USAMO 1982 Problem 5

$\mathrm{O}$ is the center of a sphere $\mathrm{S}$. Points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are inside $\mathrm{S}, \mathrm{OA}$ is perpendicular to $\mathrm{AB}$ and $\mathrm{AC}$, and there are two spheres through $A, B$, and $\mathrm{C}$ which touch $\mathrm{S}$. Show that the sum of their radii equals the radius of $\mathrm{S}$.