Question 39
Let $ABC$ be a triangle. $D$ is the foot of the internal bisector of the angle $A$. The perpendicular from $D$ to the tangent $AT$ ($T$ belongs to $BC$) to the circumscribed circle of $ABC$ intersects the altitude $AH_a$ at the point $I$ ($H_a$ belongs to $BC$). If $P$ is the midpoint of $AB$ and $O$ is the circumcenter of $\triangle ABC$, $TI$ intersects $AB$ at $M$ and $PT$ intersects $AD$ at $F$, prove that $MF$ is perpendicular to $AO$.