For the three spheres you get the 3 functions:

I:   ( x - x0 )^2 + ( y - y0 )^2 + ( z - z0 )^2 - r0 ^2
II:  ( x - x1 )^2 + ( y - y1 )^2 + ( z - z1 )^2 - r1 ^2
III: ( x - x2 )^2 + ( y - y2 )^2 + ( z - z2 )^2 - r2 ^2

with <xn,yn,zn> as the middle points and rn the radii

When you set I == II and I == III and simplify this you get two
equations of the following form:

A*x+B*y+C*z==D

(with A,B,C,D as constants)

These are the equations for the planes in which the cutting circles lie
in. Now you just need to calcualte the cutting line of these two planes.
And finally cut one of your inital sphere with this line... then you get
your desired points.

I didn't care about the specal cases here (0/1/infitely many points)
because you said you were only interested in the 2 points-case.

I don't know if there is a more easy way but it should work this way.
You can also get a general formula from this.

Hope this helps
- Micha

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