Fuss's problem

A polygon which has both a circumcircle (which touches each vertex) and an incircle (which is tangent to each side). All triangles are bicentric with R^2 - x^2 = 2R r, where R is the circumradius, r is the inradius, and x is the separation of centers. For bicentric quadrilaterals, a result sometimes known as Fuss's problem, the circles satisfy 2r^2(R^2 + x^2) = (R^2 - x^2)^2 (Dörrie 1965, Salazar 2006) or, in another form, 1/(R - x)^2 + 1/(R + x)^2 = 1/r^2 (Davis; Durége 1861; Casey 1888, pp. 109-110; Johnson 1929; Dörrie 1965).

bicentric quadrilateral | bicentric triangle | circumcircle | incircle | polygon | Poncelet's porism | Poncelet transverse | tangential quadrilateral | triangle | Weill's theorem