What condition must fulfill the inradius, the circumradius, the incenter and the circumcenter so that two circles allow a bicentric polygon between them? Nina Guindilla answered:
That condition depends on the number of sides of the polygon. I've searched and I've found two theorems. Euler's theorem in Geometry provides the condition for triangles and Fuss's theorem provides the condition for bicentric quadrilaterals:
If r is the inradius, R is the circumradius and d is the distance between the incenter and the circumcenter then...
Euler's theorem: 1/r = 1/(R+d) + 1/(R–d)
Fuss's theorem: 1/r2 = 1/(R+d)2 + 1/(R–d)2