Vivianijev izrek in njegove posplošitve

Ceferin, Terezija (author), Repovš, Dušan (mentor), Cencelj, Matija (co-mentor)

V magistrskem delu predstavljamo Vivianijev izrek, ki velja v enakostraničnem trikotniku in pravi, da je vsota razdalj med poljubno točko in stranicami enaka višini trikotnika oziroma vsota razdalj je konstantna. V delu ugotavljamo, ali je vsota razdalj od točke do stranic neenakostraničnega trikotnika tudi enaka kateri izmed višin trikotnika oziroma raziskujemo, ali obstaja kakšno drugo razmerje med vsoto razdalj in višinami. Nadalje preučujemo posplošitve izreka na mnogokotnike in poliedre. Koncept posplošitve na izbranih primerih prikažemo z uporabo različnih metod. V ta namen ločeno preučujemo konveksne in konkavne mnogokotnike (oziroma poliedre). V zaključnem delu navedemo konkretne primere obravnave izreka pri matematiki v šoli in primer njegove uporabe pri risanju diagramov, ki imajo obliko enakostraničnega trikotnika.

• Language: Slovenian
• Keywords: Vivianijev izrek
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2016
• PID: 20.500.12556/RUL-87143
• COBISS.SI-ID: 11340873
• Publication date in RUL: 04.09.2017

Secondary language

• Language: English
• Title: Viviani's theorem and its generalizations

Abstract

The present master’s thesis deals with Viviani’s theorem valid in an equilateral triangle and stating that the sum of the distances between any interior point and the sides equals the triangle’s altitude i.e. that the sum of the distances is constant. In the paper it is investigated whether the sum of the distances from an interior point to the sides of a nonequilateral triangle also equals any of the triangle’s altitudes or whether there exists any other relation between the sum of the distances and the altitudes. A further investigation refers to a generalisation of the theorem to other polygons and polyhedra. The generalisation concept on chosen examples is shown by the use of various methods. To this end, convex and concave polygons (or polyhedra) are investigated separately. The conclusion gives concrete examples of dealing with the theorem in class and an example of its use in the drawing of diagrams having the form of an equilateral triangle.

• Keywords: Viviani’s theorem