Problem 18. On the diagonals $A B_{1}$ and $C A_{1}$ of the lateral faces of the prism $A B C A_{1} B_{1} C_{1}$, points $D$ and $E$ are located such that the lines $D E$ and $B C_{1}$ are parallel. Find the ratio of the segments $D E$ and $B C_{1}$.
Solution. Represent the lower base of the prism as an equilateral triangle $ABC$, and the line $BC_{1}$ as a point $B=C_{1}$ (Fig. 9). Then the segment $DE$, parallel to the line $BC_{1}$, will be represented by the point $D=E$ (at point $E$, the lines $B_{1}A$ and $A_{1}C$ intersect). Since the segments $DE$ and $BC_{1}$ are parallel, the lines $C_{1}E$, $BD$, and $AA_{1}$ intersect at point $M$. Since $MD:DB=1:2$, then $DE:BC_{1}=1:2$.