Author | Redakce |

Abstract | Not available |

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## Current Issue

### Content

### A group over the set of Pascal points on the sides of a convex quadrilateral

Author | D. Fraivert, D. Fraivert |

Abstract | The theory of a convex quadrilateral and a circle that forms Pascal points is a new topic in Euclidean geometry. This theory defines the concepts of "pairs of Pascal points on the sides of a convex quadrilateral" and "a circle that forms Pascal points". In the present paper, we shall find all the existing pairs of Pascal points and show that it is possible to define a group structure over the pairs of Pascal points on the sides of a convex quadrilateral. |

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### Constructions of Ovals

Author | M. Holešová |

Abstract | Ovals are often used in technical practice to approximate the ellipse. We will show interesting constructions of ovals from authors such as S. Serlio, G. Guarini and F.S. Meyer. We will present new ovals that they are result from the modification of the already known constructions. |

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### Buchberger algorithm and systems of polynomial equations for MO problem solvers [CZ]

Author | J. Hora, S. Königsmarková |

Abstract | In the mid-1960s B. Buchberger discovered an algorithm for finding the Gröbner base and thus opened a new way for solving algebraic equations. Would it be possible for talented students that participate mathematical competitions, to learn the basics of this method in a limited amount of time? This article proposes a procedure that allows the students to understand the complex theory, using simple examples based only on the lexicographical ordering of terms. |

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### Modeling in ProgeCAD in Teaching Constructive Geometry at the Faculty of Forestry and Wood Technology at Mendel University in Brno [CZ]

Author | A. Králová |

Abstract | This paper deals with implementation of 3D modeling in progeCAD into the Constructive Geometry course taught at The Faculty of Forestry and Wood Technology of the Mendel University in Brno. I introduce some teaching materials used during the lessons and also assignments required of students to pass the course. I demonstrate how to create 3D model of a solid and discuss some problems connected with 3D modeling in progeCAD. |

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### Absolute Circle (Sphere) Geometry by Reflection

Author | E. Molnár |

Abstract | In this survey we refresh and slightly extend the classical circle geometry and circle inversion as basic transformation, so that they can be extended to non-Euclidean planes, mainly to the Bolyai-Lobachevsky hyperbolic plane. These involve analytic discussions and extension to higher dimensions, based on projective polarity (metric), and lead to generalized Poincaré (conformal) models of non-Euclidean geometries. Novelties arise that we reformulate and extend (slightly renew) our German papers, and “introduce” them into the English electronic literature. We mainly concentrate on plane geometries, higher dimensional cases cause “only technical” (but tiresome) difficulties. |

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### Some Summation Techniques and the Possibilities of Their Use in Training Future Teachers of Mathematics [CZ]

Author | D. Tyr |

Abstract | The article deals with final summaries, especially summaries including combinatorial numbers. The author presents well-known and less-known summation techniques. They can be divided into two groups - techniques feasible without the use of algebraic software and techniques feasible by algebraic software (summation algorithms). Methods of determining the sums sought are demonstrated on suitably chosen examples, some of which can be solved using secondary school mathematics without the need for further mathematical knowledge. Others require basic knowledge of mathematical analysis, algebra, or the ability to control algebraic software. The content of the article is intended for students of mathematics teaching (future teachers). |

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### Apollonius’ tasks in navigation [CZ]

Author | Š. Voráčová |

Abstract | Navigation is defined as the science of getting a craft or person from one place to another. The historical development of navigation embraces great geometrical principles, the practical design of instruments of observation and wide-ranging methods of calculation. The geometrical nature of methods has not changed since the days of sail and offers an interesting historical perspective for any applied mathematics course. Similarly, the problem of Apollonius is closely related to the location problems in radio and satellite navigation. These ancient problems are still important in context of new technologies and new navigation systems. |

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### Frégier Points Revisited

Author | G. Weiss |

Abstract | Given a conic c and a point O on c, then the hypotenuses of right-angled triangles inscribed to c and having common vertex P intersect in one single point F, the Frégier point to O with respect to c. In the following this property will be referred to as “Theorem 1” and it is the starting point for further investigation. Varying O on c results in a point set {F} of a curve f of 2nd order. In general, f is a conic concentric and similar to c. In some sense, Frégier’s point of view is a variation of Thales’ theorem, and it can therefore also be applied to the Theorem of the Angle at Circumference as well as to higher-dimensional analogues of the mentioned Frégier points and conics. Even so the topic roots in Projective Geometry, an elementary geometric treatment might also be of interest and suites also as Maths course material for high schools. |

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