In an isosceles triangle $ABC$ in which $AC = BC$ and $\angle ABC < 60^o$, $I$ and $O$ are the centers of the inscribed and circumscribed circles, respectively. For the triangle $BIO$, the circumscribed circle intersects the side $BC$ again at $D$. Prove that: i) lines $AC$ and $DI$ are parallel, ii) lines $OD$ and $IB$ are perpendicular.