Question 195
Let $ABC$ be an acute triangle with fixed points $B, C$, where point $A$ moves on the major arc $BC$ of $(O) = (ABC)$ such that $AB \neq AC$. The incircle $(I)$ of triangle $ABC$ touches $BC$ at $D$. Let $I_a$ be the $A$-excenter of triangle $ABC$. $I_aD$ cuts $OI$ at $L$, and $E$ lies on $(I)$ such that $DE$ is parallel to $AI$.
a) $LE$ cuts $AI$ at $F$. Prove that $AF = AI$.
b) Let $M$ be the point on the circumcircle $(J)$ of triangle $I_aBC$ such that $I_aM$ is parallel to $AD$. The line $MD$ cuts $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.
a) $LE$ cuts $AI$ at $F$. Prove that $AF = AI$.
b) Let $M$ be the point on the circumcircle $(J)$ of triangle $I_aBC$ such that $I_aM$ is parallel to $AD$. The line $MD$ cuts $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.