42. On the lateral sides $K L$ and $M N$ of an isosceles trapezoid $K L M N$, points $P$ and $Q$ are chosen respectively such that the segment $P Q$ is parallel to the bases of the trapezoid. It is known that a circle can be inscribed in each of the trapezoids $K P Q N$ and $P L M Q$, and the radii of these circles are $R$ and $r$ respectively. Determine the bases $|L M|$ and $|K N|$.
42. Drop perpendiculars from the centers of the circles to one of the lateral sides and draw a line through the center of the smaller circle parallel to this side. This will form a right triangle with the hypotenuse \( R + r \), one leg \( R - r \), and an acute angle at this leg \(\alpha\), which is equal to the acute angle at the base of the trapezoid. Thus,

\[
\cos \alpha = \frac{R - r}{R + r}
\]

The larger base is

\[
2 R \operatorname{ctg} \frac{\alpha}{2} = 2 R \sqrt{\frac{1 + \cos \alpha}{1 - \cos \alpha}} = 2 R \sqrt{\frac{R}{r}}
\]

The smaller base is

\[
2 r \operatorname{tg} \frac{\alpha}{2} = 2 r \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} = 2 r \sqrt{\frac{r}{R}}
\]