HMMT 2017 T10

Let $L B C$ be a fixed triangle with $L B=L C$, and let $A$ be a variable point on arc $L B$ of its circumcircle. Let $I$ be the incenter of $\triangle A B C$ and $\overline{A K}$ the altitude from $A$. The circumcircle of $\triangle I K L$ intersects lines $K A$ and $B C$ again at $U \neq K$ and $V \neq K$. Finally, let $T$ be the projection of $I$ onto line $U V$. Prove that the line through $T$ and the midpoint of $\overline{I K}$ passes through a fixed point as $A$ varies.