| Abstract | Ovaj rad opisuje odnos racionalne trigonometrije s klasičnom Euklidovom geometrijom. U Euklidovoj geometriji osnovni pojmovi su udaljenost i kut, dok u racionalnoj trigonometriji koristimo kvadrat udaljenosti i razmak. Racionalna trigonometrija koristi kvadratne vrijednosti i zbog toga se rješenja mogu prikazati u racionalnom obliku tako da se sve može izračunati primjenom aritmetike i algebre te su na taj način dobivena jednostavnija rješenja. U radu je iskazano i dokazano pet glavnih teorema racionalne trigonometrije kao i još neki potrebni za razvitak teorije. U njima se povezuje šest elemenata trokuta, a to su uvjet kolinearnosti, Pitagorin poučak, teorem o razmaku kao racionalan analogon poučka o sinusima, teorem o dualnom razmaku koji odgovara poučku o kosinusu i formula tri razmaka koja predstavlja teorem da je zbroj kutova u trokutu 180◦. Nakon svakog teorema dan je primjer njegove primjene u geometriji i njegova usporedba s Euklidovom geometrijom. U svakom od poglavlja predstavljeni su osnovni pojmovi i oznake potrebni za razvoj dane teorije. Također su definirane nove funkcije, npr. Arhimedova funkcija te su uvedeni i novi pojmovi poput dualnog razmaka i zakreta koje možemo povezati s vektorskim i skalarnim produktom. Neki od teorema koji su dokazani su sljedeći: teorem o polovištu, teoremi o paralelogramu, teoremi o simetrali dužine i teoremi o simetrali kuta. Na kraju rada dani su primjeri zadataka iz geometrije i fizike u kojima je korištena teorija racionalne trigonometrije. |
| Abstract (english) | This thesis describes the relationship between rational trigonometry and classical Euclidean geometry. In Euclidean geometry, the basic terms are distance and angle, while in rational trigonometry we use quadrance and spread. Rational trigonometry uses quadratic values and, because of this, solutions of the geometric problems can be presented in a rational form, so that everything can be calculated using arithmetic and algebra, and in this way simpler solutions are obtained. Thesis presents and proves five main theorems of rational trigonometry, as well as some other results necessary for the development of the theory. They connect six elements of the triangle, namely Triple quad formula, Pythagoras’ theorem, Spread law as a rational analog of the law of sines, Cross law which corresponds to the law of cosines, and Triple spread formula, which represents the theorem that the sum of the angles in a triangle is 180◦. After each theorem, an example of its geometry application is given and a comparison with Euclidean geometry is given. In each of the chapters, the basic terms and symbols necessary for the development of the given theory are presented. New functions were also defined, e.g. the Archimedes function, and new concepts such as cross and twist were introduced, which we can associate with the vector and scalar product. Some of the theorems that have been proved are as follows: Bisector theorem, Parallelogram theorems and theorems on vertex and perpendicular bisectors. At the end of the thesis, examples of problems from geometry and physics were given in which the theory of rational trigonometry can be applied. |
