Construction of Square on Given Straight Line

Theorem

On any given line segment, it is possible to construct a square.

Proof

Let $AB$ be the given line segment.

Construct $AC$ perpendicular to $AB$.

By Construction of Equal Straight Lines from Unequal, place $D$ on $AC$ so $AD = AB$.

Construct $DE$ parallel to $AB$, and construct $BE$ parallel to $AD$.

So $ADEB$ is a parallelogram.

So $AB = DE$ and $AD = BE$ from Opposite Sides and Angles of Parallelogram are Equal.

So the parallelogram is equilateral.

Also, as $AD$ falls on the parallels $AB$ and $DE$, from Parallel Implies Supplementary Interior Angles we have that $\angle BAD + \angle ADE$ equals two right angles.

But as $\angle BAD$ is right, so is $\angle ADE$.

And, from Opposite Sides and Angles of Parallelogram are Equal, so are $\angle ABE$ and $\angle BED$.

So $ADEB$ is equilateral and equiangular, and therefore, by definition, a square.

$\blacksquare$

Historical Note

This is Proposition 46 of Book I of Euclid's The Elements.