In this master’s thesis bicentric pairs of triangle points have been analysed. A table which currently contains 119 bicentric pairs is guided by Clark Kimberling. In the first part of the thesis, trilinear and barycentric coordinates, center of triangle, bicentric pairs of points have been defined. Connections between trilinear and barycentric as well as trilinear and Cartesian coordinates have been given. Best known centers of triangle and their trilinear coordinates and some bicentric pairs of triangle points are given. Central part of the thesis contains definitions, theorems, and proofs linked to the best known bicentric pair of triangle points as well as the first and the second Brocard points. Trilinear coordinates of the Brocard points are derived and some characteristics are given to the Brocard angle. Definitions and theorems connected to the Brocard circle as well as the first and the second Brocard triangle are given. In the closing part of this thesis we describe another bicentric pair of points, the socalled Yff’s points and prove some of its properties.