Steiner's 10 theorems on complete quadrilaterals
| Autor: | Šumiga, Ines |
|---|---|
| Přispěvatelé: | Šiftar, Juraj |
| Jazyk: | chorvatština |
| Rok vydání: | 2021 |
| Předmět: |
Desargues' theorem
nine-point circle Euclidean geometry Simson-Wallaceov pravac ortogonalni pramenovi kružnica kružnica devet točaka orthogonal pencils of circles PRIRODNE ZNANOSTI. Matematika Simson-Wallace line geometrija euklidske ravnine polar circle of a triangle polarna kružnica trokuta NATURAL SCIENCES. Mathematics Desarguesov teorem |
| Popis: | Steinerov teorem o potpunom četverostranu sastoji se od 10 međusobno povezanih tvrdnji, a objavljen je u obliku zadatka, dakle bez dokaza, u izdanju časopisa "Annales de Mathematiques Pures et Applique'es", skraćeno zvanog "Journal de Gergonne" za 1827./1828. godinu. Cilj je ovog rada izložiti dokaze svih 10 propozicija. U tu svrhu, uvodno prvo poglavlje sadrži pregled različitih pojmova i činjenica iz geometrije euklidske ravnine koje su potrebne za dokaz. Između ostalih, to su Simson-Wallaceov pravac, kružnica devet točaka, polarna kružnica trokuta, ortogonalni pramenovi kružnica te Desarguesov teorem i harmonička četvorka, kao teme bliskije projektivnoj geometriji. U drugom poglavlju dokazano je redom svih 10 stavaka, djelomično grupiranih prema njihovom sadržaju i metodi dokaza. Steiner polazi od četiri trokuta koji su određeni s po tri stranice potpunog četverovrha, promatra njihove opisane kružnice i ortocentre te uočava nekoliko položaja kolinearnih i konciklićkih točaka čime dolazi do daljnjih točaka, pravaca i kružnica s posebnim svojstvima. U završna tri stavka analizira se složena struktura generirana središtima upisanih i pripisanih kružnica početna četiri trokuta. Uočavaju se daljnje pravilnosti dobivenih konfiguracija te se na kraju dobiva novi opis iste točke o kojoj govori prvi stavak, a to je zajednička točka četiri kružnice opisane promatranim trokutima. Steiner's theorem on a complete quadrilateral consists of 10 interrelated statements, and it was published in the form of a problem or a challenge, that is, without proof, in the 1827./1828. volume of the journal "Annales de Mathematiques Pures et Applique'es", also known as "Journal de Gergonne". The main goal of this paper is to present proofs of all 10 statements. To that purpose, the introductory chapter is a brief recapitulation of numerous notions and facts from Euclidean geometry that will be used in the proofs. Some of those concepts are the Simson-Wallace line, the nine-point circle, the polar circle of a triangle, orthogonal pencils of circles, as well as several topics more related to projective geometry, such as the Desargues' theorem and harmonic quadruple of points. In the second chapter, all 10 statements are proven. Some proofs are grouped according to the contents of the statement or the method used in the demonstration. Steiner starts by observing the four triangles associated to the complete quadrilateral, together with their circumcircles and orthocenters. By noticing some specific sets of collinear and concyclic points, he points out further specific points, lines and circles with particular properties. The last three statements are based on the analysis of a complex structure generated by incenters and centers of excircles of the initial triangles. An examination of the obtained configurations leads to the final statement, describing from a completely different perspective the point which occurred in the first proposition as a common point of all four circumcircles of associated triangles. |
| Databáze: | OpenAIRE |
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