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master's thesis
Opisana i upisana kružnica trokuta
Zagreb: University of Zagreb, Faculty of Science, 2017. urn:nbn:hr:217:397179
Perković, Marijana
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Perković, M. (2017). Opisana i upisana kružnica trokuta (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:397179

Perković, Marijana. "Opisana i upisana kružnica trokuta." Master's thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:397179

Perković, Marijana. "Opisana i upisana kružnica trokuta." Master's thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:397179

Perković, M. (2017). 'Opisana i upisana kružnica trokuta', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 November 2024, https://urn.nsk.hr/urn:nbn:hr:217:397179

Perković M. Opisana i upisana kružnica trokuta [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2017 [cited 2024 November 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:397179

M. Perković, "Opisana i upisana kružnica trokuta", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2017. Available at: https://urn.nsk.hr/urn:nbn:hr:217:397179

Metadata
TitleOpisana i upisana kružnica trokuta
AuthorMarijana Perković
MentorSanja Varošanec (mentor)
Committee memberSanja Varošanec (predsjednik povjerenstva)
Committee memberOzren Perše (član povjerenstva)
Committee memberFranka Miriam Bruckler (član povjerenstva)
Committee memberMarko Vrdoljak (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2017-07-17, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U ovom su radu najprije prikazana osnovna svojstva nekoliko vrsta kružnica koje su povezane s trokutom. S pojmom upisane kružnice povezana je i poznata Heronova formula koja je dokazana koristeći svojstva vezana uz polumjer upisane kružnice trokuta. Važnu ulogu u geometriji ima i Eulerov teorem koji govori o udaljenosti središta opisane kružnice i središta upisane kružnice trokuta. U radu je dan njegov dokaz te je korišten u rješavanju mnogih zadataka. Kružnica devet točaka je također povezana s pojmom upisane i opisane kružnice trokuta te je dan niz važnih tvrdnji vezanih uz kružnicu devet točaka, od kojih je najvažnija Feuerbachov teorem. Nadalje, dane su konstrukcije trokuta u kojima je jedan od zadanih elemenata polumjer opisane ili upisane kružnice trokuta ili položaj središta tih kružnica, ali i dokazana nemogućnost izvodljivosti nekih konstrukcija ravnalom i šestarom. U završnom dijelu rada dane su neke nejednakosti vezane uz duljine stranica te upisanu i opisanu kružnicu trokuta.
Abstract (english)
In this thesis, the basic properties of several types of circles related to triangle are presented first. The incircle is related to the well-known Heron's formula which has been proven using the properties of the inradius. Euler's formula is one of the most important theorems of geometry, which relates the circumradius, the inradius and the distance between the circumcenter and the incenter of a triangle. We proved Euler's formula and we used it in solving many tasks. Every triangle has a nine point circle which is connected to its inscribed circle and circumscribed circle. Important properties of nine-point circle are given, including the most important Feuerbach's theorem. Furthermore, there are given constructions of a triangle given by the circumradius, the inradius or the center position of these circles. We also proved the impossibility of some constructions using a compass and a straightedge. In the last chapter, theorems on geometric inequalities for the sides and the radii of a triangle are obtained.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:397179
Study programmeTitle: 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 resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2018-03-14 12:38:46