10.182. A circle is circumscribed around a square with side $a$. A square is inscribed in one of the resulting segments. Determine the area of this square.
## Solution.

The diagonal of the square is the diameter of the circle (Fig. 10.151). $(2 R)^{2}=2 a^{2}$, hence $R=\sqrt{2} a$. Let the side of the smaller square be $x$. Then $P M^{2}+P O^{2}=O M^{2}, P M=\frac{x}{2}, P O=\frac{a}{2}+x$. Therefore, $\frac{x^{2}}{4}+\left(\frac{a}{2}+x\right)^{2}=\frac{a^{2}}{2}$, from which $x=\frac{a}{5}$. The area of the smaller square $S=x^{2}=\frac{a^{2}}{25}$.

Answer: $\frac{a^{2}}{25}$.