Two circles with radii 5 and 4 touch each other externally. A line tangent to the smaller circle at point $A$ intersects the larger circle at points $B$ and $C$, and

$A B = B C$. Find $A C$.
Let a circle of radius 4 with center $O_{1}$ and a circle of radius 5 with center $O_{2}$ touch each other externally at point $D$ (see the left figure). Then

$O_{1} O_{2}=O_{1} D+O_{2} D=9$

Drop perpendiculars $O_{2} M$ to the chord $B C$ and $O_{1} F$ to the line $O_{2} M$. Then $A O_{1} F M$ is a rectangle, so $M F=4$ and $O_{1} F=A M$.

Let $B M=M C=x$. Then $A B=B C=2 x, A M=3 x$. By the Pythagorean theorem, $O_{2} M^{2}=25-x^{2}, O_{1} F^{2}+O_{2} F^{2}=81$, or $9 x^{2}+\left(\sqrt{25-4 x^{2}}-4\right)^{2}=81$. From this, $x^{2}=9$. Therefore, $A C=4 x=12$.
![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-33.jpg?height=386&width=1544&top_left_y=-2&top_left_x=260)

## Answer

12. 

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