KURSCHAK 2015 Problem 2

Consider a triangle $ABC$ and a point $D$ on its side $\overline{AB}$ . Let $I$ be a point inside $\triangle ABC$ on the angle bisector of $ACB$ . The second intersections of lines $AI$ and $CI$ with circle $ACD$ are $P$ and $Q$ , respectively. Similarly, the second intersection of lines $BI$ and $CI$ with circle $BCD$ are $R$ and $S$ , respectively. Show that if $P\neq Q$ and $R\neq S$ , then lines $AB$ , $PQ$ and $RS$ pass through a point or are parallel.