Centre of the circle of intersection of two spheres ($c=a+\frac{d^2+R^2-S^2}{2d^2}(b-a)$)

For two spheres with distinct centres $a,b$, radii $R,S>0$ and $d=|b-a|$, subtracting their equations gives a plane perpendicular to $b-a$. Its projection of $a$ is the displayed centre, and the circle radius squared is $R^2-[(d^2+R^2-S^2)/(2d){]}^2$. A nondegenerate circle exists exactly when $|R-S|<d<R+S$; equality gives tangency. Coincident equal spheres are not a circle and are excluded by $d>0$.