Question 15
Given a non-isosceles triangle $ABC$, inscribed in circle $(O)$ with $X, Y, Z$ are midpoints of the major arcs $BC, CA, AB$ of $(O)$. Let $D, E, F$ be the tangent points of the incircle $(I)$ of $ABC$ on $BC, CA, AB$ respectively. Suppose that line $XE$ intersects $(O)$ again at $M$ and cuts $YD$ at $P$; line $XF$ intersects $(O)$ again at $N$ and intersects $ZD$ at $Q$. Let $T$ be the intersection of $QE, PF$ and $XT$ cuts $(O)$ again at $K$. Prove that the circumcircle of triangle $AOK$ bisects the segment $MN$.