Question 3
A circle $(O)$ has a fixed chord $BC$ that is not a diameter. A point $A$ moves on the major arc $BC$, with $AB\ne AC$. Draw the angle bisector $AD$ and median $AM$ of $\triangle ABC$, where $D,M\in BC$. The circumcircle of $ADM$ meets the lines $AB,AC$ again at $E,F$, respectively.
a) Prove that $BE=CF$.
b) Let $I$ be the midpoint of $EF$, and let line $MI$ meet $AB,AC$ at $H,G$, respectively. Prove that the circumcircle of $AHG$ always passes through a fixed point as $A$ moves on the major arc $BC$.
a) Prove that $BE=CF$.
b) Let $I$ be the midpoint of $EF$, and let line $MI$ meet $AB,AC$ at $H,G$, respectively. Prove that the circumcircle of $AHG$ always passes through a fixed point as $A$ moves on the major arc $BC$.