Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$ with a side length of 4 cm. Two tangents are drawn from point $C$ to this circle. Find the radius of the circle if the angle between the tangents is $60^{\circ}$.
# Solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_b2236406d6db6e8875c1g-07.jpg?height=445&width=576&top_left_y=1431&top_left_x=817)

Fig. 1

The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. The diagonal of the square $A B C D A C=4 \sqrt{2}$. If radii of the circle are drawn to the points of tangency, a square with side $R$ is formed. Then $O A=R \sqrt{2}$, $OC = \frac{O K}{\sin 30^{\circ}}=2 R$. As a result, we get the equation $2 R=R \sqrt{2}+4 \sqrt{2}$, from which $R=4(\sqrt{2}+1)$.

Answer. $4(\sqrt{2}+1)$.