BD-BDMO 2016 Problem 6

$\triangle ABC$ 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 $IB$ ?