Konika devet točaka

master's thesis

• Author: Tajana Berić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Sonja Štimac (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Teorem o konici devet točaka poopćenje je poznatog teorema o kružnici devet točaka. U tom poopćenju ulogu ortocentra trokuta \(ABC\) preuzima bilo koja točka \(P\) u ravnini trokuta koja ne pripada spojnicama njegovih vrhova. Tvrdnja glasi da postoji konika koja prolazi kroz sljedećih devet točaka: polovišta stranica trokuta, polovišta dužina koje spajaju vrhove trokuta s točkom \(P\) te sjecišta pravaca tih spojnica sa suprotnim stranicama trokuta. Prvi navedeni dokaz sastoji se u primjeni afine transformacije koja prikladno izabrani trokut i njegov ortocentar preslika u zadane točke \(A\), \(B\), \(C\) i \(P\). Time se tražena konika dobiva kao afina slika jedne kružnice devet točaka. Nadalje, primjenom analitičke metode u realnoj afinoj ravnini dokazuje se da je konika devet točaka geometrijsko mjesto centara svih konika pramena određenog četverovrhom \(ABCP\). U trećem, najopsežnijem poglavlju, prethodne tvrdnje interpretiraju se i poopćavaju u terminima projektivne geometrije. U tu svrhu potrebno je primijeniti niz pojmova i ključnih rezultata projektivne geometrije. Umjesto geometrijskog mjesta centara konika jednog pramena promatra se, općenito, polarna konika pridružena bilo kojem pravcu \(p\) s obzirom na pramen zadan osnovnim četverovrhom. Polarna konika može prolaziti još dvjema istaknutim točkama pravca \(p\), čime postaje konika jedanaest točaka. Klasični teorem o kružnici devet točaka odatle slijedi u slučaju kad su te dvije točke tzv. apsolutne točke realne projektivne ravnine.

Abstract (english)

The nine-point conic theorem is a generalization of the well-known nine-point circle theorem. In this generalization, the role of the orthocenter of the triangle \(ABC\) is taken over by any point \(P\) in the plane of the triangle, which does not belong to the lines joining the vertices of the triangle. The claim is that there is a conic that passes through the following nine points: the midpoints of the sides of the triangle, the midpoints points of the segments joining \(P\) to the vertices of the triangle and the points where the lines joining \(P\) to the vertices intersect the opposite sides. The first proof consists of the application of an affine transformation which maps an appropriately chosen triangle and its orthocenter to the points \(A\), \(B\), \(C\) and \(P\). The required conic is then obtained as the affine image of the nine-point circle. Furthermore, applying the analytical method in the real affine plane, it is proven that the nine-point conic is the locus of centers of the conics belonging to the pencil determined by the quadrilateral \(ABCP\). In the third, most comprehensive chapter, previous assertions are interpreted and generalized in terms of projective geometry. To this purpose, it is necessary to apply a number of concepts and key results of projective geometry. The locus of centers of conics belonging to a pencil is replaced, more generally, by the polar conic associated to any line \(p\) with respect to the observed pencil of conics. The polar conic may be incident with two more prominent points of the line \(p\), thus becoming the eleven-point conic. The classical nine-point circle theorem then follows in the special case when these two points are the so called absolute points of the real projective plane.

• Keywords: konika devet točaka; kružnica devet točaka; projektivna geometrija
• Keywords (english): nine-point conic theorem; nine-point circle theorem; projective geometry
• Language: croatian
• URN:NBN: urn:nbn:hr:217:814530
• 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: 2020-01-10 13:30:21