Problem

Source: Ukraine TST 2012 p2

Tags: geometry, circumcircle, Cyclic, Concyclic, circles



$E$ is the intersection point of the diagonals of the cyclic quadrilateral, $ABCD, F$ is the intersection point of the lines $AB$ and $CD, M$ is the midpoint of the side $AB$, and $N$ is the midpoint of the side $CD$. The circles circumscribed around the triangles $ABE$ and $ACN$ intersect for the second time at point $K$. Prove that the points $F, K, M$ and $N$ lie on one circle.