Correct solutions from outside the Bradley University community were received from Jens Voβ, Lou Cairoli, Dan Dima, Jérôme Lefebvre, Paul Lee.

Imagine that the equilateral triangle has been drawn with one vertex on the given point P, one at the point L on the given line, and one at the point C on the given circle.  If you rotate the point P through 60 degrees, you will reach the point C.  Therefore a necessary and sufficient condition for the equilateral triangle to exist is that the given line when rotated 60 degrees intersect the circle.  It is easy to construct configurations where there are 0, 1, 2, 3 or 4 intersections. 

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