Bicentrični mnogokuti

master's thesis

• Title (english): Bicentric polygons
• Author: Ana Matijević
• Mentor: Marija Galić (mentor)
• Committee member: Marija Galić (predsjednik povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu bavili smo se bicentričnim mnogokutima. Dobro je poznato da se svakom trokutu kružnica može i upisati i opisati. Eulerova formula za trokut, koju smo i dokazali, daje nam vezu između polumjera njemu upisane i opisane kružnice. Kao uvod u temu bicentričnih četverokuta, na kojima je u ovome radu naglasak, bilo je važno barem ukratko opisati i dati neke osnovne karakterizacije dviju dobro nam znanih klasa četverokuta, a to su tangencijalni i tetivni četverokuti. Bicentrični četverokuti su četverokuti koji su istovremeno tetivni i tangencijalni. Kao takvi, imaju neke zanimljive karakterizacije, a i rezultate koji se između ostalog odnose na vezu između radijusa upisane i opisane kružnice bicentričnih četverokuta. Prirodno se nametnulo dokazati analogon Eulerove formule za trokute poznat pod imenom Fussov teorem, kao i Yunovu nejednakost koja povezuje unutarnje kutove četverokuta te oba radijusa. Izvanredan teorem koji nam govori o postojanju četverokuta koji je istovremeno upisan jednoj, a opisan drugoj kružnici dao je francuski matematičar Jean-Victor Poncelet, a teorem je poznat pod imenom Ponceletov porizam. On je primjenjiv na sve mnogokute te zbog njega znamo da ukoliko postoji neki mnogokut koji je upisan vanjskoj kružnici i opisan unutarnjoj, onda postoji beskonačno mnogo takvih mnogokuta.

Abstract (english)

In this thesis, our main theme was bicentric polygons. It is well known that every triangle possesses both an incircle and a circumcircle. Euler’s triangle formula, which we have proven, gives us a relation between the inradius and the circumradius. As an introduction to the main focus of this paper, bicentric quadrilaterals, it was important to, at least briefly, describe and give some basic characterizations of two well-known classes of quadrilaterals: tangential and cyclic quadrilaterals. Bicentric quadrilaterals are quadrilaterals that are cyclic and tangential at the same time. As a result, they have some interesting characterizations and results that, among others, relate to the connection between both the radii of the incircle and circumcircle of bicentric quadrilaterals. Naturally, it was necessary to prove the analogue of Euler’s triangle formula known as Fuss’s theorem, as well as Yun’s inequality, which connects the interior angles of the quadrilateral and both radii. A remarkable theorem that tells us about the existence of a quadrilateral that is simultaneously inscribed in one circle and circumscribed around another one was discovered by the French mathematician Jean-Victor Poncelet, and the theorem is known as Poncelet’s porism. It can be applicable to all polygons and that theorem gives us a statement that if there exists a polygon of n-sides which is inscribed in a given circle and circumscribed about another circle, then infinitely many such polygons exist.

• Keywords: trokut; kružnica; Eulerova formula za trokut; tangencijalni četverokuti; tetivni četverokuti; Fussov teorem; Yunovu nejednakost; Ponceletov porizam
• Keywords (english): triangle; circle; Euler’s triangle formula; tangential quadrilaterals; cyclic quadrilaterals; Fuss's theorem; Yun's inequality; Poncelet porism
• Language: croatian
• URN:NBN: urn:nbn:hr:217:423042
• 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: 2022-11-02 14:02:11