If the distance between the centres of two circles is less than the sum of their raddi, the circles will intersect at two points.

Suppose we have circlesand

The circles have radiiandrespectively.

The centres of the circles are atandrespectively. The distance between the centres isso the circles intersect in two points.

Expanding the brackets for each circles gives

Subtraction gives

Substitution intogives

Expanding the brackets and simplifying gives

The solutions to this equations areandrespectively.

ThenandThe circles cross at the pointsand