1. Two circles touch each other and the sides of two adjacent angles, one of which is $60^{\circ}$. Find the ratio of the radii of the circles.
1. Answer: $r: R=1: 3$ Solution. The centers of the circles lie on the bisectors $A K$ and $A N$ of the adjacent angles,

$\alpha=\square B A C=60^{\circ}$

$\square K A N=90^{\circ}, \square C A K=\square A Q O=30^{\circ}, A M=h$. $r=h \tan \frac{\alpha}{2}, R=h \cot \frac{\alpha}{2} \rightarrow r: R=\tan^2 \frac{\alpha}{2}=\frac{1}{3}$