## Task 20/73

Given a circle with radius $r$ and a regular $n$-sided polygon inscribed in the circle. The task is to find the sum of the squares of the distances from any point $A$ on the circumference of the circle to the vertices of the $n$-sided polygon.
We first consider the case where $n=2 k$ is even. Then for each vertex $P_{m}$ of the n-gon, there is a diametrically opposite point $P_{m+k}(m \leq k)$. The points $P_{m}, P_{m+k}, A$ thus form a triangle over the diameter, and by the theorem of Thales, this is a right-angled triangle at $A$. Therefore, we have:

$$
A P_{m}^{2}+A P_{m+k}^{2}=4 r^{2}
$$

Thus, for the sum of all squared distances (since $k=\frac{n}{2}$ such pairs of points exist), $S=$ $4 k r^{2}=2 n r^{2}$.

If now $n=2 k+1$ is odd, we first form a regular $2 n$-gon by inserting an additional point between each pair of vertices. For this 2n-gon, the same result yields $S=4 n r^{2}$. Returning to the $n$-gon, this sum must be halved. Therefore, it holds generally that $S=2 n r^{2}$.