VMO 2016 Problem 6

Given a triangle $ABC$ inscribed by circumcircle $(O)$ . The angles at $B,C$ are acute angle. Let $M$ on the arc $BC$ that doesn't contain $A$ such that $AM$ is not perpendicular to $BC$ . $AM$ meets the perpendicular bisector of $BC$ at $T$ . The circumcircle $(AOT)$ meets $(O)$ at $N$ ( $N\ne A$ ).

a) Prove that $\angle{BAM}=\angle{CAN}$ .

b) Let $I$ be the incenter and $G$ be the foor of the angle bisector of $\angle{BAC}$ . $AI,MI,NI$ intersect $(O)$ at $D,E,F$ respectively. Let ${P}=DF\cap AM, {Q}=DE\cap AN$ . The circle passes through $P$ and touches $AD$ at $I$ meets $DF$ at $H$ ( $H\ne D$ ).The circle passes through $Q$ and touches $AD$ at $I$ meets $DE$ at $K$ ( $K\ne D$ ). Prove that the circumcircle $(GHK)$ touches $BC$ .