Let C_{1}, C_{2} be the centers and r_{1}, r_{2} be the radii of two circles S = 0 and S’ = 0 respectively. Further \(\overline{{{C}_{1}}{{C}_{2}}}\) represents the line segment from C_{1} to C_{2}. The following cases arises with regard to the relative position of two circles.

1. C_{1 }+ C_{2} > r_{1 }+ r_{2}

In this case the two circles do not intersect and one circle will be away from the other circle.2. C_{1}C_{2} = r_{1} + r_{2}

In this case the two circles touch each other externally.3. |r_{1} – r_{2}| < C_{1}C_{2} < r_{1} + r_{2 }

In this case the two circles intersect in two distinct points.4. C_{1}C_{2} = |r_{1} – r_{2}|

The two circles touch each other internally in this case.5. C_{1}C_{2} < |r_{1} – r_{2}|

In this case the two circles do not intersect/touch and one circle will be completely inside the other.**Note:** If C_{1}C_{2} = 0 then the centers of the two circles coincide and they are concentric circles.