I have been thinking about this problem a little differently, by
comparing the length of the line-segment vector projected onto the
radius in the same plane as the line-segment vector to the value of R
or perhaps the reverse length (of the radius vector projected onto the
line-segment), but I can't quite get beyond that starting thought.

Can anyone take my idea and tell me if it's really a good start?

-- 
(B=)
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