Question 6
Let $ABC$ be inscribed in a circle $(O)$ with center $O$, with the angles at $B,C$ acute. Choose $M$ on the arc $BC$ not containing $A$ such that $AM$ is not perpendicular to $BC$. Let $T$ be the intersection of $AM$ and the perpendicular bisector of $BC$. The circumcircle of $AOT$ meets $(O)$ again at $N\ne A$.
a) Prove that $\angle BAM=\angle CAN$.
b) Let $I$ be the incenter and $G$ the foot of the internal bisector at $A$. The lines $AI,MI,NI$ meet $(O)$ at $D,E,F$, respectively. Set $P=DF\cap AM$ and $Q=DE\cap AN$. The circle through $P$ tangent to $AD$ at $I$ meets $DF$ at $H$ ($H\ne D$); the circle through $Q$ tangent to $AD$ at $I$ meets $DE$ at $K$ ($K\ne D$). Prove that the circumcircle of $GHK$ is tangent to $BC$.
a) Prove that $\angle BAM=\angle CAN$.
b) Let $I$ be the incenter and $G$ the foot of the internal bisector at $A$. The lines $AI,MI,NI$ meet $(O)$ at $D,E,F$, respectively. Set $P=DF\cap AM$ and $Q=DE\cap AN$. The circle through $P$ tangent to $AD$ at $I$ meets $DF$ at $H$ ($H\ne D$); the circle through $Q$ tangent to $AD$ at $I$ meets $DE$ at $K$ ($K\ne D$). Prove that the circumcircle of $GHK$ is tangent to $BC$.