Nožišni poligoni

master's thesis

• Author: Ivan Bošković
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu prikazani su raznovrsni rezultati o nožišnim poligonima, dakle poligonima čiji su vrhovi nožišta okomica spuštenih iz neke točke u ravnini na stranice nekog poligona. U večini slučajeva riječ je o četverokutima, no ima rezultata koji se odnose na n-terokute općenito. U euklidskoj geometriji trokuta odavno je poznato mnogo teorema o nožišnim trokutima pa je u prvom poglavlju izložen dio takvih rezultata. Istaknimo nejednakost poznatu kao Erdoes-Mordellov teorem, budući da se kasnije u radu dokazuje teorem za tetivne četverokute koji se može smatrati nekom vrstom analogona te nejednakosti. Istaknimo još dokaz svojstva četverokuta, različitog od paralelograma, da postoji točno jedna točka sa svojstvom da su nožišta okomica na stranice četverokuta kolinearne točke. Po analogiji s poznatim svojstvom iz geometrije trokuta, ta se jedinstvena točka naziva Simsonovom točkom, a pravac na kojem leže nožišta okomica naziva se Simsonov pravac četverokuta. Dokazane su još različite nejednakosti za n-terokute koje su analogoni poznatim rezultatima za četverokute te Steinerov teorem o kružnici koju čine točke čiji pridruženi nožišni poligoni imaju konstantnu površinu.
Erdos-Mordellov

Abstract (english)

This paper presents various results on foot polygons, ie polygons whose vertex tips are perpendiculars lowered from a point in the plane to the sides of a polygon. In most cases, these are quadrilaterals, but there are results that apply to n-terracots in general. Many theorems on knife triangles have long been known in the Euclidean geometry of triangles, so a part of such results is presented in the first chapter. Let us emphasize the inequality known as the Erdoes-Mordell theorem, since later in the paper we prove a theorem for tendon quadrilaterals that can be considered as a kind of analogue of this inequality. Let us point out another proof of the property of a quadrilateral, different from a parallelogram, that there is exactly one point with the property that the blades of the perpendiculars on the sides of the quadrilateral are collinear points. By analogy with a known property from the geometry of a triangle, this unique point is called the Simson point, and the direction on which the perpendicular blades lie is called the Simson direction of the quadrilateral. Different inequalities for n-terracots that are analogous to the known results for quadrilaterals and Steiner's theorem on a circle formed by points whose associated knife polygons have a constant surface have been proved.

• Keywords: nožišni poligoni; n-terokuti; nožišni trokuti; Erdoes-Mordellov teorem; Simsonov pravac
• Keywords (english): foot polygons; n-terracots; triangles; Erdoes-Mordell theorem; Simson direction
• Language: croatian
• URN:NBN: urn:nbn:hr:217:234076
• 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: 2020-11-13 08:18:17