Question 175
Given a circumcircle $O$ and two fixed points $B, C$ on $O$ ($BC$ is not the diameter of $O$). A point $A$ is moving on $O$ such that $ABC$ is an acute triangle. Let $E, F$ be the feet of the altitudes from $B, C$ of triangle $ABC$. Let $I$ be an arbitrary circle passing through $E$ and $F$.
a) Let $I$ touch $BC$ at $D$. Prove that $\frac{DB}{DC} = \sqrt{\frac{\cot B}{\cot C}}$.
b) Suppose that $I$ intersects $BC$ at $M$ and $N$. Let $H$ be the orthocenter of triangle $ABC$ and $P, Q$ be the intersections of $I$ and the circumcircle of triangle $HBC$. Let $K$ be the circle passing through $P, Q$ and touches $O$ at $T$ ($T$ is on the same side with $A$ with respect to $PQ$). Prove that the interior angle bisector of $\angle MTN$ passes through a fixed point.
a) Let $I$ touch $BC$ at $D$. Prove that $\frac{DB}{DC} = \sqrt{\frac{\cot B}{\cot C}}$.
b) Suppose that $I$ intersects $BC$ at $M$ and $N$. Let $H$ be the orthocenter of triangle $ABC$ and $P, Q$ be the intersections of $I$ and the circumcircle of triangle $HBC$. Let $K$ be the circle passing through $P, Q$ and touches $O$ at $T$ ($T$ is on the same side with $A$ with respect to $PQ$). Prove that the interior angle bisector of $\angle MTN$ passes through a fixed point.