Prostorová zobecnění vlastností trojúhelníku
Spatial generalizations of the properties of the triangle
rigorous thesis (RECOGNIZED)
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http://hdl.handle.net/20.500.11956/32894Identifiers
Study Information System: 96157
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- Kvalifikační práce [10932]
Author
Faculty / Institute
Faculty of Mathematics and Physics
Discipline
Training Teachers of Mathematics and Descriptive Geometry at Higher Secondary Schools
Department
Department of Mathematics Education
Date of defense
21. 1. 2011
Publisher
Univerzita Karlova, Matematicko-fyzikální fakultaLanguage
Czech
Grade
Recognized
Keywords (Czech)
Prostorové zobecnění, trojúhelník, čtyřstěn, Mongeův bod, kulová plocha dvanácti bodů, kulová plocha osmi bodů, Eulerova přímka, Lemoinův bod, isodynamické body, Lemoinova rovina, Brocardova osaKeywords (English)
Spatial generalization, triangle, tetrahedron, Monge point, twelve-point sphere, eight-point sphere, Euler line, Lemoine point, isodynamic point, Lemoine plane, Brocard axisNA' ZEV PRA' CE Prostorova' zobecneňı' vlastnostı' troju'helnı'ku AUTOR Jirˇı' Sřubarˇ SˇKOLITEL Prof. RNDr. Adolf Karger, DrSc. SˇKOLI'CI' PRACOVISŤEˇ Katedra didaktiky matematiky ABSTRAKT V pra' ci jsou popsa' ny zajı'mave' vlastnosti troju'helnı'ku, neˇktere' vsěobecneˇ zna' me', jine' me'neˇ zna' me'. Cı'lem bylo popsat analogicke' vlastnosti cťyršteňu a tyto vlastnosti doka' zat. Prˇi du˚kazech prostorovy'ch vztahu˚ jsou pouzˇity syn- teticka' i vy'pocětnı' metoda, preferovana' je ale synteticka' metoda vzhledem k jejı' na' zornosti. Pra' ce je rozdeľena do dvou cˇa' stı'. V prvnı' cˇa' sti jsou popsa' ny ty vlastnosti cťyršteňu, ktere' odpovı'dajı' pojmu˚m težˇisťeˇ a ortocentrum troju'helnı'ku. Jsou odvozeny podmı'nky pro existenci ortocentra cťyršteňu. Da' le je pro cťyršteňy bez ortocentra zaveden Mongeu˚v bod, ktery' ma' vlastnosti ortocentru odpo- vı'dajı'cı'. V druhe' cˇa' sti pra' ce jsou zkouma' ny neˇktere' dalsˇı' vlastnosti troju'helnı'ku - - Simsonova prˇı'mka, Longchampu˚v bod, kruzňice devı'ti bodu˚, Eulerova prˇı'mka, Lemoinu˚v bod, isodynamicke' body, Lemoinova osa a Brocardova osa. Jako hlavnı' vy'sledek te'to pra' ce jsou definova' ny a je doka' za' na exis- tence prostorovy'ch analogiı' uvedeny'ch vlastnostı' troju'helnı'ku - Longcham- pova bodu...
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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