Help in solving the geometry question

Question

I having the following question with me which is a part of the 2013 IMO shortlist

"Let $ABC$ be an acute-angled triangle with orthocenter $H$, and let $W$ be a point on side $BC$. Denote by $M$ and $N$ the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ the circumcircle of $BW N$. Analogously, denote by $\omega_2$ the circumcircle of $CWM$.Let $P$ be the point on $\omega_1$ such that $W P$ is parallel to $CN$, and let $Q$ be the point on $\omega_2$ such that $W Q$ is parallel to $BM$. Prove that $P$, $Q$ and $H$ are collinear if and only if $BW = CW$ or $AW$ is perpendicular to $BC$."

I tried a lot but failed in coming up with a nice synthetic solution. Could anyone please help me out with it

Answer 1

Too long for a comment (and it has several pictures)

Now I know why this problem sounds so familiar to me. This was the original proposal of the first geometry Problem from the IMO shortlist 2013! The solution to the modified problem might be helpful.

Nevertheless, the problem selected by the committee for the IMO (see $4^{th}$ problem of IMO $2013$) was apparently "more suitable for the competition":

Problem 4, IMO 2013

Let $\triangle ABC$ be an acute-angled triangle with orthocenter $H$, and let $W$ be a point on side $BC$. Denote by $M$ and $N$ the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega{1}$ the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ which is diametrically opposite to $W$. Analogously, denote by $\omega_2$ the circumcircle of $CWM$, and let $Y$ be the point on $\omega_2$ which is diametrically opposite to $W$. Prove that $X, Y$ and $H$ are collinear.

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You can find the official solution in the pdf I linked above and alternative solutions here

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I personally think that the best solution is provided by Evan Chen in his excellent Euclidean Geometry in Mathematical Olympiads.

But first the hints:

$15$. Do you see a pair of perpendicular lines

$106$. Add a Miquel point

$157$. Show that $X, H, P$ are collinear, where $P$ is said Miquel point.

Finally

Finally, I've made a picture of your problem in Geogebra that might help other users to figure out how to solve it