If a cyclic polygon with at least four sides has an incircle, must it be regular?

Question

Suppose that the vertices of a polygon with four or more sides lie on a circle, and that another (possibly non-concentric) circle touches each of its sides. Intuitively, it seems to me that the polygon has to be regular, and the circles concentric, but I don't see a way to go about proving it. Does anyone have an idea how to do this?

Answer 1

No. The four-sided case is given below, followed by a mathematical-induction approach that can be used to generate bicentric polygons with more sides.

To see what happens with four sides, you first observe that a convex quadrilateral with sides $a,b,c,d$ in rotational order can always be made cyclic by adjusting either diagonal until one vertex hits the same circle as the other three.

Now suppose that $a+c=b+d$, not necessarily equal. For instance, you might select $a=b\not=c=d$, a kite. Then the quadrilateral always has an incircle, but we have just seen it can be made cyclic too by adjusting the diagonals (in the case of a kite, you end up with one pair of opposing angles being right angles, but the others need not be so). So you have a quadrilateral with both an incircle and a circumcircle. Such quadrilateral is called bicentric.

Now suppose that, having constructed a bicentric quadrilateral $ABCD$, you hold the circumcircle fixed and move the incircle, without changing its radius, towards vertex $A$. Vertices $B$ and $D$ are adjusted along the circumcircle so that $\overline{AB}$ and $\overline{AD}$ remain tangent to the incircle. Similarly $C$ is adjusted so that $\overline{BC}$ is tangent to the incircle, but the point on the circumcircle that makes "$\overline{CD}$" tangent to the incircle is no longer $C$ but a fifth point $C'$ such that a pentagon $ABCC'D$ is formed. Initially $\overline{CC'}$ will lie outside the incircle, but ultimately it must pass through as the incircle approaches tangency at $A$ and the pentagon collapses to that point. So there must be an intermediate position where $\overline{CC'}$ just touches the incircle and you now have a bicentric pentagon with the same incircle/circumcircle radius ratio as the original quadrilateral.

A similar procedure can be used to add another side to the pentagon, then the hexagon, and so on.