Question 7
Let $B,C$ be fixed points on a circle $(O)$, with $BC$ not passing through $O$. Let $A$ vary on the major arc $BC$, with $AB\ne AC$. The incircle $(I)$ of $ABC$ touches $BC$ at $D$. Let $I_a$ be the $A$-excenter, let $L=I_aD\cap OI$, and choose $E$ on $(I)$ such that $DE\parallel AI$.
(a) Let $F=LE\cap AI$. Prove that $AF=AI$.
(b) On the circumcircle $(J)$ of $I_aBC$, choose $M$ such that $I_aM\parallel AD$, and let $MD$ meet $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.
(a) Let $F=LE\cap AI$. Prove that $AF=AI$.
(b) On the circumcircle $(J)$ of $I_aBC$, choose $M$ such that $I_aM\parallel AD$, and let $MD$ meet $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.