1992 USAMO 第 4 题

Chords $AA'$, $BB'$, and $CC'$ of a sphere meet at an interior point $P$ but are not contained in the same plane. The sphere through $A$, $B$, $C$, and $P$ is tangent to the sphere through $A'$, $B'$, $C'$, and $P$. Prove that $AA'=BB'=CC'$.

球体的弦 $AA'$、$BB'$ 和 $CC'$ 在内点 $P$ 处相交，但不包含在同一平面中。通过 $A$、$B$、$C$ 和 $P$ 的球体与通过 $A'$、$B'$、$C'$ 和 $P$ 的球体相切。证明 $AA'=BB'=CC'$。