All triangles are bicentric, i.e., possess both an incircle and a circumcircle. This is not necessarily the case for polygons with four or more sides. The inradius r and circumradius R of a triangle are connected by 1/(R + d) + 1/(R - d) = 1/r, where d is the distance between the incenter and circumcenter.

bicentric polygon | bicentric quadrilateral | Euler triangle formula