MathNet 2025-part-01 — 187 problems (EN–VI) Question 157

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$. Let $\omega$ be the circle with diameter $BC$, and suppose it intersects the segment $AD$ at point $K$ inside triangle $ABC$. On ray $KD$, let $L$ be a point such that $KA = KL$. Let lines $BL$ and $CL$ intersect the circle $\omega$ again at points $P$ and $Q$, respectively. Prove that lines $AD$, $QE$, and $PF$ are concurrent.

(Bilegdemberel Bat-Amgalan)