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.