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master's thesis
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Miquel's Theorem
Eva Kalčić (2019)
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Applied Mathematics
Applied Mathematics Research Group
Cite this document

Kalčić, E. (2019). Miquelov teorem (Master's thesis). Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:860283

Kalčić, Eva. "Miquelov teorem." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2019. https://urn.nsk.hr/urn:nbn:hr:126:860283

Kalčić, Eva. "Miquelov teorem." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2019. https://urn.nsk.hr/urn:nbn:hr:126:860283

Kalčić, E. (2019). 'Miquelov teorem', Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 26 January 2020, https://urn.nsk.hr/urn:nbn:hr:126:860283

Kalčić E. Miquelov teorem [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2019 [cited 2020 January 26] Available at: https://urn.nsk.hr/urn:nbn:hr:126:860283

E. Kalčić, "Miquelov teorem", Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2019. Available at: https://urn.nsk.hr/urn:nbn:hr:126:860283

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Metadata
TitleMiquelov teorem
Title (english)Miquel's Theorem
AuthorEva Kalčić
MentorZdenka Kolar-Begović (mentor)
Committee memberMihaela Ribičić Penava (predsjednik povjerenstva)
Committee memberZdenka Kolar-Begović (član povjerenstva)
Committee memberTomislav Marošević (član povjerenstva)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
(Chair of Applied Mathematics)
(Applied Mathematics Research Group)
Osijek
Defense date and country2019-09-06, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Geometry and Topology
Abstract
Miquelov teorem kaže da ako na svakoj stranici trokuta ili njihovim produženjima odaberemo po jednu točku, tada se kružnice određene vrhom trokuta i dvjema točkama na stranicama koje prolaze tim vrhom sijeku u jednoj točki. Tu točku nazivamo Miquelovom točkom polaznog trokuta, trokut određen odabranim točkama na stranicama Miquelov trokut polaznog trokuta, a kružnice određene vrhom trokuta i dvjema točkama na stranicama koje prolaze tim vrhom Miquelovim kružnicama polaznog trokuta. U radu je teorem dokazan na dva načina koristeći svojstvo koncikličnosti točaka i primjenom inverzije. Osim Miquelovog teorema, dokazali smo i brojne tvrdnje koje govore o položaju Miquelove točke u odnosu na Miquelov trokut, o jednakostima kutova koje vrijede u polaznom i Miquelovom trokutu, o trokutima sličnim polaznom trokutu, o karakterističnim točkama Miquelovog trokuta. U radu su razmatrane i zanimljive tvrdnje o koncikličnosti točaka u kojima se spominju Miquelovi elementi trokuta.
Abstract (english)
Miquel’s theorem is concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. The point of intersection is called Miquel point relative to the initial triangle, the triangle connecting the side points is called Miquel triangle relative to the initial triangle and circles passing through each polygon vertex and its neighboring side points is called Miquel circles relative to the initial triangle. We have proved the theorem in two ways, using properties of concyclic points and by applying inversion. Apart from Miquel’s theorem, we have also proved many propositions about position of Miquel point in relation to the Miquel triangle, equality of angles in initial and Miquel’s triangle, characteristic points of Miquel’s triangle. Also, in this paper are considered interesting claims about concyclic points in which the Miquel elements of the triangle are mentioned.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:860283
Study programmeTitle: Mathematics and Computer Science
Study programme type: university
Study level: integrated undergraduate and graduate
Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of usehttp://rightsstatements.org/vocab/InC/1.0/
Created on2019-09-13 11:15:34