VN-TST 2012 Problem 1

Consider a circle $(O)$ and two fixed points $B,C$ on $(O)$ such that $BC$ is not the diameter of $(O)$ . $A$ is an arbitrary point on $(O)$ , distinct from $B,C$ . Let $D,J,K$ be the midpoints of $BC,CA,AB$ , respectively, $E,M,N$ be the feet of perpendiculars from $A$ to $BC$ , $B$ to $DJ$ , $C$ to $DK$ , respectively. The two tangents at $M,N$ to the circumcircle of triangle $EMN$ meet at $T$ . Prove that $T$ is a fixed point (as $A$ moves on $(O)$ ).