If the distance between the centres of two circles is less than the sum of…

Concept. The position of two circles is fixed by comparing the distance d between their centres with two lengths: the sum of the radii (r1 + r2) and the difference of the radii (|r1 − r2|). Each configuration has its own characteristic condition.

• d > r1 + r2 : the circles lie completely apart (no common point).

• d = r1 + r2 : the circles touch externally.

• |r1 − r2| < d < r1 + r2 : the circles intersect (boundaries cross at two points).

• d = |r1 − r2| : the circles touch internally.

• d < |r1 − r2| : one circle lies wholly inside the other.

Application. Here the distance is less than the sum of the radii, i.e. d < r1 + r2. This is exactly the condition that carries the circles past the "apart / externally touching" boundary, so the two boundaries are now close enough to cross. Hence the circles intersect each other.

Cross-check. Take r1 = 5, r2 = 3, d = 6. Then r1 + r2 = 8 and d = 6 < 8, and the two circles do cut each other at two points. The touching cases need equalities (d = 8 for external, d = 2 for internal) and containment needs d < 2 — none of which is the condition stated here.