What will be the radius of the common circle formed by intersection of two spheres of radii $r_1$ and $r_2$ that cut orthogonally.

Question

Two spheres of radii $r_1$ and $r_2$ intersect each other orthogonally. Prove that the circle formed by the intersection of the two spheres has a radius $$\frac{r_1 r_2}{\sqrt{r_1^{2} + r_2^{2}}}.$$

Answer 1

HINT.

See below a section of the spheres, passing through their centers $A$ and $B$. They intersect each other orthogonally if radii $AC$ and $BC$ are perpendicular.

It follows that $ABC$ is a right triangle with legs $r_1$ and $r_2$. And its altitude $CH$ is the radius of the intersection circle.