On the plane, there are two circles with radii 4 and 3, with centers at points $O_1$ and $O_2$, touching a certain line at points $M_1$ and $M_2$ and lying on opposite sides of this line. The ratio of the segment $O_1 O_2$ to the segment $M_1 M_2$ is $\frac{2}{\sqrt{3}} \cdot$ Find $O_1 O_2$.
Drop a perpendicular $O 1 F$ from the center of the first circle to the extension of the radius $O 2 M 2$ of the second circle. Then

$$
O 1 F=M 1 M 2, M 2 F=O 1 M 1=4, O 2 F=O 2 M 2+M 2 F=3+4=7
$$

From the right triangle $O 1 F O 2$, we find that

$$
\cos \angle F O 1 O 2=\frac{O_{1} F}{O_{1} O_{2}}=\frac{M I_{1} M I_{2}}{O_{1} O_{2}}=\frac{\sqrt{3}}{2}
$$

which means, $\angle F O 1 O 2=30^{\circ}$. Therefore, $O 1 O 2=2 O 2 F=2 \cdot 7=14$.

## Answer

14.