7. [6] Let $C_{1}$ and $C_{2}$ be externally tangent circles with radius 2 and 3 , respectively. Let $C_{3}$ be a circle internally tangent to both $C_{1}$ and $C_{2}$ at points $A$ and $B$, respectively. The tangents to $C_{3}$ at $A$ and $B$ meet at $T$, and $T A=4$. Determine the radius of $C_{3}$.
Answer: 88 Let $D$ be the point of tangency between $C_{1}$ and $C_{2}$. We see that $T$ is the radical center of the three circles, and so it must lie on the radical axis of $C_{1}$ and $C_{2}$, which happens to be their common tangent $T D$. So $T D=4$.
We have
$$
\tan \frac{\angle A T D}{2}=\frac{2}{T D}=\frac{1}{2}, \quad \text { and } \quad \tan \frac{\angle B T D}{2}=\frac{3}{T D}=\frac{3}{4} .
$$

Thus, the radius of $C_{3}$ equals to
$$
\begin{aligned}
T A \tan \frac{\angle A T B}{2} & =4 \tan \left(\frac{\angle A T D+\angle B T D}{2}\right) \\
& =4 \cdot \frac{\tan \frac{\angle A T D}{2}+\tan \frac{\angle B T D}{2}}{1-\tan \frac{\angle A T D}{2} \tan \frac{\angle B T D}{2}} \\
& =4 \cdot \frac{\frac{1}{2}+\frac{3}{4}}{1-\frac{1}{2} \cdot \frac{3}{4}} \\
& =8 .
\end{aligned}
$$