Pappusov teorem: razni dokazi i varijacije

Papusov teorem je jednostavan, ali u isto vrijeme jako bitan i koristan teorem pripisan Papusu iz Aleksandrije, posljednjem velikom matematičaru Aleksandrijske škole. Ovaj teorem pravilno iskazan sastoji se od samo devet točaka i devet pravaca i smatra se jednim od prvih velikih teorema projektivne geometrije. Fokus ovoga rada dokazi su Papusovog teorema u namjeri prezentiranja raznih metoda i razlika među njima, istodobno prikazujući njegove generalizacije i varijacije. U ovome radu naglasit ćemo posebnosti ovoga teorema koji svoju punu općenitost ima u projektivnoj geometriji. Iskazat ćemo i na dva načina dokazati jednu njegovu euklidsku varijaciju koja je zapravo specijalizacija Papusovog teorema kada se Papusov pravac nalazi u beskonačnosti. Papus je originalno dokazao ovaj teorem primjenom euklidskih metoda, ali u ovome radu ćemo također dati i dva projektivna načina kako ga dokazati koristeći homogene koordinate. Istaknuti ćemo i tri varijacije Papusovog teorema: Pascalov teorem, Chayley-Bacharach-Chasles teorem i Miquelov teorem. Svaki od ovih teorema ćemo iskazati i dokazati te objasniti u kakvoj je vezi s Papusovim teoremom. Rad ćemo završiti algebarskim dokazom Papusovog teorem u njegovoj punoj općenitosti tako što ćemo Papusov teorem izraziti pomoću vektorskih produkata i determinante. U tom na prvi pogled jednostavnom dokazu nužna je upotreba računalnog programa. Pappus’s Theorem is simple yet very important and useful theorem contributed to Pappus of Alexandria, the last great mathematician of the Alexandrian School. The statement of this theorem, when properly stated, consists only of nine point and nine lines and we consider it one of the first great theorems of projective geometry. This thesis focuses on proofs of Pappus’s Theorem with intention of giving variety of methods and discussing differences between them while also presenting his generalizations and variations. In this thesis, we will emphasize special properties of this theorem which attains its full generality in projective geometry. We will state one of his Euclidean versions which is actually the specialization of the Pappus’s theorem when Pappus’s line is send to infinity. Pappus originally proved this theorem using Euclidean methods but in this thesis we will also describe two projective ways how to prove it using homogeneous coordinates. We will also present three variations of Pappus’s Theorem: Pascal’s Theorem, ChayleyBacharach-Chasles Theorem and Miquel’s Theorem. We will state and prove each of these theorems and explain what connections do they have with Pappus’s Theorem. We will end this thesis with algebraic proof of Pappus’s Theorem in its full generality by expressing the statements of Pappus’s Theorem using cross-products and a determinant. At first that proof seems to be very simple but the help of the computer program is essential here.