Question 26
Let $ABC$ be a scalene triangle. The incircle $(I)$ of triangle $ABC$ touches $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. The line passing through $E$ and perpendicular to $BI$ cuts $(I)$ again at $K$ and the line passing through $F$ and perpendicular to $CI$ cuts $(I)$ again at $L$. The point $J$ is the midpoint of $KL$.
a) Prove that $D$, $I$, and $J$ are collinear.
b) Suppose that $B, C$ are fixed and $A$ is a moving point such that $\frac{AB}{AC} = k$ ($k$ is a given constant). Line $IE, IF$ cut $(I)$ again at $M, N$ respectively ($M \neq E, N \neq F$). Line $MN$ cuts $IB, IC$ at $P, Q$. Prove that the perpendicular bisector of the segment $PQ$ passes through a fixed point.
a) Prove that $D$, $I$, and $J$ are collinear.
b) Suppose that $B, C$ are fixed and $A$ is a moving point such that $\frac{AB}{AC} = k$ ($k$ is a given constant). Line $IE, IF$ cut $(I)$ again at $M, N$ respectively ($M \neq E, N \neq F$). Line $MN$ cuts $IB, IC$ at $P, Q$. Prove that the perpendicular bisector of the segment $PQ$ passes through a fixed point.