# Straightedge-only construction of segment of length $\sqrt{7}$, given regular hexagon with unit sides

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.

• Do you mean a regular hexagon of unit side length? – Arpan Mar 24 '15 at 9:03
• Perhaps you could explain your thoughts about the problem so far? – Travis Willse Mar 24 '15 at 9:04

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 = $\sqrt 7$. Added after Blue comment. All lines can be drown with an unmarked straightedge.

• 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.) – Blue Mar 24 '15 at 10:03
• Use minor diagonals, and it is not difficult. – Emilio Novati Mar 24 '15 at 10:41

Here's a variation on @Emilio's answer: Extend two hexagon edges to get $A$; then connect $A$ to $B$.