Example 16. To completely cover a square with a side length of 1 using three circles of the same radius, what is the minimum radius of these equal circles?

To completely cover a square with a side length of 1 using three circles of the same radius, the minimum radius of these equal circles is to be determined.
Solve $A H=\sqrt{1+(2 x)^{2}}$, 
$M G=\sqrt{\left(\frac{1}{2}\right)^{2}+(1-2 x)^{2}}$.
Also, $A H=M G$,
Solving for $x$ gives $x=\frac{1}{16}$.
$$
\therefore A E=\sqrt{\left(\frac{1}{2}\right)^{2}+x^{2}}=\frac{1}{16} \sqrt{65} \text {. }
$$

Thus, these three semicircles have a radius of $\frac{\sqrt{65}}{16}$, and they cover the rectangles $A B H G$, $G N M D$, and $N H C M$, thereby covering the entire square.