AT-OEMO-REG 2015 Problem 4

Let $ABC$ be an isosceles triangle with $AC = BC$ and $\angle ACB < 60^\circ$ . We denote the incenter and circumcenter by $I$ and $O$ , respectively. The circumcircle of triangle $BIO$ intersects the leg $BC$ also at point $D \ne B$ .

(a) Prove that the lines $AC$ and $DI$ are parallel. (b) Prove that the lines $OD$ and $IB$ are mutually perpendicular.

(Walther Janous)