I'm always a fan of Brute Force.
Put your circles on a set of coordinate axes, once circle (radius r and center (0,0)) at the origin and the other (radius R and center (a,0)) on the positive x-axis.
Points on the origin circle look like this: (b,c) where
Points on the remote circle look like this: (d,e) where
Just to emphasize that the circles do not intersect, let's also define f > 0 so that a = r + f + R
The line has equation
You can use calculus or or algebra to show that the line is perpendicular to a radius at the points of tangency.
That's enough information to find the points. The angles aren't too tricky after that. Let's see what you get. It will be fun!
Note: I have no doubt there is an elegant way to do this, like tocbol's geometry solution. I couldn't do that because I did not use the fact that the radii were the same. My solution is more general. If nothing else, this little demonstration should encourage you that you ALWAYS should be able to find SOME approach. It may be ugly, but it's better than the frustration inherent in just staring at it. Once you have the solution, you can search for elegance.