Posledici Carnotovega izreka : delo diplomskega seminarja

Šadl Praprotnik, Ada (author), Vavpetič, Aleš (mentor)

Carnotov izrek nam pove, kdaj šest točk, po dve na vsaki stranici poljubnega trikotnika, leži na isti stožnici. Predstavili in dokazali bomo Carnotov izrek v evklidski in projektivni ravnini ter dve njegovi posledici. Prva posledica je definirana v evklidski ravnini in nam predstavi, kako s pomočjo danega trikotnika in izbrane točke konstruiramo šest točk na nosilkah stranic trikotnika, ki ležijo na isti stožnici. To stožnico imenujemo Cevova stožnica. Druga posledica je definirana v projektivni ravnini in nam pove, da točke na stranicah poljubnega trikotnika - ki jih dobimo tako, da iz vsakega oglišča tega trikotnika narišemo dve tangenti na dano stožnico, nato pa tangenti sekamo z nasprotno stranico - ležijo na isti stožnici.

• Language: Slovenian
• Keywords: Carnotov izrek, Pascalov izrek, projektivna ravnina, Cevova stožnica, Gergonnova točka
• Work type: Final seminar paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-108764
• UDC: 514
• COBISS.SI-ID: 18690137
• Publication date in RUL: 20.07.2019

Secondary language

• Language: English
• Title: Two applications of the theorem of Carnot

Abstract

The theorem of Carnot gives us a necessary and sufficient condition for six points, two on each side of a given triangle, to be on a conic. We will explain the theorem in both Evclidean and projective plane and then explain two of its applications. The first one is called the construction of the Cevian conic. It tells us how to construct six points on the sides or the carriers of the sides of a triangle so that they will form a conic, a Cevian conic. The second application is defined in the projective plane and tells us that the tangent lines from the vertices of a given triangle to an arbitrary conic intersect the carriers of the opposite sides of the triangle in six points that are on a conic.

• Keywords: Carnot theorem, Pascal theorem, projective plane, Cevian conic, Gergonne point