Question 77
In right triangle $A B C$, $\angle A C B = 90^{\circ}$ and $\tan A \gt \sqrt{2}$. $M$ is the midpoint of $A B$, $P$ is the foot of the altitude from $C$, and $N$ is the midpoint of $C P$. Line $A B$ meets the circumcircle of $C N B$ again at $Q$. $R$ lies on line $B C$ such that $Q R$ and $C P$ are parallel, $S$ lies on ray $C A$ past $A$ such that $B R = R S$, and $V$ lies on segment $S P$ such that $A V = V P$. Line $S P$ meets the circumcircle of $C P B$ again at $T$. $W$ lies on ray $V A$ past $A$ such that $2 A W = S T$, and $O$ is the circumcenter of $S P M$. Prove that lines $O M$ and $B W$ are perpendicular.