In the following figure, the circle with center $B$ is tangent to the circle with center $A$ at $X$. The circle with center $C$ is tangent to the circle with center $A$ at $Y$. Additionally, the circles with centers $B$ and $C$ are also tangent. If $A B=6, A C=5$ and $B C=9$, what is the measure of $A X$?

![](https://cdn.mathpix.com/cropped/2024_05_01_10107d2fba0ec26bc845g-05.jpg?height=711&width=740&top_left_y=593&top_left_x=752)

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Solution

Let $r_{a}, r_{b}$, and $r_{c}$ be the radii of the circles centered at $A, B$, and $C$, respectively. If $Z$ is the point of tangency of the circles centered at $B$ and $C$, the data from the problem allows us to set up the following system of equations:

$$
\begin{aligned}
A B & =A X-B X \\
6 & =r_{a}-r_{b} \\
A C & =A Y-C Y \\
5 & =r_{a}-r_{c} \\
B C & =B Z+Z C \\
9 & =r_{b}+r_{c}
\end{aligned}
$$

Then

$$
\begin{aligned}
6+5 & =\left(r_{a}-r_{b}\right)+\left(r_{a}-r_{c}\right) \\
& =2 r_{a}-\left(r_{b}+r_{c}\right) \\
& =2 r_{a}-9 \\
r_{a} & =10
\end{aligned}
$$

Therefore, $A X=10$.