Question 57
Let $ABCD$ be a quadrilateral inscribed in a unit circle with center $O$. Suppose that $\angle AOB = \angle COD = 135^{\circ}$, $BC = 1$. Let $B'$ and $C'$ be the reflections of $A$ across $BO$ and $CO$ respectively. Let $H_1$ and $H_2$ be the orthocenters of $AB'C'$ and $BCD$, respectively. If $M$ is the midpoint of $OH_1$, and $O'$ is the reflection of $O$ about the midpoint of $MH_2$, compute $OO'$.
