| Abstract | U ovom radu bavili smo se karakterizacijama trapeza. Najprije smo definirali trapez, a zatim iskazali i dokazali njegova dobro poznata svojstva. Također smo dokazali Eulerov teorem za četverokute kao i poseban slučaj kad je četverokut trapez. Kroz analizu trapeza kao geometrijskog lika sustavno smo pokazali tri metode za određivanje površine. Podjelom na jednostavnije likove, nadopunjavanjem i transformacijskom geometrijom naveli smo različite načine za dobivanje poznate formule za izračunavanje površine trapeza ako su nam poznate duljine osnovica i duljina visine trapeza. Zatim smo uveli pojam trokuta dijagonalnih točaka i objasnili vezu takvog trokuta s nekim karakterizacijama trapeza. Osim karakterizacija vezanih uz duljine i odnose stranica trapeza, također smo uveli pojam Garfieldovog trapeza i objasnili po čemu je on zanimljiv. Dokazali smo i iskazali poopćenje teorema koji se pripisuje Napoleonu. Bavili smo se i najpravilnijim trapezom - jednakokračnim. Pokazali smo kakve su dijagonale jednakokračnog trapeza te kako možemo izračunati duljinu dijagonale. Na kraju, iskazali smo i dokazali Steinerov teorem za trapez koji kaže da u svakom trapezu polovišta osnovica, točka sjecišta dijagonala i točka u kojoj se sijeku produžeci krakova, leže na istom pravcu. |
| Abstract (english) | In this thesis, we have explored the characterizations of trapezoid. First, we defined what a trapezoid is and then stated and proved its well-known properties. We also proved Euler’s theorem for quadrilaterals and its special case when a quadrilateral is a trapezoid. Through the analysis of the trapezoid as a geometric figure, we systematically presented three methods for determining its area. By dividing it into simpler shapes, complementing, and using transformational geometry, we outlined various approaches to derive the well-known formula for calculating the area of a trapezoid, when the lengths of the bases and the height are known. Then, we introduced the concept of the diagonal point triangle and explained its connection to some characterizations of trapezoids. In addition to characterizations related to lengths and side ratios of trapezoid, we also introduced the concept of Garfield’s trapezoid and explained its significance. We also have proven and stated a generalization of the theorem attributed to Napoleon. Furthermore, we explored the most regular trapezoid, the isosceles trapezoid, demonstrating properties of its diagonals and providing methods for calculating their lengths. Finally, we presented and proved Steiner’s theorem for trapezoid, which asserts that the midpoints of the bases, the intersection point of the diagonals, and the point where the extensions of the legs intersect, lie on the same line. |
