Question 114
Let $ABC$ be an arbitrary triangle, and $A_{1}$, $B_{1}$ and $C_{1}$ three points, $A_{1} \in (BC)$, $B_{1} \in (AC)$, $C_{1} \in (AB)$, such that $\frac{BA_{1}}{A_{1}C} = \frac{CB_{1}}{B_{1}A} = \frac{AC_{1}}{C_{1}B}$. Prove that the lengths of the segments $[AA_{1}]$, $[BB_{1}]$ and $[CC_{1}]$ can serve as the side lengths of a triangle.