$\underline{115625}$ topics: [  Tangent circles [Mean proportionals in a right triangle]

Two circles touch each other externally at point $C$. A line is tangent to the first circle at point $A$ and to the second circle at point $B$. The line $A C$ intersects the second circle at point $D$, different from $C$. Find $B C$, if $A C=9$, $C D=4$.

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Let the common tangent to the circles passing through point $C$ intersect segment $A B$ at point $M$. Then $M A = M C = M B$, i.e., the median $C M$ of triangle $A B C$ is equal to half of side $A B$, which means $\angle A C B = 90^{\circ}$.

The adjacent angle to $\angle A C B$, $\angle B C D$, is also $90^{\circ}$, so $B D$ is the diameter of the second circle. Since $A B$ is tangent to this circle, $B D \perp A B$.

In the right triangle $A B D$, segment $B C$ is the altitude dropped to the hypotenuse, therefore,

$$
B C = \sqrt{A C \cdot C D} = \sqrt{3 \cdot 4} = 6
$$

## Answer

6.