[ Homothetic Circles]

Inside an angle, there are three circles $\mathrm{S}_{1}, \mathrm{~S}_{2}, \mathrm{~S}_{3}$, each of which touches the two sides of the angle, and circle $S_{2}$ touches circles $S_{1}$ and $S_{3}$ externally. It is known that the radius of circle $S_{1}$ is 1, and the radius of circle $S_{3}$ is 9. What is the radius of circle $S_{2}$?
Homothety with the center at the vertex of the angle, translating circle $S_{1}$ to circle $S_{2}$, also translates circle $S_{2}$ to circle $S_{3}$.

## Solution

Consider the homothety $\mathrm{H}$ with the center at the vertex of the angle, translating circle $\mathrm{S}_{1}$ to circle $\mathrm{S}_{2}$. Circle $\mathrm{S}_{2}$, when this homothety is applied, transitions to a circle that is tangent to the sides of the angle and externally tangent to the circle $\mathrm{H}\left(\mathrm{S}_{1}\right)=\mathrm{S}_{2}$; thus, circle $\mathrm{S}_{2}$, when the homothety is applied, transitions to circle $S_{3}$. Denote the radius of circle $S_{i}$ by $R_{i}, i=1,2,3$, and the homothety coefficient of $H$ by $k$. Then $\mathrm{R}_{2}=\mathrm{kR}_{1}$ and $\mathrm{R}_{3}=\mathrm{kR} \mathrm{R}_{2}$, from which $\mathrm{R}_{2}{ }^{2}=\mathrm{R}_{1} \mathrm{R}_{3}$. Substituting the values $\mathrm{R}_{1}=1$ and $\mathrm{R}_{3}=9$, we finally get $\mathrm{R}_{2}{ }^{2}=9$ and $\mathrm{R}_{2}=3$.

## Answer

3.00