Bicentric polygon

[1] 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.

, 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.

[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.

The radius of the inscribed circle is the apothem (the shortest distance from the center to the boundary of the regular polygon).

The fact that it will always do so is implied by Poncelet's closure theorem, which more generally applies for inscribed and circumscribed conics.

[6] Moreover, given a circumcircle and incircle, each diagonal of the variable polygon is tangent to a fixed circle.