Apolonijev problem

master's thesis

• Title (english): Apollonius' problem
• Author: Martina Hanževački
• Mentor: Mea Bombardelli (mentor)
• Committee member: Mea Bombardelli (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Matej Mihelčić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu proučava se i rješava Apolonijev problem koji glasi: Dani su tri objekta, krug, pravac ili točka. Konstruiraj kružnicu koja dodiruje sva tri dana objekta. Problem je je podijeljen u deset zasebnih konstruktivnih problema. Rad je podijeljen na tri poglavlja. U uvodu su navedeni svih deset problema te neke osnovne definicije, teoremi i konstrukcije. U prvom poglavlju opisane su konstrukcije Apolonijevih problema u kojima nema kružnice tj. u kojima su svi zadani elementi ili točke ili pravci. Poglavlje je podijeljeno na četiri potpoglavlja od kojih je svako rješenje jednog konstruktivnog problema. U drugom poglavlju opisane su konstrukcije Apolonijevih problema u kojima su jedan ili dva dana elementa kružnice, a preostali točke i/ili pravci. Poglavlje je podijeljeno na pet potpoglavlja od kojih je svako rješenje jednog konstruktivnog problema. U trećem poglavlju opisane su konstrukcije originalnog Apolonijevog problema u kojima su sva tri dana elementa kružnice. U svim poglavljima proučava se metodika rješavanja konstruktivne zadaće, etape rješavanja (analiza, konstrukcija, dokaz i rasprava) te su neki problemi riješeni i na više načina. Detaljno su opisane etape rješavanja svih konstrukcija. Neki problemi podijeljeni su na više slučajeva ovisno o položajima zadanih objekata i zasebno riješeni kao konstruktivne zadaće. Zadatci su poredani prema složenosti i popraćeni su nizom slika konstruiranih u GeoGebri.

Abstract (english)

The thesis studies and solves the Apollonius’ problem, which is: Three objects are given, a circle, a line, or a point. Construct a circle that touches all three given objects. The problem is subdivided into ten constructive problems. All ten problems, some basic definitions, theorems and simpler construction problems needed for the thesis are listed in the introduction. The thesis consists of three chapters. The first chapter describes the constructions of Apollonius’ problems in which there are no circles, or in other words, in which all given elements are either points or lines. The chapter is divided into four subchapters - each containing a solution to one constructive problem. The second chapter describes the constructions of Apollonius’ problems in which one or two given elements are circles and others are points and/or lines. This chapter is divided into five subchapters with each containing a solution to one constructive problem. The third chapter describes the construction of the original Apollonius’ problem, in which all three given elements are circles. All chapters study the methods of solving a constructive problem and stages of solving one (analysis, construction, proof, and discussion). Also, some problems are solved in more than one way. The stages of solving the constructive problems are described in detail. Some problems are divided into several cases depending on the positions of given objects and are solved separately as constructive problems. The problems are arranged by complexity and are substantiated with a list of images constructed using GeoGebra.

• Keywords: krug; pravac; točka; konstruktivni problemi; GeoGebra
• Keywords (english): circle; line; point; constructive problem; GeoGebra
• Language: croatian
• URN:NBN: urn:nbn:hr:217:473127
• 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: 2022-10-26 11:49:03