Trokuti s cjelobrojnim stranicama i trisektibilnim kutovima

master's thesis

• Author: Tanja Krog
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Trokuti s cjelobrojnim stranicama i trisektibilnim kutovima posebna su klasa trokuta. Kosinusi kutova takvih trokuta su racionalni brojevi s dodatnim svojstvom da se mogu izraziti pomoću stanovitog polinoma 3. stupnja. Pokazuje se da ne postoje jednakostranični i jednakokračni pravokutni takvi trokuti. U ovom radu naglasak smo stavili na raznostranične trokute s cjelobrojnim stranicama i trisektibilnim kutovima, no osvrnuli smo se i na postojanje jednakokračnih koji nisu pravokutni te pravokutnih takvih trokuta. Pokazuje se da ključnu ulogu za postojanje razmatrane vrste trokuta ima tzv. rezidual, a to je pozitivni cijeli broj koji je na jednostavan način povezan s racionalnom vrijednosti kosinusa. Preciznije, dva šiljasta kuta s racionalnim kosinusima imaju isti rezidual ako i samo ako postoji trokut s cjelobrojnim stranicama koji sadrži ta dva kuta. Nadalje, pomoću reziduala dolazi se do opće formule za stranice promatranih raznostraničnih trokuta koji nemaju pravi kut. Glavni rezultat ovog rada otkriva da za svaki kvadratno slobodan pozitivan cijeli broj \(r\) postoji beskonačno mnogo različitih trokuta s cjelobrojnim stranicama i trisektibilnim kutovima kojima je zajednički rezidual jednak \(r\). U radu su izložene jednostavne metode za generiranje takvih trokuta te niz primjera za odgovarajuće duljine triju stranica. Konačno, pokazuje se da su šiljasti kutovi primitivnog Pitagorinog trokuta su trisektibilni ako i samo ako je hipotenuza potpuni kub.

Abstract (english)

Integer-sided triangles with trisectible angles are a special class of triangles. Cosines of angles of such triangles are rational numbers with the additional property of representation by a specific third degree polynomial. It is shown that no equilateral triangles and no right angled isosceles triangles belong to that class. In this paper, the emphasis is placed on scalene integer-sided triangles with trisectible angles, but the existence of right triangles with these properties is thoroughly investigated, too, as well as the inclusion of isosceles triangles without a right angle in that class. It is shown that the key feature for the existence of the considered triangle type is the so-called residual. Residual is a positive integer that is in a simple way associated to the rational value of the cosine. More precisely, two acute angles with rational cosines have the same residual if and only if there is an integer-sided triangle containing these two angles. Furthermore, a general formula based on residuals is derived for the sidelengths of the observed scalene triangles without a right angle. The main result of this paper reveals that for each square-free positive integer r there exist infinitely many distinct scalene integersided triangles with trisectible angles and with r as their common residual. Some simple methods for generating such triangles are presented and examples for the corresponding sidelengths are given. Finally, it is shown that the acute angles of primitive Pythagorean triangles are trisectible if and only if their hypotenuse is a perfect cube.

• Keywords: trokut; trisektibilni kutovi; rezidual
• Keywords (english): triangle; trisectible angles; residual
• Language: croatian
• URN:NBN: urn:nbn:hr:217:338406
• 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: 2019-05-27 12:30:28