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Let's consider a regular hexagon with unit side length. Draw a line segment of length $\sqrt{7}$ using nothing except a straightedge (that is, an unmarked ruler). The position of the segment may be chosen as you wish.

My thoughts: I've drawn all sorts of lines and tried to get the segment in one of the inside triangles, but to no avail.

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    $\begingroup$ Do you mean a regular hexagon of unit side length? $\endgroup$ – Arpan Mar 24 '15 at 9:03
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    $\begingroup$ Perhaps you could explain your thoughts about the problem so far? $\endgroup$ – Travis Willse Mar 24 '15 at 9:04
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The rectangle that has as basis a diagonal of the hexagon and as height the orthogonal median ( joining mean points of opposite sides) has diagonal = $\sqrt 7$.

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Added after Blue comment. enter image description here

All lines can be drown with an unmarked straightedge.

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  • $\begingroup$ How do you propose to construct the vertical blue and red lines using only an unmarked straightedge? (With a small adjustment, the basic idea can be salvaged.) $\endgroup$ – Blue Mar 24 '15 at 10:03
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    $\begingroup$ Use minor diagonals, and it is not difficult. $\endgroup$ – Emilio Novati Mar 24 '15 at 10:41
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Here's a variation on @Emilio's answer:

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Extend two hexagon edges to get $A$; then connect $A$ to $B$.

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