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