VN-TST 2018 Problem 1

Let $ABC$ be a acute, non-isosceles triangle. $D,\ E,\ F$ are the midpoints of sides $AB,\ BC,\ AC$ , resp. Denote by $(O),\ (O')$ the circumcircle and Euler circle of $ABC$ . An arbitrary point $P$ lies inside triangle $DEF$ and $DP,\ EP,\ FP$ intersect $(O')$ at $D',\ E',\ F'$ , resp. Point $A'$ is the point such that $D'$ is the midpoint of $AA'$ . Points $B',\ C'$ are defined similarly. a. Prove that if $PO=PO'$ then $O\in(A'B'C')$ ; b. Point $A'$ is mirrored by $OD$ , its image is $X$ . $Y,\ Z$ are created in the same manner. $H$ is the orthocenter of $ABC$ and $XH,\ YH,\ ZH$ intersect $BC, AC, AB$ at $M,\ N,\ L$ resp. Prove that $M,\ N,\ L$ are collinear.