[ Angle between a tangent and a chord $\quad]$ [ Auxiliary similar triangles ]

Points $A, B$ and $C$ lie on the same line. Segment $A B$ is the diameter of the first circle, and segment $B C$ is the diameter of the second circle. A line passing through point $A$ intersects the first circle at point $D$ and is tangent to the second circle at point $E, B D=9, B E=12$. Find the radii of the circles.
Let $O_{1}$ and $O_{2}$ be the centers of the first and second circles, respectively, and $R_{1}$ and $R_{2}$ be their radii. Up to symmetry, there are three possible cases for the positions of points $A, B$, and $C$ on the line.

1) Point $A$ lies between points $B$ and $C$. Then $A$ is inside the second circle, and there is no line passing through $A$ that is tangent to the second circle.
2) Point $B$ lies between points $A$ and $C$. Then $\angle B E C=\angle A E O_{2}=90^{\circ}, \angle D E B=\angle E C B$. Triangles $B E C$ and $BDE$ are similar by two angles, so

$B C: B E=B E: B D$, from which $2 R_{2}=B C=B E^{2} /{ }_{B D}=16$.

Segments $B D$ and $E O_{2}$ are parallel, so $B D<E O_{2}$, and since $B D=9$ and $E O_{2}=R_{2}=8$, this case is impossible.

3) Point $C$ lies between points $A$ and $B$. Similarly to the previous case, we get $R_{2}=8$. Triangles $A D B$ and

$A O_{2}: A B=E O_{2}: D B$, or $\left(2 R_{1}-8\right): 2 R_{1}=8: 9$, from which $R_{1}=36$.

## Answer

36 and 8.