Question 23
A triangle $ABC$ with $AB \lt BC$ and an obtuse angle at vertex $B$ is given. The incircle of the triangle $ABC$ with incentre $I$ touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. The line $AI$ intersects the side $BC$ at point $K$. The ray $IB$ intersects the circumcircle of the triangle $KIF$ at point $P$ ($P \neq I$), whereas the ray $IC$ intersects the circumcircle of the triangle $KIE$ at point $Q$ ($Q \neq I$). Prove that the line $PQ$ bisects the line segment $DK$.