Question 4
Let $(O)$ be a circle and $B,C$ fixed points on it such that $BC$ is not a diameter. A point $A$ varies on $(O)$ so that triangle $ABC$ is acute. Let $E,F$ be the feet of the altitudes from $B,C$, respectively. Let $(I)$ be a variable circle through $E,F$, with center $I$.
a) Suppose $(I)$ is tangent to $BC$ at $D$. Prove $\dfrac{DB}{DC}=\sqrt{\dfrac{\cot B}{\cot C}}$.
b) Suppose $(I)$ meets segment $BC$ at $M,N$. Let $H$ be the orthocenter of $ABC$, and $P,Q$ the intersections of $(I)$ with the circumcircle of $HBC$. A circle $(K)$ passes through $P,Q$ and is tangent to $(O)$ at $T$, where $T$ and $A$ are on the same side of $PQ$. Prove that the internal bisector of $\angle MTN$ always passes through a fixed point.
a) Suppose $(I)$ is tangent to $BC$ at $D$. Prove $\dfrac{DB}{DC}=\sqrt{\dfrac{\cot B}{\cot C}}$.
b) Suppose $(I)$ meets segment $BC$ at $M,N$. Let $H$ be the orthocenter of $ABC$, and $P,Q$ the intersections of $(I)$ with the circumcircle of $HBC$. A circle $(K)$ passes through $P,Q$ and is tangent to $(O)$ at $T$, where $T$ and $A$ are on the same side of $PQ$. Prove that the internal bisector of $\angle MTN$ always passes through a fixed point.