62. (9th grade) Three circles with radii $a, b$, and $c$ are pairwise externally tangent. Find the length of the chord cut off by the third circle from the common internal tangent of the first two circles.
62. Construct $O_{3} M \| A C$, where $A C$ is the common tangent of circles $O_{1}$ and $O_{2}$. In triangles $O_{2} O_{3} M$ and $O_{1} O_{3} M$ (Fig. 33), the leg $O_{3} M$ is common. Assuming $C M=x$, we form the equation:

![](https://cdn.mathpix.com/cropped/2024_05_21_ec9ddf54a7c86775293eg-131.jpg?height=420&width=483&top_left_y=1383&top_left_x=358)

Fig. 33

$$
\begin{gathered}
(b+c)^{2}-(b-x)^{2}=(a+c)^{2}-(a+x)^{2} \\
x=\frac{c(a-b)}{a+b} ; K O_{3}=x=\frac{c(a-b)}{a+b} ; A B= \\
=\frac{4 c \sqrt{a b}}{a+b}
\end{gathered}
$$