Droz-Farnyjev teorem

master's thesis

• Author: Ines Huić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Luka Grubišić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-17, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Švicarski matematičar Arnold Droz-Farny objavio je 1899. godine, bez dokaza, tvrdnju iz geometrije trokuta u euklidskoj ravnini koja je u matematičkoj literaturi postala poznata kao Droz-Farnyjev teorem: Polovišta segmenata koje na stranicama trokuta odsijecaju dva međusobno okomita pravca koji prolaze njegovim ortocentrom tri su kolinearne točke. U ovom radu prikazan je prvi sintetički dokaz osnovnog Droz-Farnyjevog teorema, koji je objavio Aymé 2004. godine, zatim dokazi nekih njegovih poopćenja, među kojima je najvažniji Goormaghtighov teorem iz 1930. godine te još neki zanimljivi rezultati povezani s osnovnim teoremom. Primjerice, Droz-Farnyjevi pravci pridruženi jednom trokutu s obzirom na različite izbore parova okomitih pravaca kroz njegov ortocentar određuju familiju trokuta koji su svi upisani u jednu kružnicu, a opisani jednoj konici te imaju zajednički Eulerov pravac i još neke karakteristične točke. Time je ustanovljeno svojstvo tipa kakav se u geometriji često naziva porizmom, a ovdje je to i poseban slučaj glasovitog Ponceletovog porizma. Na kraju su istaknuti bitni zaključci aksiomatske analize Droz-Farnyjevog i Goormaghtighovog teorema koju su 2016. proveli R. i H. Struve, s gledišta zasnivanja geometrije na pojmu zrcaljenja.

Abstract (english)

The Swiss mathematician Arnold Droz-Farny published in 1899, without a proof, the statement on a property of a triangle in the Euclidean plane geometry that became known in mathematical literature as the Droz-Farny Line Theorem: Two perpendicular lines through the orthocenter of a triangle intercept on the sides of the triangle three segments whose midpoints are collinear. In this paper we present the first purely synthetic proof of the basic Droz-Farny theorem as it was published by Aymé in 2004, as well as proofs of some of its generalizations, among which the theorem of Goormaghtigh from 1930 turned out to be the most important one. We also exhibit some interesting results related to the basic theorem. For example, Droz-Farny lines generated in a triangle by various choices of two perpendicular lines through the orthocenter constitute a family of triangles that all share the circumcircle and an inscribed conic, having the Euler line and some characteristic points in common, too. In this way a certain property is established of the type that in geometry is often called a porism. This one is also a special case of the renowned Poncelet porism. Finally, we indicate the essential conclusions of the axiomatic analysis of the theorems of Droz-Farny and Goormaghtigh as published by R. and H. Struve in 2016, from the viewpoint of the geometry of reflections.

• Keywords: Droz-Farny; Aymé; Goormaghtighov teorem; Ponceletova porizma; Struve
• Keywords (english): Droz-Farny; Aymé; Goormaghtighov theorem; Poncelet porism; Struve
• Language: croatian
• URN:NBN: urn:nbn:hr:217:281830
• 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: 2018-11-28 10:50:07