Given (see figure 1 and figure 2): , , and .
- If , then no solutions; one circle completely outside the other.
- If , then no solutions; one circle completely inside the other.
- If d = 0 and , then infinite number of solutions; circles are coincident.
Subtract both equations:
How much is a when the two circles touch at one point, i.e. ?
Then substituting a in  yields:
Similar triangles in figure 3:
Or for the other intersection point (not visible in figure 3):
And substituting in the previous equation for results in:
And similar for the y-coordinates:
So, this yields two xy-coordinates , one for each intersection point. If both circles touch in one point (h = 0), then .