Q:

As shown in the diagram, each circle's radius is 1, AG is tangent to O3, what's the length of EF?

I've been stuck on this problem for awhile now. I can't figure out how to use Power of a Point to get EF, and the only lengths I have are AG, GO3, AO3, and that $AE\cdot AF=8$. Please help me out, and thank you!

A:

Let $M$ be the midpoint of chord $EF$. $O_2M \perp AG$ and $O_3G\perp AG$.
By similar triangles $\triangle O_2AM \sim \triangle O_3AG$,
$$\begin{align*}
\frac{O_2 M}{O_3G} &= \frac{AO_2}{AO_3}\\
O_2M &= \frac 35
\end{align*}$$
Then $EF$ is
$$\begin{align*}EF &= 2 EM\\
&= 2\sqrt{O_2E^2 - O_2M^2}\\
&= 2\sqrt{1^2 - \left(\frac 35\right)^2}\\
&= \frac 85
\end{align*}$$

