Moments in a Peculiar Cyclic Polygon

Source Problem

Let $A_1 A_2\ldots A_n\,$ be a cyclic polygon inscribed in the circle $w=w(O,R),\,$ with center $O\,$ and radius $R.\,$ If its centroid coincides with $O,\,$ then for any point $P\in w,\,$ the following identity holds

$\displaystyle \sum_{k=1}^nPA_k^2=2nR^2.$

Solution 1

Assume $R=1.\,$ In complex numbers, we choose $A_k=a_k,\,$ $k=1,2,\ldots,n,\,$ and $P=a,\,$ so that $|a|=1,\,$ as well as $|a_k|=1,\,$ $k=1,2,\ldots,n.\,$ Moreover, $\displaystyle \sum_{k=1}^na_k=0.\,$ We have

\displaystyle \begin{align} \sum_{k=1}^nPA^2 &= \sum_{k=1}^n(a-a_k)(\overline{a-a_k})= \sum_{k=1}^n(a-a_k)(\overline{a}-\overline{a_k})\\ &=\sum_{k=1}^n(1-a\overline{a_k}-\overline{a}a_k+1)\\ &=2n-a\sum_{k=1}^n\overline{a_k}-\overline{a}\sum_{k=1}^na_k\\ &=2n. \end{align}

Remark 1 (Marian Dinca): For $n=3,\,$ a triangle with the aforementioned property is necessarily equilateral.

Remark 2 (Leo Giugiuc): For $n\ge 4,\,$ the polygon need not be regular. For example, any rectangle has that property.

Remark 3: It is easy to obtain $2n$-gons with that property. The vertices of such a polygon can be obtained by randomly selecting $n\,$ points on a circle and combining them with their antipodes, their reflections in the center of the circle.

Solution 2

The problem is a particular case of a well-known theorem of Lagrange:

The moment of inertia of a system of material points relative to an arbitrary point $P\,$ is the sum of two quantities: the moment of inertia of the system relative to its center of gravity and the moment of inertia of the total mass placed at the center of gravity relative to $P\,$.

Indeed, in the case at hand, both components on the "right-hand side" equal $nR^2.$

Acknowledgment

A problem posted at the Peru Geometrico facebook group, concerning a regular pentagon, elicited a comment from Leo Giugiuc to the effect that the announced property is shared by all cyclic polygons whose centroid coincides with their circumcenter. This problem has been reposted at the group and is the subject of the present page. The solution is by Leo Giugiuc. An identical solution has been submitted by Marian Dinca.

Earlier the problem has been posed for regular $n$-gons. 