Sounds like a calculus problem to me!
Anyways, I think you can break it down like this:
For one segment to be twice as long as the other, you need to break the line at the 1/3 or 2/3 marks. Thus, I believe you can 'simplify' the problem by breaking the line as such:
1. At the 1/3 and less -- line segment one will be at least twice the second.
2. Between 1/3 and 2/3 -- line segment one will be less than twice the second, or vice versa.
3. At the 2/3 and more -- line segment two will be at least twice the first.
So, three cases, two of which satisfy your criteria.
Does 2 / 3 sound right?