Primjena koordinatne metode u planimetriji

master's thesis

• Author: Katarina Barać
• Mentor: Sanja Varošanec (mentor)
• Committee member: Sanja Varošanec (predsjednik povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Mladen Vuković (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

U ovom radu opisujemo koordinatnu metodu u planimetriji te pomoću nje dokazujemo razne teoreme iz različitih područja matematike. Na samom početku definiran je koordinatni sustav na pravcu i koordinatni sustav u ravnini te je dana kratka povijesna crtica o nastanku Kartezijevog koordinatnog sustava. Definirali smo i koordinatnu metodu. Navedeni su i osnovni teoremi analitičke geometrije koje ćemo koristiti u radu. Teoreme o četiri karakteristične točke trokuta (težište, ortocentar, središte opisane kružnice, središte upisane kružnice) te teorem o Eulerovom pravcu smo dokazali koordinatnom metodom. Prikazali smo i zadatke s matematičkih natjecanja na kojima je moguća primjena koordinatne metode. Teoreme o trokutu i četverokutu također smo dokazali koordinatnom metodom. Na kraju smo naveli neka svojstva krivulja drugog reda, tj. svojstva elipse, hiperbole i parabole te ih dokazali primjenjujući koordinatnu metodu.

Abstract (english)

In this thesis, the coordinate method in planimetry is described and it is used to prove several theorems from various fields of mathematics. At the beginning of the paper the Cartesian coordinate system on line and on plane are defined, and the short history overview of the Cartesian coordinate system is mentioned. The coordinate method is defined. The basic theorems of analytic geometry are also mentioned. Theorems of triangle concurrency points (centroid, orthocenter, circumcenter, incenter) and Euler line theorem are proved with coordinate method. Mathematical competition problems where application of coordinate method is possible are shown. Theorems of the triangle and quadrilateral are also proved with coordinate method. At the end of this thesis, some properties of the second-order curves, i. e. properties of ellipse, hyperbola, parabola are shown and proved with coordinate method.

• Keywords: koordinatna metoda; planimetrija; Kartezijev koordinatni sustav; karakteristične točke trokuta; težište; ortocentar; središte opisane kružnice; središte upisane kružnice; Eulerov pravac; krivulje drugog reda
• Keywords (english): coordinate method; planimetry; Cartesian coordinate system; triangle concurrency points; centroid; orthocenter; circumcenter; incenter; Euler line theorem; second-order curves
• Language: croatian
• URN:NBN: urn:nbn:hr:217:545967
• 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: 2019-03-04 09:59:02