Question 70
Let $ABCDEF$ be a convex hexagon such that $\angle ACE = \angle BDF$ and $\angle BCA = \angle EDF$. Let $A_1 = AC \cap FB$, $B_1 = BD \cap AC$, $C_1 = CE \cap BD$, $D_1 = DF \cap CE$, $E_1 = EA \cap DF$ and $F_1 = FB \cap EA$. Suppose $B_1, C_1, D_1, F_1$ lie on the same circle $\Gamma$. The circumcircles of $\triangle BB_1F_1$ and $\triangle ED_1F_1$ meet at $F_1$ and $P$. The line $F_1P$ meets $\Gamma$ again at $Q$. Prove that $B_1D_1$ and $QC_1$ are parallel. (Here, we use $\ell_1 \cap \ell_2$ to denote the intersection point of lines $\ell_1$ and $\ell_2$.)