Pappusov teorem o šesterovrhu

master's thesis

• Author: Ana Lučić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Damir Bakić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Pappusov teorem o šesterovrhu jedan je od najvažnijih teorema projektivne geometrije. On opisuje konfiguraciju od devet točaka i devet pravaca koja nastaje tako što se parovi nasuprotnih stranica šesterovrha, čiji vrhovi leže naizmjence na dva različita pravca, sijeku u trima kolinearnim točkama. U radu je izloženo osam različitih dokaza Papposovog teorema, s ciljem da se upoznaju i usporede različite metode pristupa. Posebno se razmatraju prednosti projektivno-geometrijskog pristupa u odnosu na osnovnu formulaciju teorema u okviru euklidske ravnine. Istaknuta je veza s Pascalovim teoremom i, preko Cayley-Bachrach-Chaslesovog teorema, dublja povezanost s algebarskom geometrijom. S druge strane, projektivnom interpretacijom euklidskog pojma kružnice pomoću para imaginarnih točaka na neizmjerno dalekom pravcu, dobiva se Miquelov teorem iz geometrije kružnica. Završno, naznačena je mogućnost formalnog algebarskog dokaza, preko homogenih koordinata, kojim se Pappusov teorem obuhvaća u svojoj punoj općenitosti.

Abstract (english)

The hexagon theorem of Pappus is one of the most important theorems in projective geometry. It describes the configuration of nine lines and nine points which is generated by pairs of opposite sides of the hexagon, whose vertices lie alternatively on two different lines, intersect in three collinear points. In this paper we present eight proofs of Pappus’s theorem, with the aim of learning about different methods of approach to the problem and making a comparison between them. In particular, we expose the advantage of the projective geometry setting over the initial Euclidean formulation. We emphasize the connection to the Pascal’s theorem and, through the Cayley-Bachrach-Chasles theorem, the profound connection with Algebraic geometry. On the other hand, applying the projective interpretation of a circle, which is basically a Euclidean concept, by using a pair of imaginary points on the line at infinity, we derive Miquel’s theorem from the geometry of circles. Finally, we indicate the possibility of a formal algebraic proof using homogeneous coordinates, which encompasses Pappus’s theorem in its full generality.

• Keywords: Pappus; šesterovrh; projektivna geometrija; Pascalov teorem; Cayley-Bachrach-Chaslesov teorem; Miquelov teorem
• Keywords (english): Pappus; hexagon; projective geometry; Pascal’s theorem; Cayley-Bachrach-Chasles theorem; Miquel’s theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:865273
• 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: 2018-02-02 13:36:37