Question 11
Let $A_1$, $B_1$ and $C_1$ be the feet of the altitudes in the acute triangle $ABC$. On the line segments $B_1C_1$, $C_1A_1$, $A_1B_1$ one considers the points $X$, $Y$, and $Z$, respectively, such that
$$
\frac{C_1X}{XB_1} = \frac{b \cos C}{c \cos B}, \quad \frac{A_1Y}{YC_1} = \frac{c \cos A}{a \cos C} \quad \text{and} \quad \frac{B_1Z}{ZA_1} = \frac{a \cos B}{b \cos A}.
$$
Prove that the lines $AX$, $BY$ and $CZ$ are concurrent.
$$
\frac{C_1X}{XB_1} = \frac{b \cos C}{c \cos B}, \quad \frac{A_1Y}{YC_1} = \frac{c \cos A}{a \cos C} \quad \text{and} \quad \frac{B_1Z}{ZA_1} = \frac{a \cos B}{b \cos A}.
$$
Prove that the lines $AX$, $BY$ and $CZ$ are concurrent.