Dva svojstva simetrala kutova trokuta i Steiner-Lehmusov teorem

master's thesis

• Author: Ana Prosenečki
• Mentor: Mario Krnić (mentor)
• Committee member: Mario Krnić (predsjednik povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-11-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog diplomskog rada je izučavanje nekih značajnih svojstava simetrala kutova trokuta. Rad se sastoji od četiri poglavlja. Nakon uvodnog poglavlja, u drugom poglavlju bavimo se teoremom o simetrali kuta u trokutu. Dajemo nekoliko dokaza tog teorema kao i njegove inačice. Kao primjenu, dobivamo eksplicitne formule za duljine simetrala kutova trokuta. Nadalje, u istom poglavlju izvodimo nekoliko zanimljivih nejednakosti koje uključuju simetrale kutova trokuta. U trećem poglavlju dajemo nekoliko primjena jednostavne činjenice da simetrala kuta trokuta raspolavlja odgovarajući luk opisane kružnice trokuta. Nadalje, primjenom glavnih rezultata iz drugog i trećeg poglavlja rješavamo neke složenije geometrijske probleme s brojnih matematičkih natjecanja za srednjoškolce. Konačno, u četvrtom poglavlju dajemo nekoliko dokaza Steiner - Lehmusovog teorema koji kaže da ako trokut ima dvije simetrale kutova jednakih duljina, onda je on jednakokračan.

Abstract (english)

In the present thesis, we study some interesting properties of angle bisectors in a triangle. The thesis is divided into four chapters as follows: after an introductory chapter, in Chapter 2 we give several variants and proofs of an angle bisector theorem. As an application, we derive formulas for lenghts of angle bisectors of a triangle. Further, we also study some inequalities involving angle bisectors of a triangle. In Chapter 3 we utilize a simple fact that an angle bisector of a triangle bisects the corresponding arc of a circumscribed circle. As an application of main results from Chapters 2 and 3, we also solve some geometry problems appearing on numerous mathematical competitions for high school students. Finally, in Chapter 4 we give several proofs of the Steiner - Lehmus theorem which asserts that if the triangle has two equal angle bisectors, then it is isosceles.

• Keywords: trokut; simetrala kuta; Steiner-Lehmusov teorem
• Keywords (english): triangle; angle bisector; Steiner-Lehmus theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:225279
• 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: 2019-06-12 12:05:26