Example (11)
If three circles with radius $r^{\prime}\left(r^{\prime}<r\right)$ can cover $\odot(O, r)$, how large does $r^{\prime}$ need to be at least?
We divide $\odot(O, r)$ into three equal sectors, each with a central angle of $120^{\circ}$ (Figure 1.17). Construct a circle with chord $A B$ as its diameter. Since $\angle A O B = 120^{\circ} > 90^{\circ}$, $\triangle O A B$ lies inside the constructed circle. Similarly, the segment $A m B$ also lies within the constructed circle, thus this circle covers the entire sector $O A m B$.
Construct another two circles with $B C$ and $C A$ as their diameters, respectively. These three circles cover $\odot(O, r)$, and their radii $r^{\prime} = \frac{\sqrt{3}}{2} r < r$.
It is not hard to see that $r^{\prime}$ cannot be any smaller, because at least one of the three smaller circles must cover one-third of the circumference of $\odot(O, r)$, thus covering a chord of length $\sqrt{3} r$, so the radius of the smaller circle must be $\geqslant \frac{\sqrt{3}}{2} r$.