Centri trokuta i glavni centri trokuta

master's thesis

• Author: Marko Begić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Mirko Polonijo (član povjerenstva)
• Committee member: Ivica Nakić (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-02-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu izloženi su neki pojmovi i rezultati iz područja geometrije trokuta povezani s konceptom centra trokuta kao poopćenja klasičnih karakterističnih točaka trokuta. Rezultati ovog smjera istraživanja u posljednjih tridesetak godina naročito je vidljiv u „Enciklopediji centara trokuta“ Clarka Kimberlinga. Za definiranje i proučavanje centara trokuta najpogodnije su homogene koordinate, a ovdje se služimo posebnim tipom takvih koordinata koje se nazivaju trilinearne koordinate. U prvom dijelu rada upoznajemo se s tim koordinatnim sustavom i primjenjujemo ga na klasične centre trokuta, primjerice centri upisane i opisane kružnice, ortocentar. Zatim se općenito definira centre trokuta kao točke čije se trilinearne koordinate mogu izraziti kao funkcije kutova i duljina stranica. Nadalje, uočavaju se tzv. bicentrični parovi točaka kakve su npr. Brocardove točke i operacije s kojima se od bicentričnog para dobiva centar. Poseban je naglasak na glavnim centrima trokuta koji se uvode s motivom da se poznatim postupkom dobije novi trokut koji je perspektivan polaznom. Složenost problema ne dopušta jednostavno eksplicitno rješenje, no svaki glavni centar, a to je centar čije se trilinearne koordinate mogu izraziti u obliku \(f(a):f(B):f(C)\) pripada skupu rješenja.

Abstract (english)

In this master's thesis we showed some terms and results from field of triangle geometry that are linked with the concept of triangle center as generalization of classic characteristic points of triangle. Results of this field’s research can be seen in “Encyclopedia of triangle centers” by Clark Kimberling. To define and study centers of triangles we find homogeneous coordinates to be the most suitable. Here we used special kind of those coordinates that are called trilinear coordinates. In first part of the thesis we get to know around that coordinate system and apply it to classic triangle centers like incenter, centroid and orthocenter. Then we defined triangle centers as points whose trilinear coordinates can be expressed as functions of angels and side lengths. Furthermore, we noticed, so called, bicentric pairs of points such as Brocard points and operations that make center from bicentric pair. We will also saw major triangle centers which were introduced to show how we can get a new triangle which is perspective to the first one. Complexity of the problem does not allow simple explicit solution, but every major center, and that is center whose trilinear coordinates can be expressed in terms \(f(a):f(B):f(C)\), is part of the set of solution.

• Keywords: centri trokuta; geometrija trokuta; karakteristične točke trokuta; trilinearne koordinate; klasični centri trokuta; Brocardove točke
• Keywords (english): triangle center; triangle geometry; characteristic points of triangle; trilinear coordinates; classic triangle centers; Brocard points
• Language: croatian
• URN:NBN: urn:nbn:hr:217:934785
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-05-23 10:36:23