I'd like to add that the original premise about the solution-by-intersection didn't sit right with me. I took out the old pencil and paper and drew out a lead-to-lag and lag-to-lead just to make sure.
When you're on a lead LOS, there is no intersection; the lines are close to parallel. On an OverLead LOS (lead LOS but you have significantly more speed than the target), the lines will intersect beyond the target. On a Lag LOS, the lines intersect somewhere in front of the target. Changing from one to another doesn't necessarily generate an intersection, although if you go from lead to lag (or from lag to overlead), you will begin to get an intersection from the lag/overlead that you weren't getting before.
Is the target "near" those intersections? It depends entirely on the speed differential. The faster you are going relative to the target, the closer together the minimum possible range (intersection in lag) and maximum possible range (intersection in overlead) become.
So, I think it's better to think of the intersections as maximum or minimum possible ranges, and to guestimate how much closer/farther the target is based on whether you think the target is at patrol, tactical, or transit speeds, as well as your own speed.
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