Two straight pipes (circular cylinders), with radii $1$ and $\frac19$, lie parallel and in contact on a flat floor. Looking at them head-on, what is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?

1. Two circles of radii $r_1,r_2$ tangent to each other and to a line touch the line at points $2\sqrt{r_1r_2}$ apart (the horizontal leg of a right triangle with hypotenuse $r_1+r_2$ and other leg $|r_1-r_2|$).
2. The given pipes touch the floor at points $2\sqrt{1\cdot\frac19}=\frac23$ apart. A third pipe of radius $r$ touches the floor at distance $2\sqrt r$ from the big pipe and $2\sqrt{r/9}=\frac23\sqrt r$ from the small pipe.
3. \emph{Between the pipes:} $2\sqrt r+\frac23\sqrt r=\frac23$, so $\sqrt r=\frac14$ and $r=\frac1{16}$.
4. \emph{On the other side of the small pipe:} $2\sqrt r=\frac23+\frac23\sqrt r$, so $\sqrt r=\frac12$ and $r=\frac14$. On the other side of the big pipe the equation $\frac23\sqrt r=\frac23+2\sqrt r$ has no solution.
5. The sum is $\frac1{16}+\frac14=\boxed{\frac5{16}}$.