1982 USAMO 第 5 题

$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.

$A、B$和$C$是球体$S$的三个内点，使得$AB$和$AC$通过$A$垂直于$S$的直径，并且可以通过均与$S$相切的$A$、$B$和$C$构建两个球体。证明它们的半径之和等于$S$的半径。