12.65*. A square is inscribed in a circle, and another square is inscribed in the segment cut off from the circle by one of the sides of the first square. Find the ratio of the lengths of the sides of these squares.
12.65. Let $2a$ and $2b$ 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, which lie on the circle, is $\sqrt{(a+2b)^{2}+b^{2}}$. On the other hand, this distance is $\sqrt{2}a$. Therefore, $(a+2b)^{2}+b^{2}=2a^{2}$, i.e., $a=2b \pm \sqrt{4b^{2}+5b^{2}}=(2 \pm 3)b$. Only the solution $a=5b$ is suitable for us.