Squaring a line segment

Question

I dont know whether I am being silly or not, but my question is:

Given a line segment (say length $l$), how can you draw a line segment of of length $l^2$ using straight edge and compass?

I absolutely have no idea...

Answer 1

Provided you also have a line segment of length $1$, then yes. Let $\overline{AB}$ have length $AB = l$, and $\overline{AC}$ have length $AC = 1$. Extend $\overline{AB}$ and $\overline{AC}$ indefinitely (this is possible in principle, which is what we care about).

Draw $\overline{BC}$. Also construct $D$ on $\overset{\longrightarrow}{AC}$ such that $AD = l$. Then a line through $D$ parallel to $\overline{BC}$; this line will intersect $\overset{\longrightarrow}{AB}$ at $E$. We now have, by similar triangles, $\frac{AB}{AC} = \frac{AD}{AE}$, or $\frac{1}{l} = \frac{l}{AE}$, which yields $AE = l^2$.