AUSTRIAN-POLISH 1987 Problem 6

Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max} \underset{i}{min}\delta (P_i, d)$ , where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$ . Let $F_n$ be the family of all such sets $A$ . Determine $D_n = \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$ .