12.1.7 Olympic geometry

Problem: On sides \( A B, B C, C D, D E, E F, F A \) of the regular hexagon \( A B C D E F \) with the area \( S \) points \( A_{1}, B_{1}, C_{1}, D_{1}, E_{1}, F_{1} \) are marked so, that \[ \frac{A A_{1}}{A_{1} B}=\frac{B B_{1}}{B_{1} C}=\frac{C C_{1}}{C_{1} D}=\frac{D D_{1}}{D_{1} E}=\frac{E E_{1}}{E_{1} F}=\frac{F F_{1}}{F_{1} A}=\frac{1}{4} . \]