B3. A figure consists of a square $A B C D$ and a semicircle with diameter $A D$ outside the square. The side of the square has length 1. What is the radius of the circumscribed circle of the figure?

![](https://cdn.mathpix.com/cropped/2024_04_17_fae9e9df1e99e6590d6bg-2.jpg?height=343&width=346&top_left_y=1753&top_left_x=1392)
B3.

Let the center of the circumscribed circle be $O$, the point of tangency with the smaller circle be $E$, and the midpoint of $B C$ be $M$. By the Pythagorean theorem, we have $|O C|^{2} = |M C|^{2} + |O M|^{2}$. Since $|O M| = |E M| - |O E| = \frac{3}{2} - |O C|$, it follows that $|O C|^{2} = \left(\frac{1}{2}\right)^{2} + \left(\frac{3}{2} - |O C|\right)^{2}$. Solving this equation gives $|O C| = \frac{5}{6}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_eb67361d5b5f4c2be0e0g-2.jpg?height=337&width=326&top_left_y=1596&top_left_x=1499)