Mutual arrangement of two circles

Given two circles with centers $O_1$ and $O_2$, and radii $r_1$ and $r_2$, their mutual arrangement depends on the distance $d = |O_1O_2|$ between the centers:

• Separate (no intersection): $d > r_1 + r_2$
• Externally tangent: $d = r_1 + r_2$
• Intersecting: $|r_1 - r_2| < d < r_1 + r_2$
• Internally tangent: $d = |r_1 - r_2|$
• One inside the other (no intersection): $d < |r_1 - r_2|$

Important properties

• The number of intersection points is 0, 1, or 2.

• Tangency occurs when the circles touch at exactly one point.

• If $r_1 = r_2$ and $O_1 = O_2$, the circles coincide (are the same).