Steinerjeva trikotniku včrtana elipsa

diplomsko delo

Klavdija Majcen (Author), Dušan Pagon (Mentor)

Abstract

V diplomskem delu je predstavljena elipsa, včrtana trikotniku. Osredotočimo se na Steinerjevo elipso, ki je včrtana v trikotnik in se dotika vseh stranic trikotnika v njihovih razpoloviščih. Oglišča trikotnika naj ležijo v kompleksni ravnini in so ničle polinoma. Ničli odvoda tega polinoma sta potem gorišči Steinerjeve trikotniku včrtane elipse. Elipsa ima središče v težišču trikotnika. Premica, ki se najbolj približa ogliščem trikotnika, poteka skozi gorišči trikotniku včrtane elipse in jo imenujemo premica najboljšega prileganja. Diplomsko delo ima v šestih poglavjih. V prvem poglavju je predstavljen Mardenov izrek, ki ga lahko dokažemo šele v kombinaciji z Bôcherjevim delom dokaza. V drugem poglavju smo se osredotočili na Mardenov in Bôcherjev dokaz, ter kombinacijo teh dveh podkrepili s primeri. Mardenov izrek velja za polinome tretje stopnje s kompleksnimi koeficienti; te povezave med ničlami kubičnega polinoma in ničlami njegovega odvoda so opisane v naslednjem, tretjem poglavju. V tem delu je vpeljan tudi Steinerjev izrek, po katerem se imenuje elipsa, včrtana trikotniku. Steinerjev izrek uporabimo pri afini in linearni transformaciji; obe preslikavi sta predstavljeni v četrtem poglavju, v petem poglavju pa smo dokazali Steinerjev izrek in opisali Steinerjevo včrtano elipso. Diplomsko delo smo zaključili z zadnjim, šestim poglavjem, kjer smo se posvetili dokazovanju izreka iz tretjega poglavja. V diplomskem delu sem si pomagala s programom GeoGebra, s pomočjo katerega sem narisala vse priložene slike.

Keywords

diplomska dela;trikotniku včrtana elipsa;Steinerjev izrek;Mardenov dokaz;Bôcherejv dokaz;ničle polinoma;afina transformacija;linearna transformacija;

Data

Language:	Slovenian
Year of publishing:	2013
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[K. Majcen]
UDC:	514.135(043.2)
COBISS:	20138248
Views:	1535
Downloads:	145
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The Steiner inellipse of a triangle
Secondary abstract:	In our thesis we present the ellipse inscribed within triangle. We have focused on the Steiner's ellipse which is inscribed within triangle and it touches all the sides of a triangle in their midpoints. Vertices of the triangle are placed in the complex plane and they are the zeros of a polynomial. Zeros of the derivative of this polynomial are the focal points of the Steiner's ellipse inscribed into the triangle. The center of this ellipse is at the center of gravity of the original triangle. The line that passes closest to all corners of the given triangle, contains the focal points of the ellipse inscribed within this triangle and is called the best-fit straight line. The thesis is divide into six chapters. In the first chapter Marden's theorem (that can be proved only in combination with the Bôcher's part of the proof) is presented. In second chapter we focus on Marden's and Bôcher's parts of the complete proof and we support the combination of these two proofs with some examples. Marden's theorem applies to third-degree polynomials with complex coefficients; connections between the zeros of cubic polynomial and zeros of its derivative are investigated in the following, third chapter. In this chapter we then introduce the Steiner's theorem, due to whose author the ellipse inscribed within the triangle is also named. We use the Steiner's theorem in affine and linear transformations, both investigated in the fourth chapter. In the fifth chapter we have proved Steiner's theorem and studied Steiner's inscribed ellipse in details. We are completing our thesis with the last, sixth chapter, where we focus on the proof of main theorem from the third chapter. Through the thesis we have used the program GeoGebra, which helped us at the preparation of all included illustrations.
Secondary keywords:	Marden's proof;Bôcher's proof;Steiner's theorem;the ellipse inscribed within triangle;zeros of a cubic polynomial;linear and affine transformation;best-fit straight line;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	39 f.
ID:	8727567