Question 166
Let $I$ be the incenter of the scalene triangle $ABC$, with $AB \lt AC$, and let $I'$ be the reflection of point $I$ in the line $BC$. The angle bisector $AI$ meets the side $BC$ at $D$ and the circumcircle of $ABC$ at $E$. The line $EI'$ meets the circumcircle at $F$. Prove that $\frac{AI}{IE} = \frac{ID}{DE}$ and $IA = IF$.