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Pythagorean triangles
Sláma, Michal ; Jančařík, Antonín (advisor) ; Kvasz, Ladislav (referee)
Title: Pythagorean triangles Author: Michal Sláma Department: Department of mathematics and mathematical education Supervisor: RNDr. Antonín Jančařík, Ph.D. Abstract: The thesis considers usable problems concerning pythagorean triangles. In the first part, there is described an expression of a parametrization of pythagorean triangles. Next parts are dedicated to an expression of properties of edges and radii of the incircle and excircles. Next chapters describe Heronian triangles and their decomposition to pythagorean triangles, as a way to solution of the problem of a Heronian triangle where all heights are integer numbers. In addition are given some examples of triangles for school practice. Keywords: Pythagorean triangles, Heronian triangles, incircle and excircles, triangle heights as integer numbers
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Spatial generalizations of the properties of the triangle
Šrubař, Jiří ; Karger, Adolf (advisor) ; Boček, Leo (referee) ; Lávička, Miroslav (referee)
TITLE Spatial generalizations of the properties of the triangle AUTHOR Jirˇı' Sřubarˇ SUPERVISOR Prof. RNDr. Adolf Karger, DrSc. DEPARTMENT Department of mathematics education ABSTRACT The present thesis describes various interesting properties of a triangle. The aim is to find and prove similar properties of its spatial generalization - a tetrahedron. Even though both synthetic and computational methods are used for proving spatial relations, synthetic approach is preferred whenever possible. The thesis is divided into two parts. In the first part, the properties of the tetrahedron analogous to the centroid and the orthocenter of the triangle are described. Also, conditions on the existence of the orthocenter of the tetrahedron are derived. Moreover, for tetrahedrons without an orthocenter, the so-called Monge point is introduced as its generalization. In the second part of the thesis, some further properties of the triangle are studied - - the Simson line, the de Longchamps point, the nine-point circle, the Euler line, the Lemoine point, the isodynamic points, the Lemoine axis and the Brocard axis. As the main contribution of the present thesis we define and prove the existence of spatial analogues of the above mentioned properties for the tetrahedron - the de Longchamps point, the twelve-point and...
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Relations among elements of a triangle
Machovcová, Lucie ; Zhouf, Jaroslav (advisor) ; Dvořák, Petr (referee)
This thesis summarizes not only most of the basic components and qualities of a triangle but it also brings pieces of information about triangles which are not usually taught at elementary and high schools. The aim of the thesis is to make readers acquainted with those components and to show differences between them. There are solved constructional and arithmetical problems in the last chapter. These problems utilize relations described in the thesis. The topic of these relations goes beyond mathematical curriculum of high schools. A part of the thesis is made of pictures to imagine better the terms described and the relations of components. All of the pictures are designed in a geometrical program called GeoGebra. Key words: triangle, side, angle, altitude, median, area, perimeter.
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Winery Marcinčák Novosedly
Janovská, Tereza ; Todorov, Petr (referee) ; Boleslavská, Yvona (advisor) ; Petříček, Tomáš (advisor)
The subject of the bachelor's thesis for the academic year 2020/21 was the design of a representative and production space for winemaker Petr Marcinčák, the owner of the Marcinčák winery, whose current object does not correspond to the required representation. The newly created building is located on the outskirts of the village Novosedly, on an elevated hill Stara hora. This plot offers a unique view of the Palava wine region and the nearby wine town of Mikulov, which is dominated by the local castle. The land is situated uphill. Its terrain is sloping using the so-called terraces, where the individual terraces are spaced up to four meters in height. With the help of these height spacings, I could afford to design a partially recessed building. The representation part and the production part are located separately so that the production does not disturb possible guests of the winery. The production hall is designed in the lower part of the plot. Structurally, it is a skeletal system with a vegetation roof. The representative building is designed as a system of three blocks folded into a triangle with the height difference of the individual blocks. The space was created between the blocks and is used for communication. The first floor, also the entrance, serves as a reception for newly accommodated guests, a tasting room and accommodation. The second floor is used for catering and wine representation. The first underground floor serves as an accommodation. The whole building is designed from monolithic reinforced concrete. The main part of the tasting and catering is solved by reinforced concrete frames due to the large span. The roofs are used as terraces or vegetation roofs, with the exception of the roofing of the last above-ground floor, where it is a classic single-skin roof. The facade of the building is designed individually depending on the height of the floor. Stone cladding is chosen for the underground floor. The first floor is
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Architectural study of the sacral object Brno - Líšeň
Bortlová, Kateřina ; Marek, Jiří (referee) ; Šindlar, Jiljí (advisor)
The content of the diploma thesis was an architectural study of a sacral building in Brno - Líšeň. The subject of the study was the elaboration of an ideological design of a sacral complex, which consisted of a church and a parish. The main operator of the whole complex will be the Salesian community, whose background is in close proximity to the area. The basic idea of the design was to design a spiritual place, which in its form conveys clear information about the use of the object and at the same time supports the development of ideas of the Salesian community. It is a structure that conveys the mission itself, not by one element, but by a set of several elements that together create a story, vision, mission. The ideological solution in this case thus becomes an equilateral triangle, which symbolizes the Father, the Son and the Holy Spirit. With its dominant vertical direction facing the sky, the object symbolically points to the spiritual path and self-realization. The given area is solved as a complex, the elements of which complement each other and thus create a harmonious whole. The design creates a space for both liturgical celebrations in the interior and exterior, which is used by the outdoor presbytery, which is complemented by residential stairs. These elements are intended to encourage residents to meet and establish friendly relations.
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Spatial generalizations of the properties of the triangle
Šrubař, Jiří
TITLE Spatial generalizations of the properties of the triangle AUTHOR Jirˇı' Sřubarˇ SUPERVISOR Prof. RNDr. Adolf Karger, DrSc. DEPARTMENT Department of mathematics education ABSTRACT The present thesis describes various interesting properties of a triangle. The aim is to find and prove similar properties of its spatial generalization - a tetrahedron. Even though both synthetic and computational methods are used for proving spatial relations, synthetic approach is preferred whenever possible. The thesis is divided into two parts. In the first part, the properties of the tetrahedron analogous to the centroid and the orthocenter of the triangle are described. Also, conditions on the existence of the orthocenter of the tetrahedron are derived. Moreover, for tetrahedrons without an orthocenter, the so-called Monge point is introduced as its generalization. In the second part of the thesis, some further properties of the triangle are studied - - the Simson line, the de Longchamps point, the nine-point circle, the Euler line, the Lemoine point, the isodynamic points, the Lemoine axis and the Brocard axis. As the main contribution of the present thesis we define and prove the existence of spatial analogues of the above mentioned properties for the tetrahedron - the de Longchamps point, the twelve-point and...
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Worksheets for teaching Geometry in the 7th grade of elementary school
NĚMCOVÁ, Věra
Bachelor work is composed from worksheets, which are divided into five main themes: consistency, center symmetry, quadrilateral, triangle and trapezoid and prism. Each chapter is subdivided by sub-topics. Each sub-topics contains a theoretical worksheet, a exercise worksheet and test. At the end of each chapter is included a summary exercise worksheet and summary test of the themes.
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