IR-TST 2019 Problem 15

In triangle $ABC$ , $M,N$ and $P$ are midpoints of sides $BC,CA$ and $AB$ . Point $K$ lies on segment $NP$ so that $AK$ bisects $\angle BKC$ . Lines $MN,BK$ intersects at $E$ and lines $MP,CK$ intersects at $F$ . Suppose that $H$ be the foot of perpendicular line from $A$ to $BC$ and $L$ the second intersection of circumcircle of triangles $AKH, HEF$ . Prove that $MK,EF$ and $HL$ are concurrent.

Proposed by Alireza Dadgarnia