HMMT 2009 U6

Let $A B C$ be a right triangle with hypotenuse $A C$. Let $B^{\prime}$ be the reflection of point $B$ across $A C$, and let $C^{\prime}$ be the reflection of $C$ across $A B^{\prime}$. Find the ratio of $\left[B C B^{\prime}\right]$ to $\left[B C^{\prime} B^{\prime}\right]$.