Question 48
Let $\triangle ABC$ be an acute-angled triangle such that $|AB| \lt |AC|$, with circumscribed circle $\Gamma$ with centre $O$. The points $D$, $E$ and $F$ are constructed as the feet of the altitudes from $A$, $B$ and $C$, respectively. We call the intersection point of the tangents to $\Gamma$ through $B$ and $C$ $P$. The line through $P$ perpendicular to $EF$ intersects the line $AD$ at the point $Q$. Let $R$ be the perpendicular projection of $A$ onto $EF$.
Prove that the lines $DR$ and $OQ$ are parallel.
Prove that the lines $DR$ and $OQ$ are parallel.