In the case of two dimensions, the intuition is as follows: For any line segment "xy", consider the possible range of lengths of "xv", where "v" is any point on the perpendicular bisector of "xy". It is apparent that while there is no upper bound to the length of "xv", there is a lower bound, which occurs when "v" is the midpoint of "xy". So if "xy" is shorter than or equal to "zu", then the range of possible lengths of "xv" will be a superset of the range of possible lengths of "zw", where "w" is any point on the perpendicular bisector of "zu".