$\triangle ABC$ is an isosceles triangle with $AC = BC$ and $\angle ACB < 60^{\circ}$. $I$ and $O$ are the incenter and circumcenter of $\triangle ABC$. The circumcircle of $\triangle BIO$ intersects $BC$ at $D \neq B$. (a) Do the lines $AC$ and $DI$ intersect? Give a proof. (b) What is the angle of intersection between the lines $OD$ and $IB$?