87. Four dogs. Four dogs $A, B, C$, and $D$ stand at the corners of a square meadow and suddenly start chasing each other as indicated by the arrows in Fig. 2.

Each dog runs straight towards the next: $A$ towards $B$, $B$ towards $C$, $C$ towards $D$, and $D$ towards $A$. The side of the meadow is 100 m, and the speed of the dogs is $10 \mathrm{m} / \mathrm{s}$.

After what time interval will the dogs meet? Will their paths intersect and where? What is the length of each path?

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-042.jpg?height=340&width=342&top_left_y=1298&top_left_x=1371)

Fig. 2.
87. Since each dog runs at a right angle to the direction of the dog chasing it, and the one chasing runs straight toward the fleeing one, the chasing dog approaches the next dog at a speed of $10 \mathrm{m} /$ s and will catch it after 10 seconds. As a result, the path of each dog is $100 \mathrm{~m}$. At every moment, the four dogs form a square. This square rotates and shrinks: its sides decrease uniformly at a speed of $10 \mathrm{m} /$. . The paths will intersect at the center $S$ of the initial square. These will be curved lines (logarithmic spirals). They will not intersect earlier, because if any dog had crossed another's path, it would mean that it had been in that place earlier, which is impossible due to the equal distance of all dogs from $S$ at every moment.