In geometry, a bicentric polygon is a tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triangles and all regular polygons are bicentric. On the other hand, a rectangle with unequal sides is not bicentric, because no circle can be tangent to all four sides.

Every triangle is bicentric. In a triangle, the radii r and R of the incircle and circumcircle respectively are related by the equation

where x is the distance between the centers of the circles. This is one version of Euler's triangle formula.

Not all quadrilaterals are bicentric (having both an incircle and a circumcircle). Given two circles (one within the other) with radii R and r where ${\displaystyle R>r}$, there exists a convex quadrilateral inscribed in one of them and tangent to the other if and only if their radii satisfy

where x is the distance between their centers. This condition (and analogous conditions for higher order polygons) is known as Fuss' theorem.

A complicated general formula is known for any number n of sides for the relation among the circumradius R, the inradius r, and the distance x between the circumcenter and the incenter. Some of these for specific n are:

where ${\displaystyle p={\tfrac {R+x}{r}}}$ and ${\displaystyle q={\tfrac {R-x}{r}}.}$

Every regular polygon is bicentric. In a regular polygon, the incircle and the circumcircle are concentric—that is, they share a common center, which is also the center of the regular polygon, so the distance between the incenter and circumcenter is always zero. The radius of the inscribed circle is the apothem (the shortest distance from the center to the boundary of the regular polygon).