Question 40
For any point $X$ inside an acute-angled triangle $ABC$ we define
$$
f(X) = \frac{AX}{A_1X} \cdot \frac{BX}{B_1X} \cdot \frac{CX}{C_1X}
$$
where $A_1, B_1$, and $C_1$ are the intersection points of the lines $AX, BX$, and $CX$ with the sides $BC, AC$, and $AB$, respectively.
Let $H, I$, and $G$ be the orthocenter, the incenter, and the centroid of the triangle $ABC$, respectively.
Prove that $f(H) \ge f(I) \ge f(G)$.
$$
f(X) = \frac{AX}{A_1X} \cdot \frac{BX}{B_1X} \cdot \frac{CX}{C_1X}
$$
where $A_1, B_1$, and $C_1$ are the intersection points of the lines $AX, BX$, and $CX$ with the sides $BC, AC$, and $AB$, respectively.
Let $H, I$, and $G$ be the orthocenter, the incenter, and the centroid of the triangle $ABC$, respectively.
Prove that $f(H) \ge f(I) \ge f(G)$.