2. A regular hexagon with side length $a$ is given. Using only a ruler, construct a segment with length $\frac{a}{n}$, for $n=2,3,4, \ldots$.
Solution. Let a regular hexagon $A B C D E F$ with side $a$ be given. First, we construct the intersections $K$ and $L$ of the line $D E$ with the lines $B C$ and $A C$, respectively, then the intersection $M$ of the lines $A B$ and $C D$, and finally the intersections $N$ and $O$ of the line $C F$ with the lines $M L$ and $A D$, respectively (see the diagram on the right). It is easily proven that the hexagon $B M N K D O$ is regular with side $a$ and that $F N=3 F O=3 a$. By continuing the process, we can determine a point $S$ on the line $F O$ for a given natural number $N$ such that $F S=n F O=n a$. Now, if $T$ is the intersection of the lines $E S$ and $O D$, then from the similarity of triangles $F S E$ and $O S T$ it follows that $O T=\frac{n-1}{n} a$, i.e., $T D=\frac{1}{n} a$.

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