Newtonovi četverokuti

master's thesis

• Author: Darija Dabić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; Zagreb
• Defense date and country: 2020-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Newtonov četverokut je konveksan četverokut upisan u kružnicu čija je jedna stranica promjer kružnice. Rješavajući problem pronalaska promjera d kružnice kojoj je upisan konveksni četverokut čije su zadane tri stranice a,b i c, Newton je došao do relacije poznate kao Newtonova jednadžba. Newton se nije bavio rješavanjem vlastite jednadžbe, međutim, istu je izveo na šest različitih načina. Pozitivna cjelobrojna rješenja Newtonove jednadžbe pronašao je P. Bachmann. Sva cjelobrojna rješenja Newtonove jednadžbe pronašao je A. Oppenheim, a formula koja daje beskonačno mnogo racionalnih rješenja, rezultat je matematičara N. Anninga koji je koristio elegantnu relaciju između kosinusa kutova trokuta. Sva racionalna rješenja Newtonove jednadžbe pronašao je I. A. Barnett koristeći jednostavnu i elegantnu metodu koja se temelji na linearnoj algebri. Jednake rezultate Barnettovim, postigao je M. Hajja koristeći relaciju između kosinusa kutova generaliziranog trokuta. U ovom radu prezentirana je i direktna algebarska metoda koja daje sva racionalna rješenja. Prikazan je i slučaj nekonveksnog Newtonovog četverokuta, uspostavljena su određena svojstva nultočaka Newtonovog i njemu pridruženog polinoma, iskazan je i dokazan obrat Newtonovog rezultata te je dana i izvedena formula za površinu Newtonovog četverokuta. U posljednjem poglavlju je opisano nekoliko primjera pojavljivanja Newtonovog i njemu pridruženog polinoma.

Abstract (english)

A Newton quadrilateral is a convex quadrilateral inscribed in a circle whose one side is the diameter of the circle. Solving the problem of finding the diameter d of a circle that has an inscribed convex quadrilateral given by the three sides a; b, and c, Newton discovered the relation known as the Newton equation. Newton was not concerned with solving his own equation, however, he derived it in six different ways. Positive integer solutions of the Newton equation were found by P. Bachmann. All integer solutions of Newton’s equation were found by A. Oppenheim, and the formula that produces infinitely many rational solutions was found by N. Anning, who used an elegant relation among the cosines of the angles of a triangle. All rational solutions of Newton’s equation were found by I. A. Barnett using a simple and elegant method based on linear algebra. The same results of Barnett, M. Hajja achieved using the relation between the cosines of angles of a generalized triangle. In this thesis, a direct algebraic method is presented which gives all rational solutions. Also, the case of a non-convex Newton quadrilateral is presented, certain properties of the zeros of the Newton’s polynomial and its associated polynomials are established, the converse of the Newton result is presented and proved and the formula for the area of the Newton’s quadrilateral is given. The last chapter describes several examples where Newton’s polynomial and its associated polynomial appears.

• Keywords: Newtonov četverokut; Newtonova jednadžba; P. Bachmann; A. Oppenheim; N. Anning; A. Barnett
• Keywords (english): Newton's quadrilateral; Newton's equation; P. Bachmann; A. Oppenheim; N. Anning; A. Barnett
• Language: croatian
• URN:NBN: urn:nbn:hr:217:564092
• Study programme: Title: Mathematics and Computer Science Education; specializations in: Mathematics and Computer Science Education; Course: Mathematics and Computer Science Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-11-25 10:38:03