Bicentric polygon
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 , 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.
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