Given two circles that don't intersect, $k_i$ with centers in $O_i$ and radii $r_i$, $i=1,2$, prove that the centers of all circles that touch them lie on one line. Note: this is the case where one of the circles is not inside the other.
I have no clue where to start.
What I tried: centers of any two circles that touch the given circles define a line, call it $p$. Then, given any other circle that touches $k_1$ and $k_2$, I tried to prove that its center lies on $p$, but to no avail.
Any help is appreciated.