Article:Circle-Circle Intersection

Problem:
If the problem is only determining how many intersections there are (not including same center circles), there is a simple geometric interpretation: check if the distance between both centers is no more than the sum of radii. If equal, there is exactly one intersection, if less then 2 intersections, if more then 0 intersections.

To actually find the intersection, in vector notation (which generalizes to higher dimensions)

WLOG circle 1 is centered at origin and circle 2 is centered at$P$.

By definition of circle, circle 1 is points$x : |x| = r_1$ and circle 2 is points$x: |x-P| = r_2$. Squaring both sides and substituting$|x|^2$ gives the radical line described by$r_1^2 - 2 x \cdot P + |P|^2 = r_2^2$.