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.
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.
Polygons with n > 4
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:
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).
Thus we have the following decimal approximations:
If two circles are the inscribed and circumscribed circles of a single bicentric n-gon, then the same two circles are the inscribed and circumscribed circles of infinitely many bicentric n-gons. More precisely, every tangent line to the inner of the two circles can be extended to a bicentric n-gon by placing vertices on the line at the points where it crosses the outer circle, continuing from each vertex along another tangent line, and continuing in the same way until the resulting polygonal chain closes up to an n-gon. The fact that it will always do so is implied by Poncelet's closure theorem, which more generally applies for inscribed and circumscribed conics.
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