Točke ekstrema nekih geometrijskih izraza vezanih uz trokut

master's thesis

• Title (english): Extreme points of some geometric expressions related to a triangle
• Author: Marija Kolić
• Mentor: Vjekoslav Kovač (mentor)
• Committee member: Vjekoslav Kovač (predsjednik povjerenstva)
• Committee member: Hrvoje Šikić (član povjerenstva)
• Committee member: Zoran Vondraček (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu promatrali smo neke od najpoznatijih primjera geometrijskih izraza vezanih uz trokut i njihove minimalne, odnosno maksimalne vrijednosti po svim točkama trokuta. U prve dvije tvrdnje rada kao točke ekstrema smo nalazili težište, kao jednu od najučestalijih točaka trokuta, dok su se u preostale dvije tvrdnje ekstremi postizali u ne tako poznatim točkama trokuta što se tiče sustava osnovnoškolskog i srednjoškolskog obrazovanja. Najprije smo svaku od tih točaka, odnosno težište, Fermatovu točku te Lemoineovu točku, definirali i opisali njihova svojstva, dok smo potom svaku od glavnih tvrdnji ovoga rada, odnosno teoreme o ekstremima geometrijskih izraza vezanih uz trokut iskazali i detaljno dokazali. Na kraju smo svaku od tvrdnji ilustrirali u programu GeoGebra i na taj način vizualno potkrijepili prethodne dokaze u svrhu jednostavnijeg razumijevanja i otkrivanja danih tvrdnji za same učenike.

Abstract (english)

In this thesis, we have observed some of the best known examples of geometric expressions related to the triangle and their minimum and maximum values over all points of the triangle. In the first two statements of this work the extreme points are found to be the centroid, which is one of the most frequent points of the triangle, while in the remaining two statements the extremes were achieved in not so well known points of the triangle, as far as the primary and secondary education systems are concerned. First, we defined and described each of these points and their properties, i.e., the centroid, the Fermat point and the Lemoine point, and then we stated and proved in detail each of the main statements of this paper, that is the theorems on the extremes of geometric expressions related to a triangle. Finally, we illustrated each of the claims in the GeoGebra program and thus visually enhanced the previous proofs for the purpose of easier understanding and discovery of the given claims by students on their own

• Keywords: trokut; težište; Fermatova točka; Lemoineova točka; ekstremi; geometrijski izrazi vezani uz trokut.
• Keywords (english): triangle; the centroid; the Fermat point; the Lemoine point; extremes; geometric expressions related to the triangle
• Language: croatian
• URN:NBN: urn:nbn:hr:217:417654
• 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: 2021-02-26 12:30:37