Soldo, Martina (2014) Simetrale kutova trokuta i konstruktivni problemi. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract
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.
Item Type:  Thesis (Diploma thesis) 

Supervisor:  Varošanec, Sanja 
Date:  2014 
Number of Pages:  54 
Subjects:  NATURAL SCIENCES > Mathematics 
Divisions:  Faculty of Science > Department of Mathematics 
Depositing User:  Iva Prah 
Date Deposited:  10 Jul 2015 10:15 
Last Modified:  10 Jul 2015 10:15 
URI:  http://digre.pmf.unizg.hr/id/eprint/4110 
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