Routhov teorem i njegova poopćenja

master's thesis

• Author: Tea Mršić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-11-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Središnja tema ovog rada je teorem koji se uobičajeno pripisuje istaknutom engleskom matematičaru E. J. Routhu, jer smatra se da je on prvi eksplicitno iskazao taj rezultat i to kao popratnu bilješku, bez dokaza, u svojoj knjizi iz područja statike. Ako se vrhovi trokuta \(ABC\) spoje s odabranim trima točkama na suprotnim stranicama, međusobna sjecišta tih spojnica omeđuju trokut \(PQR\). Routh je dao formulu za omjer površina trokuta \(PQR\) i \(ABC\), koja se lako pretvori u funkciju djelišnih omjera \(r, s\) i \(t\), određenih točkama odabranima na stranicama. Mogućnost pristupa pomoću različitih metoda, kao i očigledna srodnost s klasičnim Menelajevim i Cevinim teoremom potaknule su brojne matematičare da pronađu nove dokaze i poopćenja Routhove formule. U radu je prikazan dio tih rezultata, uključujući objedinjenje Menelajevog i Cevinog teorema uvođenjem tzv. poopćenog Routhovog trokuta. U završnom poglavlju izložen je trodimenzionalni analogon Routhovog teorema, u kojem se za tetraedar dobiva rezultat koji u potpunosti odgovara osnovnoj formuli za trokut.

Abstract (english)

The central topic of this thesis is a theorem that is usually attributed to the outstanding English mathematician E. J. Routh, since he was apparently the first one to publish that particular statement. It was included as a helpful note, without a proof, in his book on statics. If the vertices of a triangle \(ABC\) are joined to three points chosen on the opposite sides, another triangle \(PQR\) is formed by mutual intersection points of these segments. Routh gave a formula for the ratio of the areas of \(PQR\) and \(ABC\), which can be easily transformed into a function of the ratios \(r, s\) and \(t\), determined by the points chosen on the sides. The possibility of different approaches by various methods, as well as its obvious connection to the classical theorems of Menelaus and Ceva, prompted numerous mathematicians to find new proofs and generalizations of Routh’s formula. Some of these results are presented here, including a unification of Menelaus’ and Ceva’s theorem by introducing a so-called generalized Routh’s triangle. The final chapter is dedicated to a three-dimensional analogue of Routh’s theorem, valid for any tetrahedron and entirely corresponding to the basic formula for a triangle.

• Keywords: Routhov teorem; trokut; Menelajev teorem; Cevin teorem
• Keywords (english): Routh’s theorem; triangle; Menelaus’ theorem; Ceva’s theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:885083
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-06-07 13:13:45