15.9. The length of one side of the square $A B C D$ is 4 units. A circle is drawn tangent to $\overline{B C}$ and passing through the vertices $A$ and $D$. Find the area of the circle.
15.9. $\frac{25 \pi}{4}$ square units
See Figure 4. Let $R$ be the radius of the circle. Using the Extended Law of Sines, we have
$$
2 R=\frac{4}{\sin 2 A}=\frac{4}{2 \sin A \cos A}=\frac{2}{\frac{2}{2 \sqrt{5}} \cdot \frac{4}{2 \sqrt{5}}}=5,
$$
giving $R=\frac{5}{2}$. Thus, we get
$$
\text { area of the circle }=\pi\left(\frac{5}{2}\right)^{2}=\frac{25}{4} \pi
$$