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A circle is circumscribed about the vertices of an arbitrary polygon with $n\geqslant2$ sides $ABCDEF...$ with perimeter $p$. What are the minimum and maximum radii of such a circle, given these parameters?

I was thinking that a minimum radius would be treating the polygon as a near circle, so $r_{min}=\frac{p}{2\pi}$.

For the maximum radius, the case would be two overlapping diameters, $A$ and $B$ so $r_{max}=\frac{p}4$.

If this is right, how can I prove this? If not, where am I going wrong?

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  • $\begingroup$ For $r_{max},$ think about one side with length almost $p/2,$ and the others very close to the line segment. This seems to indicate that $r_{max}$ is undefined. $\endgroup$ – cats Dec 9 '16 at 21:11
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For the problem of the minimum radius, this is equivalent to asking what the maximum perimeter is for a cyclic $n$-gon inscribed in a circle of radius $r.$ This is well-known to be the regular $n$-gon. A proof follows below. If $n$ is allowed to vary, then evidently the perimeter becomes arbitrarily close to $2\pi r.$ Otherwise, it is $2nr \sin \frac{\pi}{n}.$

Proof that this is optimal: this becomes equivalent to showing that if $\theta_1 + \ldots + \theta_n = \pi,$ then $\sum \sin \theta_i$ is maximized when they're all equal. This follows immediately from Jensen's.

Edit: for maximum radius, from my comment - take a circle of radius $R$ (large). Mark off an arc of length $p/2$ and make an $n$-gon with the endpoints of the arc as two consecutive vertices. This evidently has perimeter less than $p.$ Scaling up gives you an $n$-gon with perimeter $p.$ We can evidently choose $R$ to be arbitrarily large, so there is no maximum.

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