19. (GBR 6) The $n$ points $P_{1}, P_{2}, \ldots, P_{n}$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $d_{n}$ between any two of these points has its largest possible value $D_{n}$. Calculate $D_{n}$ for $n=2$ to 7 and justify your answer.
19. Suppose $n \leq 6$. Let us decompose the disk by its radii into $n$ congruent regions, so that one of the points $P$ lies on the boundaries of two of these regions. Then one of these regions contains two of the $n$ given points. Since the diameter of each of these regions is $2 \sin \frac{\pi}{n}$, we have $d_{n} \leq 2 \sin \frac{\pi}{n}$. This
value is attained if $P_{f}$ are the vertices of a regular $n$-gon inscribed in the boundary circle. Hence $D_{n}=2 \sin \frac{\pi}{n}$.
For $n=7$ we have $D_{7} \leq D_{6}=1$. This value is attained if six of the seven points form a regular hexagon inscribed in the boundary circle and the seventh is at the center. Hence $D_{7}=1$.