118. A circle is inscribed in a square, and then a new square is inscribed in the circle. Find the ratio of the areas of these squares.
118. It is clear that the second square can be placed so that its vertices fall on the points of tangency of the circle with the sides of the first square (see Fig. 18). After this, it is not difficult to verify that the desired ratio of the areas of the squares is 2.

![](https://cdn.mathpix.com/cropped/2024_05_21_a21e90374409a05a4ea7g-25.jpg?height=396&width=609&top_left_y=1775&top_left_x=338)

Fig. $\quad 18$

![](https://cdn.mathpix.com/cropped/2024_05_21_a21e90374409a05a4ea7g-25.jpg?height=460&width=474&top_left_y=1769&top_left_x=1245)

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