In this thesis, our main theme was bicentric polygons. It is well known that every triangle possesses both an incircle and a circumcircle. Euler’s triangle formula, which we have proven, gives us a relation between the inradius and the circumradius. As an introduction to the main focus of this paper, bicentric quadrilaterals, it was important to, at least briefly, describe and give some basic characterizations of two well-known classes of quadrilaterals: tangential and cyclic quadrilaterals. Bicentric quadrilaterals are quadrilaterals that are cyclic and tangential at the same time. As a result, they have some interesting characterizations and results that, among others, relate to the connection between both the radii of the incircle and circumcircle of bicentric quadrilaterals. Naturally, it was necessary to prove the analogue of Euler’s triangle formula known as Fuss’s theorem, as well as Yun’s inequality, which connects the interior angles of the quadrilateral and both radii. A remarkable theorem that tells us about the existence of a quadrilateral that is simultaneously inscribed in one circle and circumscribed around another one was discovered by the French mathematician Jean-Victor Poncelet, and the theorem is known as Poncelet’s porism. It can be applicable to all polygons and that theorem gives us a statement that if there exists a polygon of n-sides which is inscribed in a given circle and circumscribed about another circle, then infinitely many such polygons exist.