# What is the easiest way to find the radius and center of the circle of intersection between two spheres?

If given two spheres $S_1$ and $S_2$, of radius $r_1$ and $r_2$, centered at 3-space points $P_1$ and $P_2$, respectively. What is the easiest way to find the radius and center of the circle of intersection between two spheres?

Assume that you also know the distance between centers of those two spheres. Let's say it's $d$. And the radius of the common circle is $r$. The distance from the center of the new circle to two spheres are $d_1$ and $d_2$. Then,
$$d_1^2+d_2^2=(r_1^2-r^2)+(r_2^2-r^2)=d^2$$ $$\therefore r=\sqrt{\frac{r_1^2+r_2^2-d^2}{2}}$$
And then the center ($C$) of the new circle is an interpolation of centers of two spheres ($P_1$ and $P_2$). $$C=\frac{d_2P_1+d_1P_2}{d_1+d_2}$$