Problem

Source: XII International Festival of Young Mathematicians Sozopol 2023, Theme for 10-12 grade

Tags: geometry



In an acute scalene triangle ABC, the incircle ω touches the sides BC, CA, and AB at points D, E, and F, respectively. Let P be the foot of the perpendicular from F to DE. The line BP intersects segment AC at K, and the line AP intersects segment BC at L. The altitude through vertex C in ABC intersects the circumcircle of CKL at a point Q. Prove that line PQ passes through the center of ω.