In a plane, if two points are known and the radius is known, there are three conditions which could be true:

1) If the distance between the two points is less than 2*radius and greater than zero, then there will be two circles with radius R that will intersect those points.

2) if the distance between the two points is exactly equal to 2*radius then one circle with radius R intersects the two points.

3) if the distance between the two points is greater than 2*radius, then no circle with radius R will intersect both points.

In a plane, if three points are known, there either will be one circle that intersects those points OR there will be none (if the 3 points are co-linear).