Simetrale kutova trokuta i konstruktivni problemi

master's thesis

• Author: Martina Soldo
• Mentor: Sanja Varošanec (mentor)
• Committee member: Sanja Varošanec (predsjednik povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Robert Manger (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-07-11, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U prvom poglavlju se progovara o simetrali kuta te simetralama kutova trokuta. Definirana je simetrala kuta, opisana je njezina konstrukcija za dani kut te je iskazan i dokazan Teorem o simetrali kuta. Nadalje, iskazani su i dokazani teoremi koji govore o svojstvima simetrala unutarnjih kutova trokuta. Najvažniji od njih, Teorem o simetrali unutarnjeg kuta trokuta, dokazan je na četiri načina, a također je iskazan i dokazan i njegov obrat. Proučeni su i analogoni tog teorema. Jedan od analogona govori o simetrali vanjskog kuta trokuta, a drugi o simetralnoj ravnini tetraedra (analogija simetrale kuta u prostoru). Zatim su izvedene formule za duljine simetrala unutarnjih kutova trokuta i duljine odsječaka koji nastaju presjekom simetrale unutarnjeg kuta i nasuprotne stranice trokuta. Na kraju poglavlja su iskazani još neki teoremi o simetralama kutova trokuta i središtu trokutu upisane kružnice. Preostali dio rada je posvećen konstruktivnim problemima, tj. izvodljivosti konstrukcije trokuta ako je barem jedan od zadanih elemenata trokuta simetrala unutarnjeg kuta tog trokuta. U drugom su poglavlju dani dovoljni i nužni uvjeti o postojanju i jedinstvenosti trokuta kojem su zadane duljine jedne stranice i dvije simetrale kuta, opisana i upisana kružnica i duljina simetrale kuta, duljine simetrala unutarnjih kutova trokuta. Treće poglavlje je posvećeno konkretnim primjerima opisa konstrukcija trokuta. Na početku se poglavlja govori o tome sto znači riječ konstruirati te su u tablici izdvojene sve one konstrukcije trokuta u kojima su zadane tri veličine, od kojih je barem jedna od zadanih veličina duljina simetrale unutarnjeg kuta. Neke od rješivih konstrukcija su opisane, a za dvije konstrukcije koje nisu rješive je dokazana nerješivost.

Abstract (english)

The first chapter summarizes facts about the angle bisector and the triangle angle bisectors. The angle bisector is defined, the construction of the given angle is described and the Angle bisector Theorem is expressed and proven. Furthermore, the theorems which define the performances of the interior angle bisectors of a triangle are expressed and proven. The most important theorem, the Angle bisector Theorem of triangles, is proven in four ways, as is its opposite. The analogues of this theorem are also studied. One of these analogues describes property of the exterior angle bisector of a triangle, while the other deals with the bisecting plane of a tetrahedron. As well, the formulae for the lengths of an interior angle bisector of a triangle and formulae of sections which are the result of an intersection of an interior angle bisector on the opposite side of the triangle are derived. At the end of the chapter other theorems related to an angle bisector of a triangle and an incircle of a triangle are presented. The remainder of the thesis concerns constructive problems i.e feasibility of the triangle construction if at least one given element is an angle bisector of that triangle. In the second chapter the necessary and sufficient conditions of the existence and uniqueness of a triangle for which is given the following; the length of one side and two angle bisectors, the circumcircle, incircle and the length of the angle bisector, and the length of the interior angle bisectors of the triangle. The third chapter is related to concrete examples of the descriptions of the triangle’s construction. Firstly, the word ”construction” is described. Secondly, in the table, the triangle constructions which have three given elements where at least one of these elements is the length of the angle bisector is extracted. Some of the solvable constructions are described while for two constructions that are unsolvable, the unsolvability is proven.

• Keywords: simetrala kuta; Teorem o simetrali kuta; simetrale kutova trokuta; simetralna ravnina tetraedra; konstrukcije trokuta
• Keywords (english): angle bisector; Angle bisector theorem; angle bisector of a triangle; bisecting plane of a tetrahedron; triangle’s construction
• Language: croatian
• URN:NBN: urn:nbn:hr:217:886759
• 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: 2019-02-06 15:55:08