Question 182
Let $ABC$ be an acute, scalene triangle with the circumcircle ($O$) and angles $ACB$, $ABC$ are acute. Let $M$ be a point on arc $BC$ that does not contain $A$ and $AM$ is not perpendicular to $BC$. The line $AM$ meets the perpendicular bisector of $BC$ at $T$ and the circumcircle of triangle $AOT$ meets $O$ at $N$ ($N \neq A$).
a) Prove that $\angle BAM = \angle CAN$.
b) Let $I$ be the incenter of triangle $ABC$ and $G$ be the foot of the internal angle bisector of $\angle BAC$. Let $AI$, $MI$ and $NI$ intersect ($O$) at $D$, $E$ and $F$ respectively. Let $P$, $Q$ be the intersections of $DF$ and $AM$, $DE$ and $AN$. The circle passing through $P$ and touches $AD$ at $I$ meets $DF$ at $H$ ($H \neq D$). Similarly, the circle passing through $Q$ and touches $AD$ at $I$ meets $DE$ at $K$ ($K \neq D$). Prove that the circumcircle of triangle $GHK$ touches $BC$.
a) Prove that $\angle BAM = \angle CAN$.
b) Let $I$ be the incenter of triangle $ABC$ and $G$ be the foot of the internal angle bisector of $\angle BAC$. Let $AI$, $MI$ and $NI$ intersect ($O$) at $D$, $E$ and $F$ respectively. Let $P$, $Q$ be the intersections of $DF$ and $AM$, $DE$ and $AN$. The circle passing through $P$ and touches $AD$ at $I$ meets $DF$ at $H$ ($H \neq D$). Similarly, the circle passing through $Q$ and touches $AD$ at $I$ meets $DE$ at $K$ ($K \neq D$). Prove that the circumcircle of triangle $GHK$ touches $BC$.