Problem - 3147
Two circles, $O_1$ and $O_2$ are tangent. Let $AB$ be their common tangent line which touches $O_1$ at point $A$ and touches $O_2$ at point $B$. Extend $AO_1$ and intersects $O_1$ at another point $C$. Line $CD$ is tangent to circle $O_2$ at point $D$. Show that $AC=CD$.
Set a coordinate system such that $A$ is the origin and $AB$ is the $x$-axis. Let the radii of these two circles be $R$ and $r$, respectively. Then the coordinates of $C$ is $(0, 2R)$ and the coordinates of $O_2$ is $$(\sqrt{(R+r)^2 - (R-r)^2}, r) =(2\sqrt{R\cdot r}, r)$$
\begin{align*}
CD^2 &= CO_2^2 - r^2 \\
&= ((0-(2\sqrt{Rr}))^2 +(2R-r)^2) - r^2\\
&= 4Rr + 4R^2 +r^2 -4Rr - r^2\\
&= 4R^2\\
&= AC^2
\end{align*}