For example-There is a postulate that if two points are on OPPOSITE sides of a line, then the line segment joining the points must intersect the line, whereas if the points are on the SAME side of the line, the line segment joining them will not intersect the line. In other words, if the line does not pass BETWEEN the points, then the line segment will not intersect the line. Are there any postulate of this type for circle?
Denote $d$ be distance between centers and $r,R$ are the radii.
If 2 circles intersect at ONE point, then $d=R+r$ (externally) or $d=R-r$ (internally)
If 2 circles intersect at TWO point, then $R-r<d<R+r$
If there are no intersections, then $d>R+r$ (externally) or $d<R-r$ (internally)