| Abstract | U ovom radu prikazan je manje poznat teorem elementarne geometrije, Teorem o šest kružnica, i neke njegove generalizacije. Ovaj teorem jedan je od mnogih u knjizi "The seven circles theorem and other new theorems" koja je rezultat zajedničkog rada trojice prijatelja i geometrijskih entuzijasta, C. J. A. Evelyn-a, G. B. Money-Coutts-a, i J. A. Tyrrell-a. Prvo poglavlje sadrži kratku povijest podrijetla teorema o šest kružnica i dvije različite formulacije te njihove dokaze. Razmatraju se lanci kružnica upisanih u dani trokut \(P_{1}P_{2}P_{3}\): prva kružnica, \(C_{1}\), upisana je u kut pri vrhu \(P_{1}; C_{2}\), upisana je u kut pri vrhu \(P_{2}\) i dodiruje kružnicu, \(C_{1} ; C_{3}\), upisana je u kut pri vrhu \(P_{3}\) i dodiruje \(C_{2}\); i tako dalje, ciklički. Tvrdnja je teorema da je ovaj postupak periodičan, to jest, \(C_{7}=C_{1}\). Prva formulacija teorema razmatra samo lance kružnica kojima sve dodirne točke leže na stranicama trokuta, a ne na njihovim produžetcima, te je u ovom slučaju lanac 6-periodičan. Nadalje, promatraju se lanci kružnica u kojima sljedeća kružnica u lancu može dodirivati stranicu trokuta, ali i produžetak te stranice. Pokazuje se da je, općenito, lanac konačno 6-periodičan, ali može imati proizvoljno dug pretperiod. U drugom poglavlju razmatra se mogućnost poopćenja teorema o šest kružnica i na druge poligone, osim trokuta. Dokazuje se da postoji klasa nepravilnih \(n\)-terokuta za koje je sačuvana periodičnost. Konačno, razmatra se zanimljiva varijacija glavnog teorema, kada su stranice trokuta zamijenjene kružnicama. Tyrrell i Powell dokazali su da je i tada sačuvana 6-periodičnost. |
| Abstract (english) | The topic of this graduate thesis is one of lesser known gems of elementary geometry, The six circles theorem, and some of its generalizations. This theorem is one of many theorems in the book ”The seven circles theorem and other new theorems” which is a result of collaboration of three geometry enthusiasts, C. J. A. Evelyn, G. B. Money-Coutts, and J. A. Tyrrell. The first chapter contains a short history of The six circles theorem’s origins and two different forms and proofs of it. The theorem concerns chains of circles inscribed in a given triangle \(P_{1}P_{2}P_{3}\): the first circle, \(C_{1}\), inscribed in the first angle at \(P_{1}; C_{2}\), inscribed in the angle at \(P_{2}\) and tangent to the circle \(C_{1} ; C_{3}\), inscribed in the angle at \(P_{3}\) and tangent to \(C_{2}\), and so on, cyclically. The claim of the theorem is that this process is periodic, that is, \(C_{7}=C_{1}\). The first form of the theorem holds for a chain of circles for which all tangency points lie on the sides of the triangle, and not their extensions. Secondly, chains of circles are observed for which the next circle can be tangent to a side of the triangle but also to its extension. It is proven that, in general, the chain is eventually 6-periodic but may have an arbitrarily long pre-period. In the second chapter it is investigated whether The six circles theorem extends to polygons other than triangles. It is shown that there is a subclass of irregular \(n\)-gons for which periodicity holds. Finally, an interesting variation of the main theorem is considered, where the sides of a triangle are replaced by circles. It was proven by Tyrrell and Powell that the 6-periodicity persists even then. |
