Question 39
Let $M$ be a point in the interior of the triangle $ABC$. The line $AM$ intersects the circumcircle of the triangle $MBC$ once more at $D$, the line $BM$ intersects the circumcircle of the triangle $MCA$ once more at $E$, and the line $CM$ intersects the circumcircle of the triangle $MAB$ once more at $F$. Prove the inequality
$$
\frac{|AD|}{|MD|} + \frac{|BE|}{|ME|} + \frac{|CF|}{|MF|} \ge \frac{9}{2}. \quad (\text{Tajikistan 2014})
$$
$$
\frac{|AD|}{|MD|} + \frac{|BE|}{|ME|} + \frac{|CF|}{|MF|} \ge \frac{9}{2}. \quad (\text{Tajikistan 2014})
$$