In 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 centre is at the Feuerbach point of a triangle. We also explore other interesting properties of this hyperbola and its connection with some other significant elements of a triangle in an isotropic plane.

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. 2010 Mathematics Subject Classification. 20N05.

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In 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.

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.