Vrste mnogokuta

master's thesis

• Author: Ana Merkaš
• Mentor: Maja Starčević (mentor)
• Committee member: Maja Starčević (predsjednik povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U diplomskom radu su prikazane različite vrste mnogokuta te su dokazana neka njihova svojstva. Kod trokuta imamo osnovnu podjelu prema stranicama i kutovima te u ovisnosti o tome primjećujemo neka karakteristična svojstva trokuta. Postoje i neke manje poznate vrste trokuta poput pseudopravokutnog trokuta kojeg karakterizira određena relacija između kutova trokuta. Neke vrste trokuta se definiraju pomoću nekog drugog trokuta, poput nožišnog trokuta kojemu su vrhovi nožišta okomica spuštenih iz neke točke na pravce na kojima leže stranice danog trokuta. Kada se ta točka poklapa s ortocentrom danog trokuta, tada se dobiveni trokut naziva ortocentrični, a ako se točka poklapa sa središtem danom trokutu opisane kružnice, trokut se naziva medijalni. Postoji i savršeni trokut čiji su opseg i površina jednaki, a stranice prirodni brojevi te zlatni trokut koji se definira pomoću omjera zlatnog reza. Također, zlatni rez nalazimo i kod pravilnog ili zlatnog peterokuta. Nadalje, u radu su promatrane razne vrste četverokuta. Osim dobro poznatih vrsta paralelograma, postoje četverokuti kojima se može opisati ili upisati kružnica, a nazivamo ih tetivni, odnosno tangencijalni četverokut. Posebna vrsta tangencijalnog četverokuta je onaj s okomitim dijagonalama, a naziva se deltoid. Također, istražene su karakteristike nekih mnogokuta s više od četiri stranice, poput Pascalovog i Brianchonovog šesterokuta.

Abstract (english)

Various types of polygons have been shown in this graduate thesis and some of their properties have been proven. Triangles are classified in two general ways: by their sides and by their angles. On the basis of that, we can notice some of the characteristic properties of the triangles. There are also some lesser-known types of triangles such as the pseudo right triangle that is characterized by a certain relation between the angles of the triangle. Some types of triangles are defined by another triangle, such as the pedal triangle whose vertices are equal to foots of the lines perpendicular to lines that contain sides of the given triangle, passing through a given point. When that point fits with the orthocenter of the given triangle, then the resulting triangle is called orthic, and if this point is the center of the circumcircle, the triangle is called medial. There is a triangle which is called perfect whose perimeter and area are equal and the lengths of its sides are natural numbers. Also, there is the gold triangle which is defined by the golden ratio. Likewise, we find the golden ratio in the regular or gold pentagon. In this paper various types of quadrilaterals were observed. In addition to well-known types of parallelograms, there are quadrilaterals that have the circumcircle or the incircle, and we call them cyclic and tangential quadrilaterals respectively. A special kind of tangential quadrilateral is the one with perpendicular diagonals and is called deltoid. Furthermore, some polygons with five and more sides have been researched, such as Pascal and Brianchon hexagon.

• Keywords: mnogokuti; trokut; zlatni rez; paralelogram; četverokut; deltoid; peterokut; Pascalov šesterokut; Brianchonov šesterokut
• Keywords (english): polygons; triangle; golden ration; parallelogram; quadrilateral; deltoid; pentagon; Pascal hexagon; Brianchon hexagon
• Language: croatian
• URN:NBN: urn:nbn:hr:217:215141
• 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: 2019-06-07 11:21:20