Cramer-Castillonov problem

Cramer-Castillonov problem je geometrijski zadatak koji potječe još iz antičke Grčke. Promatramo kružnicu i tri zadane točke. Problem glasi: kako upisati trokut toj kružnici tako da pravci na kojima leže stranice trokuta prolaze kroz tri zadane točke? Problem se zatim poopćava na \(n\) točaka i konstruira se upisani \(n\)-terokut. Ovom problemu moguće je pristupiti analitički i geometrijski. Analitički pristup koristi takozvane Pitagorejske koordinate i Moebiusove transformacije. Do rješenja se dolazi pronalaskom prve točke na kružnici, a potom se preslikavanjima dolazi do ostalih točaka. U geometrijskom pristupu promatramo slučaj \(n=3\). Do rješenja se dolazi primjenom kompozicije triju inverzija. Koristimo Menelajev i Pascalov teorem te proširujemo Moebiusove transformacije na skup \(\mathbb{C}\) kako bismo mogli analizirati postojanje rješenja. U slučaju da je kompozicija triju promatranih inverzija identiteta, CCP ima beskonačno mnogo rješenja. Za kraj, uz prikaz efektivne konstrukcije u tipičnom slučaju s dva rješenja, komentiramo relevantnost problema koji i danas potiče zanimanje. Cramer-Castillon’s problem is a geometric task dating from ancient Greece. A circle and three points are given. The problem states: how to inscribe a triangle in the given circle so that the sides of the triangle lie on the lines passing though the given points? The problem is then generalized to an \(n\)-polygon. We can approach this problem analytically and geometrically. The analytic approach uses the so-called Pythagorean coordinates and Moebius transformations. We arrive at the solution by finding the first point on the circle, and then use a chain of mappings to arrive at other points. In the geometric approach, we focus on case \(n=3\). We find the solution by using a composition of three inversions. We use Menelaus and Pascal’s theorem, and expand the definition of Moebius transformation to the set \(\mathbb{C}\), so that we can analyse the existance of the solution. In the case that the composition is an identity, the problem has infinitely many solutions. In the end, along with an effective construction in a typical case with two solutions, we are commenting the relevancy of the problem which is still interesting today.