HMMT 2008 T10

On the circumcircle of $A B C$, let $A^{\prime}$ be the midpoint of arc $B C$ (not containing $A$ ). (a) $[\mathbf{1 0}]$ Show that $A, I, A^{\prime}$ are collinear. (b) $[\mathbf{2 0}]$ Show that $A^{\prime}$ is the circumcenter of BIC.