9 A square is inscribed in a circle, and another square is inscribed in the segment cut off from the circle by the side of this square. Find the ratio of the lengths of the sides of these squares.

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Let $2 a$ and $2 b$ be the side lengths of the first and second squares, respectively. Then the distance from the center of the circle to the vertices of the second square, lying on the circle, is $\sqrt{(a+2 b)^{2}+b^{2}}$. On the other hand, this distance is $\sqrt{2} a$. Therefore, $(a+2 b)^{2}+b^{2}=2 a^{2}$, i.e., $a=2 b \pm \sqrt{4 b^{2}+5 b^{2}}=(2 \pm 3) b$. The only solution that fits is $a=$ $5 b$.