KöMaL 2008 B4088

$P$ is an interior point of side $B_1 B_2$ of an acute-angled triangle $AB_1 B_2$. Let $Q$denote the reflection of $P$ about point $A$. Let $D_i$ denote the perpendicular projection of $P$ onto the line segment $AB_i$, and let $F_i$ be the midpoint of the line segment $D_i B_i$. Prove that if $QD_1$ is perpendicular to $PF_1$, then $QD_2$ is perpendicular to $PF_2$.