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Two Points Plus One Line

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Simple hint: how would you find the set of points that are equidistant from two given points?

When might that locus not intersect with the given line?

Harder, last part: try dropping perpendiculars from $A$ and $B$ onto the given line.

The distance between those positions might be useful.

When is $P$ between those positions and when is it outside?