Ponceletov izrek

delo diplomskega seminarja

Sara Močnik (Author), Aleš Vavpetič (Mentor)

Abstract

Ponceletov izrek pravi, da če za stožnici $S_1$ in $S_2$ obstaja $n$-kotnik, ki je včrtan stožnici $S_1$ in očrtan stožnici $S_2$, potem za $S_1$ in $S_2$ obstaja neskončno takih $n$-kotnikov. Vsaka točka na $S_1$ je oglišče kakega opisanega $n$-kotnika in vsaka točka na $S_2$ leži na stranici kakega opisanega $n$-kotnika. V realni projektivni ravnini najprej predstavimo in dokažemo poseben primer Ponceletovega izreka za trikotnike in nato še splošni izrek. Pri tem si pomagamo s Pascalovim izrekom, Brianchonovim izrekom, Carnotovim izrekom, dualom Carnotovega izreka in nekaj pomožnimi trditvami.

Keywords

matematika;Ponceletov izrek;projektivna geometrija;stožnice;

Data

Language:	Slovenian
Year of publishing:	2019
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[S. Močnik]
UDC:	514
COBISS:	18821209
Views:	1290
Downloads:	219
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Poncelet's theorem
Secondary abstract:	Poncelet's theorem states, that if $n$-sided polygon is inscribed in conic $S_1$ and circumscribed about conic $S_2$, then there exists infinitely many of such polygons. Moreover, for any point $P$ of $S_1$, there exists an $n$-sided polygon, also inscribed in conic $S_1$ and circumscribed about conic $S_2$, which has $P$ as one of its vertices, and for any point $R$ of $S_2$, there exists an $n$-sided polygon, also inscribed in conic $S_1$ and circumscribed about conic $S_2$, such that tangent to $S_2$ from $R$ is one of its lines. In real projective plane we first explain special case of Poncelet's theorem for triangles and then the general case. For that we use Pascal's theorem, Brianchon's theorem, Carnot's theorem, dual of Carnot's theorem and some other claims.
Secondary keywords:	mathematics;Poncelet theorem;projective geometry;conics;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	28 str.
ID:	11228341