Question 147
An acute triangle is a triangle that has all angles less than $90^{\circ}$ ($90^{\circ}$ is a Right Angle). Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$ meeting at $H$. The circle passing through points $D$, $E$, and $F$ meets $AD$, $BE$, and $CF$ again at $X$, $Y$, and $Z$ respectively. Prove the following inequality:
$$
\frac{AH}{DX} + \frac{BH}{EY} + \frac{CH}{FZ} \geq 3
$$
$$
\frac{AH}{DX} + \frac{BH}{EY} + \frac{CH}{FZ} \geq 3
$$