6.64*. In a regular $n$-gon ( $n \geqslant 3$ ), the midpoints of all sides and diagonals are marked. What is the maximum number of marked points that can lie on one circle?
6.64. Let first $n=2 m$. The diagonals and sides of a regular $2 m$-gon have $m$ different lengths. Therefore, the marked points lie on $m-1$ concentric circles (with $n$ points on each) or at the common center of these circles. Since different circles have no more than two common points, a circle not belonging to this family of concentric circles contains no more than $1+2(m-1)=2 m-1=n-1$ marked points.

Now let $n=2 m+1$. The diagonals and sides of a regular $(2 m+1)$-gon have $m$ different lengths. Therefore, the marked points lie on $m$ concentric circles (with $n$ points on each). A circle not belonging to this family of concentric circles contains no more than $2 m=n-1$ marked points.

In both cases, the maximum number of marked points lying on one circle is $n$.