Baricentrične koordinate - primjena na geometriju trokuta

master's thesis

• Author: Tina Klarin
• Mentor: Mea Bombardelli (mentor)
• Committee member: Mea Bombardelli (predsjednik povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Snježana Lubura Strunjak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-03-03, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Baricentrične koordinate vrsta su homogenih koordinata koje se pridružuju točki ravnine s obzirom na referentni trokut sustava. Baricentrične koordinate možemo definirati pomoću vektorskog računa ili pomoću orijentiranih površina i orijentiranih duljina što se može vidjeti u prvom dijelu ovog rada. Karakteristično svojstvo jednadžbe pravca i kružnice u baricentričnom koordinatnom sustavu je homogenost što nam uvelike olakšava njihovo određivanje i računanje njima. Budući da je upravo trokut referentni sustav baricentričnog koordinatnog sustava, baricentrične koordinate pokazuje se pogodnim izborom u dokazivanju teorema i rješavanju zadataka geometrije trokuta. U drugom poglavlju ovog rada iskazani su i dokazani neki poznati teoremi i tvrdnje geometrije trokuta, a u trećem je poglavlju koordinatnom metodom riješeno nekoliko zadataka s matematičkih natjecanja.

Abstract (english)

Barycentric coordinates are a type of homogeneous coordinates that join a point of the plane with respect to the reference triangle of the system. We can define barycentric coordinates using vector calculus or using oriented surfaces and oriented lengths, which can be seen in the first part of this paper. A characteristic property of the equation of a line and a circle in the barycentric coordinate system is homogeneity, which greatly facilitates their determination and calculation. Since the triangle is the reference system of the barycentric coordinate system, barycentric coordinates prove to be a suitable choice for proving theorems and solving triangle geometry problems. In the second chapter of this paper, some well-known theorems and claims of triangle geometry are presented and proven, and in the third chapter, several problems from mathematical competitions are solved using the coordinate method.

• Keywords: homogene koordinate; referentni trokut sustava; vektorski račun
• Keywords (english): homogeneous coordinates; reference triangle of the system; vector calculus
• Language: croatian
• URN:NBN: urn:nbn:hr:217:624163
• 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: 2023-07-25 13:48:25