Question 4
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $\overline{BC}$, strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes drawn from $B$ and $C$, respectively. Suppose $\omega_1$ is the circumcircle of triangle $BWN$ and $X$ is a point such that $\overline{WX}$ is a diameter of $\omega_1$. Similarly, $\omega_2$ is the circumcircle of triangle $CWM$ and $Y$ is a point such that $\overline{WY}$ is a diameter of $\omega_2$. Show that the points $X$, $Y$, and $H$ are collinear.