Imaginary areas are constucted around the intersecting lines and the ratio between areas (see diagram) gives the proportion of one of the lines length where the intersection occurs.
I thought of this method while experimenting trying to find quick ways of finding line intersections... but in the end it didn't look like it would be quick enough considering the data I had available at that point in my program so I used another method.
Presumably you could do a similar thing with volumes and 3D lines.