Question 169
Given a circle $(O)$ with a fixed chord $BC$ ($BC$ is not a diameter of the circle). Let $A$ move on the bigger arc $BC$ such that $ABC$ is an acute triangle and $AB \lt AC$. Let $I$ and $H$ respectively be the midpoint of $BC$ and the orthocenter of the triangle $ABC$. The ray $IH$ intersects the circle $(O)$ again at $K$, the line $AH$ intersects the line $BC$ at $D$ and the line $KD$ intersects the circle $(O)$ again at $M$. From $M$, we draw a perpendicular line to $BC$, intersecting $AI$ at $N$.
a) Prove that point $N$ belongs to a fixed circle when $A$ moves along the bigger arc $BC$.
b) The circle that goes through $A$ and touches $AK$ at $A$ intersects $AB$, $AC$ at $P$, $Q$, respectively. Let $J$ be the midpoint of $PQ$. Prove that the line $AJ$ always goes through a fixed point.
a) Prove that point $N$ belongs to a fixed circle when $A$ moves along the bigger arc $BC$.
b) The circle that goes through $A$ and touches $AK$ at $A$ intersects $AB$, $AC$ at $P$, $Q$, respectively. Let $J$ be the midpoint of $PQ$. Prove that the line $AJ$ always goes through a fixed point.