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 circles
and
The circles have radii
and
respectively.
The centres of the circles are at
and
respectively. The distance between the centres is
so 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 are
and
respectively.
Then
and
The circles cross at the points
and