Let A be a point on a line l, and let B be a point that does not lie on l. Prove that there is a unique circle that contains A and B and is tangent to l.
This is obviously true, but I am getting nowhere. Can I have any help please?
But I guess you are not.
There is a unique plane, , determined by
Look at the unique perpendicular to at that is in .
That perpendicular will intersect the perpendicular bisector of .
That is the center of your circle.