On Unit Circles Which Avoid All but Two Points of a Given Point-set

In this paper we show that if a finite set of at least two points in the plane has a diameter less than 3, then there is a unit circle passing through exactly two points of the set. I conjecture that if the diameter of the point-set is between 3 and 2, then the same statement is true with only one exception. The exceptional set must have four points such that three of the points are vertices of an acute triangle with circum-radius 1, and the fourth point is the common point of the three unit circles which go through only two vertices of this triangle.