Question 2
Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and $A$-excenter $J$. Let $D$ be the point where $l_a(I)$ touches $BC$. The bisector of $\angle A$ meets $BC$ and $(O)$ at $E$ and $F$, respectively. The circle $(DEF)$ meets $(O)$ again at $T$. Prove that $AT$ passes through an intersection of $(J)$ and $(DEF)$.
Editorial clarification: the source writes the undefined notation $l_a(I)$. It is read here as the incircle $(I)$, since $I$ is the incenter and $D$ is its contact point with $BC$; $(J)$ denotes the $A$-excircle.
Editorial clarification: the source writes the undefined notation $l_a(I)$. It is read here as the incircle $(I)$, since $I$ is the incenter and $D$ is its contact point with $BC$; $(J)$ denotes the $A$-excircle.