements concerning the triangle and its Kiepert triangles are also going to be proved.

A golden section quasigroup (shortly GS-quasigroup) is defined as an idempotent quasigroup which satisfies the mutually equivalent identities a(abc)c=b, a(abc)c=b. In this presentation identities and relations which are valid in a general GS-quasigroup will be researched. The geometrical meaning of the obtained identites will be given in the GS-quasigroup $C(1/2(1+\sqrt 5))$. Some interesting geometric concepts can be defined in a general GS-quasigroup. Namely, in a general GS-quasigroup the geometrical concept of the parallelogram, GS-trapezoid and some other geometric concepts can be introduced. The geometric concept of an affine-regular pentagon can be defined by means of GS-trapezoids. The concept of an affine-regular dodecahedron and affine-regular icosahedron can be obtained using the affine regular pentagons. Algebraic proofs of the statements about properties of the geometric concepts and the relationships between them in a general GS-quasigroup will be presented by means of the identities which are valid in a general GS-quasigroup. The geometrical representation of the introduced concepts and the obtained relations between them will be given in the GS-quasigroup $C(1/2(1+\sqrt 5))$.

In 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.

In 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.

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.

The 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.

The "geometric" concept of parallelogram is introduced and investigated in a general quadratical quasigroup and geometrical interpretation in a quadratical quasigroup $C(1/2(1+i)))$ is given. Some statements about relationships between the parallelograms and some other geometric structures in a general quadratical quasigroup will be also considered.

In 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.

The 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.

In 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 $C(1+\sqrt2/2)$.

The concept of the second Lemoine circle of the triangle in an isotropic plane is defined in this article. Some relationships between the introduced concept and some other elements of the triangle in an isotropic plane are also studied.

Some relations and statements concerning a heptagonal triangle will be studied in this lecture. The concept of the antiboutin triangle of the given heptagonal triangle will be introduced and some interesting relationships between these two triangles will be investigated. It will be also proved that the symmedian center, the Brocard diameter, the Brocard circle and the Lemoin line of the heptagonal triangle and its antiboutin triangle are coincident.

The concept of Brocard angle of the standard triangle in an isotropic plane $I_2$ is introduced. The relationships between Brocard angles of the allowable triangle and circum- Ceva's triangle of its centroid and circum- Ceva's triangle of its Feurbach point are investigated.

The concepts of the orthic axis, Lemoine line and Lngchamp's line of the triangle in an isotropic plane are defined. Some relationships between the introduced concepts and other elements of the triangle are defined.

The concept of an angle bisector of the triangle will be introduced in an isotropic plane. Some statements about relationships between the introduced concepts and some other previously studied geometric concepts about triangles will be investigated in an isotropic plane. A number of these statements seems to be new, and some of them are known in Euclidean geometry.

The isogonality with respect to the triangle and inversion with respect to the circle will be defined in an isotropic plane. The images of some lines and points with respect to these mappings will be studied.

Let $T$ be a triangle in a Euclidean plane. Let $g(T)$ be the orthic triangle of the triangle $T$, and let $g^2(T)$ be the orthic triangle of the triangle $g(T)$ ; generally let $g^{; ; n+1}; ; (T)$ be the orthic triangle of the triangle $g^{; ; n}; ; (T)$. In \cite{; ; DKN}; ; Dixmier, Kahane and Nicolas have proved, by means of trigonometric series, that for $n \rightarrow \infty$ the triangle $g^n(T)$ tends to the point $L$, a new characteristic point of the triangle $T$. If $(O, R)$ is the circle circumscribed to the triangle $T$, then it has been also shown that $|OL| \leq \frac{; ; 4}; ; {; ; 3}; ; R$ for all triangles $T$ and that $|OL| = \frac{; ; 4}; ; {; ; 3}; ; R$ if and only if the angles of $T$ are $\frac{; ; 4}; ; {; ; 7}; ; \pi$, $\frac{; ; 2}; ; {; ; 7}; ; \pi$, $\frac{; ; 1}; ; {; ; 7}; ; \pi$. This special triangle is called heptagonal triangle. It is very interesting and rare occurrence that heptagonal triangle is the extreme triangle because the extreme triangle in most of different extreme problems about triangles is equilateral triangle. It will be proved geometrically that equality in Dixmier-Kahane-Nicolas inequality $|OL| \leq \frac{; ; 4}; ; {; ; 3}; ; R$ is valid in the case of heptagonal triangle. The relationship between the initial heptagonal triangle $T$ and the obtained point $L$ will also be investigated.

The triangle with the angles $\frac{;\pi};{;7};, \frac{;2 \pi};{;7};, \frac{;4 \pi};{;7};$ is called a heptagonal triangle (according to Bankoff and Garfunkel). This triangle has many interesting properties and some of them will be presented in this talk. The relationship of the heptagonal triangle with its successive orthic triangles will be considered. The meaning of the heptagonal triangle, in the known Dixmier--Kahane--Nicholas inequality, will also be investigated.

The concept of pseudosquare in a general quadratical quasigroup is introduced and connections to some other geometrical concepts are studied. The geometrical presentations of some proved statetments are given in the quadratical quasigroup $C(\frac{;1};{;2};(1+i))$.

In this paper some interesting relations for the triangle with the angles $\frac{; \pi};{;7};$, $\frac{;2 \pi};{;7};$, $\frac{;4 \pi};{;7};$ are considered. Statements about lengths of its sides, Brocard angle and radius of circumscribed circle are proved

A GS--quasigroup is defined as an idempotent quasigroup which satisfies the mutually equivalent identities $a(ab \cdot c)\cdot c=b$, $a \cdot (a \cdot bc)c=b$. Some interesting geometric concepts can be defined in a general GS--quasigroup. The concepts of the affine regular polygons and polyhedra in a general GS--quasigroup are introduced by means of the identities and relations which are valid in a general GS--quasigroup. The geometrical representation of the introduced concepts and relations between them will be given in the GS--quasigroup $C( \frac{;1};{;2};(1+\sqrt 5 ))$.