Question 3
Let $ABC$ be a scalene triangle. Its incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$, respectively. The line through $E$ perpendicular to $BI$ meets $(I)$ again at $K\ne E$; the line through $F$ perpendicular to $CI$ meets it again at $L\ne F$. Let $J$ be the midpoint of $KL$.
(a) Prove that $D,I,J$ are collinear.
(b) Fix $B,C$ and let $A$ vary with $AB/AC=k$ constant. Let $M\ne E,N\ne F$ be the other intersections of $IE,IF$ with $(I)$. Let $P=MN\cap IB$, $Q=MN\cap IC$. Prove that the perpendicular bisector of $PQ$ always passes through a fixed point.
(a) Prove that $D,I,J$ are collinear.
(b) Fix $B,C$ and let $A$ vary with $AB/AC=k$ constant. Let $M\ne E,N\ne F$ be the other intersections of $IE,IF$ with $(I)$. Let $P=MN\cap IB$, $Q=MN\cap IC$. Prove that the perpendicular bisector of $PQ$ always passes through a fixed point.