12.63*. A chord of a circle is at a distance $h$ from the center. In each of the segments cut off by the chord, a square is inscribed such that two adjacent vertices of the square lie on the arc, and the other two - on the chord or its extension (Fig. 12.3). What is the difference in the lengths of the sides of these squares?

![](https://cdn.mathpix.com/cropped/2024_05_21_22b0ba5e8c5a1f49bd81g-277.jpg?height=388&width=376&top_left_y=698&top_left_x=1189)

Fig. 12.3
12.63. Let's denote the vertices of the squares as shown in Fig. 12.7. Let \( O \) be the center of the circle, \( H \) be the midpoint of the given chord, and \( K \) be the midpoint of the segment \( A A_{1} \). Since \(\operatorname{tg} A H B = 2 = \operatorname{tg} A_{1} H D_{1}\), the point \( H \) lies on the line \( A A_{1} \).

![](https://cdn.mathpix.com/cropped/2024_05_21_22b0ba5e8c5a1f49bd81g-286.jpg?height=407&width=393&top_left_y=257&top_left_x=541)

Fig. 12.7

Let \(\alpha = \angle A H B = \angle A_{1} H D_{1}\). Then \( A B - A_{1} D_{1} = (A H - A_{1} H) \sin \alpha = 2 K H \sin \alpha = 2 O H \sin^2 \alpha \). Since \(\operatorname{tg} \alpha = 2\) and \(1 + \operatorname{ctg}^2 \alpha = 1 / \sin^2 \alpha\), we have \(\sin^2 \alpha = 4 / 5\). Therefore, the difference in the lengths of the sides of the squares is \(8 h / 5\).