Question 2
A circle $(O)$ has a fixed chord $BC$ which is not a diameter. A point $A$ moves on the major arc $BC$ so that $ABC$ is acute and $AB\lt AC$. Let $I$ be the midpoint of $BC$ and $H$ the orthocenter of $ABC$. Ray $IH$ meets $(O)$ again at $K$. Line $AH$ meets line $BC$ at $D$, and line $KD$ meets $(O)$ again at $M$. The line through $M$ perpendicular to $BC$ meets $AI$ at $N$.
(a) Prove that $N$ belongs to a fixed circle as $A$ varies.
(b) The circle tangent to $AK$ at $A$ and passing through $N$ meets $AB,AC$ again at $P,Q$, respectively. Let $J$ be the midpoint of $PQ$. Prove that $AJ$ passes through a fixed point.
(a) Prove that $N$ belongs to a fixed circle as $A$ varies.
(b) The circle tangent to $AK$ at $A$ and passing through $N$ meets $AB,AC$ again at $P,Q$, respectively. Let $J$ be the midpoint of $PQ$. Prove that $AJ$ passes through a fixed point.