
The fixed point
Let $\displaystyle ABC$ be a triangle inscribed the circle $\displaystyle (O)$ and $\displaystyle A'$ is a fixed point on $\displaystyle (O)$. $\displaystyle P$ moves on $\displaystyle BC$, $\displaystyle K$ belongs to $\displaystyle AC$ so that $\displaystyle PK$ is always parallel to a fixed line $\displaystyle d$. The circumcircle of triangle $\displaystyle APK$ cuts the circle $\displaystyle (O)$ at a second point $\displaystyle E$. $\displaystyle AE$ cuts $\displaystyle BC$ at $\displaystyle M$. $\displaystyle A'P$ cuts the circle $\displaystyle (O)$ at a second point $\displaystyle N$. Prove that the line $\displaystyle MN$ passes through a fixed point.