Bisektrise površine i opsega trokuta

master's thesis

• Title (english): Bisectors of area and perimeter of the triangle
• Author: Ivan Mrčela
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu smo promatrali pravce koji raspolavljaju površinu odnosno opseg trokuta. Pokazano je da u svakom trokutu postoji beskonačno mnogo bisektrisa površine i beskonačno mnogo bisektrisa opsega. Ti skupovi pravaca stvaraju ovojnice koje se zbog svog oblika nazivaju deltoidama. Deltoide su opisane u trećem poglavlju, gdje su iskazana i neka zanimljiva svojstva. Posebno su promatrani pravci koji su ujedno i bisektrisa opsega i bisektrisa površine, takozvane AP-bisektrise. AP- bisektrisa ima konačno mnogo u svakom trokutu i njihov broj ovisi o međusobnim odnosima duljina stranica trokuta. Konkretno, broj AP-bisektrisa ovisi o predznaku diskriminante kvadratne jednadžbe čija su realna rješenja duljine odsječaka koje bisektrisa čini na stranicama trokuta, s obzirom na neki od vrhova trokuta. Diskusija rješenja pokazuje da za svaki trokut postoji jedna, dvije ili tri AP-bisektrise trokuta ovisno o predznaku diskriminante. Na primjerima su prikazani mogući slučajevi trokuta i broja AP-bisektrisa. Nakon dokazivanja rezultata o bisektrisama, prikazana je konstrukcija takvih pravaca. Također, istraženo je kako se mogu odrediti zanimljivi trokuti s cjelobrojnim duljinama stranica u kojima se može konstruirati AP-bisektrisa.

Abstract (english)

In this paper, we have considered the lines that bisect the area and/or perimeter of the triangle. It has been shown that in any triangle there are infinitely many bisectors of area and infinitely many bisectors of the perimeter. These pairs of lines form envelopes called deltoids because of their shape. Deltoids are described in the third chapter, where some interesting properties are also shown. The lines that are both perimeter bisectors and area bisectors are called AP-bisectors, and they are of particular interest. There is a finite number of AP-bisectors in each triangle, and this number depends on the relations between the sidelengths of the triangle. Specifically, the number of AP-bisectors depends on the sign of the discriminant of a quadratic equation whose real solutions are the length intercepts formed by the bisector on the sides of the triangle with respect to one of the vertices. A discussion of the solutions shows that for each triangle there is one, two or three AP-bisectors depending on the sign of the discriminant. Possible cases of triangles and their number of AP-bisectors are shown by examples. After proving the results about bisectors, a construction of such lines is shown. Furthermore, an additional analysis including results from number theory is carried out, in order to produce sequences of triangles whose sidelengths, as well as the aforementioned lengths of intercepts formed by AP-bisectors are integers.

• Keywords: deltoide; AP-bisektrise; diskriminante
• Keywords (english): deltoids; AP-bisectors; discriminant
• Language: croatian
• URN:NBN: urn:nbn:hr:217:051679
• 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: 2022-11-03 10:58:51