Upisana i pripisane kružnice trokuta

Abstract

U ovom radu proučavali smo trokut, njemu pridružene kružnice te njihova središta. Promatrali smo sličnosti, veze i svojstva između njih. Uz četiri najpoznatije karakteristične točke trokuta: težište, ortocentar, središte trokutu upisane kružnice i središte trokutu opisane kružnice, osvrnuli smo se i na druge, manje poznate karakteristične točke trokuta poput Gergonnove točke i Nagelove točke. Također, proučavali smo metričke relacije među karakterističnim točkama. Nadalje, definirali smo Feuerbachovu kružnicu i pokazali njezina svojstva. Na samom kraju rada dokazali smo jedan od najljepših teorema geometrije trokuta: Feuerbachov teorem koji govori da Feuerbachova kružnica dira upisanu kružnicu danog trokuta iznutra, a njegove pripisane kružnice izvana.This paper sets out to study the triangle, the circles connected with the triangle, and their centres. In the study, their properties, as well as the similarities and relationships between them, have been observed. Apart from the four points of concurrency in triangles: the centroid, the orthocentre, the incentre and the circumcentre, other, less known concurrency points, such as the Gergonne point and the Nagel point, have also been addressed. Furthermore, the Feuerbach circle has also been defined and its properties have been demonstrated. At the very end of the paper, one of the most beautiful theorems of the triangle geometry, the Feuerbach theorem, which states that the Feuerbach circle is tangent internally to the incircle and tangent externally to the triangles excircles, has been proved