Question 162
Given a triangle $ABC$ and a point $M$ such that the lines $MA$, $MB$, $MC$ meet the lines $BC$, $CA$, $AB$ (in that order) in Cut $D$, $E$ or $F$.
Prove that there are then always the numbers $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}$ from $\{-1,1\}$, so that:
$$
\varepsilon_{1} \cdot \frac{MD}{AD} + \varepsilon_{2} \cdot \frac{ME}{BE} + \varepsilon_{3} \cdot \frac{MF}{CF} = 1
$$
Prove that there are then always the numbers $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}$ from $\{-1,1\}$, so that:
$$
\varepsilon_{1} \cdot \frac{MD}{AD} + \varepsilon_{2} \cdot \frac{ME}{BE} + \varepsilon_{3} \cdot \frac{MF}{CF} = 1
$$