39. From point $K$, located outside a circle with center $O$, two tangents $K M$ and $K N$ are drawn to this circle ($M$ and $N$ are the points of tangency). A point $C(|M C|<|C N|)$ is taken on the chord $M N$. A line perpendicular to segment $O C$ is drawn through point $C$, intersecting segment $N K$ at point $B$. It is known that the radius of the circle is $R, \widehat{M K N}=\alpha,|M C|=b$. Find $|C B|$.
39. Let $A$ be the point of intersection of the lines $B C$ and $K M$. The quadrilateral $O N B C$ is inscribed $\left(\widehat{(O C B}=\widehat{O N B}=90^{\circ}\right)$, therefore, $\widehat{O B C}=\widehat{O N C}=\frac{\alpha}{2}$. Similarly, the quadrilateral $C M A O$ is inscribed and $\widehat{C A O}=\widehat{C M O}=\frac{\alpha}{2}$, i.e.
$\triangle O A B-$ is isosceles,

$$
|C B|=|A C|=\frac{|C O|}{\sin \frac{\alpha}{2}}=\frac{\sqrt{R^{2}+b^{2}-2 R b \cos \frac{\alpha}{2}}}{\sin \frac{\alpha}{2}}
$$