KöMaL 2008 A467

Let ABCD be a circumscribed trapezoid such that the lines $AD$ and $BC$ intersect at point $R$. Denote by $I$ the incenter of the trapezoid, and let the incircle touch the sides $AB$ and $CD$ at points $P$ and $Q$, respectively. Let the line through $P$, which is perpendicular to $PR$, meet the lines angle bisector $AI$ and $BI$ at points $A_1$ and $B_1$, respectively. Similarly, let the line through $Q$, perpendicular to $QR$, meet $CI$ and $DI$at $C_1$ and $D_1$, respectively. Show that $A_1 D_1$=$B_1 C_1$.