Barycentric Coordinates Of The Feuerbach Points

The famous Feuerbach theorem asserts that the nine point circle of a triangle istangent to both the incircle and the excircles. The four tangency points are called theFeuerbach points of the triangle.In this thesis we find the barycentric coordinates of the Feuerbach points usingtechniques of projective geometry. Precisely, according to a proof of Feuerbach Theoremusing the inversions, each Feuerbach point is the intersection of certain two straight lines.By extending the usual Euclidean plane to a projective plane and taking the barycentriccoordinates as the projective coordinates, we can determine the lines involved and hencefind the barycentric coordinates of the Feuerbach point.This thesis consists of three chapters. Chapter1gives the introduction. In Chapter2we briefly introduce the basic facts about the projective plane, the barycentric coordinates,as well as the planar inversions. Finally, in Chapter3we present a proof of the FeuerbachTheorem using the inversions, and furthermore, find the barycentric coordinates of theFeuerbach points using techniques of projective geometry.