Construction of Square on Given Straight Line

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Theorem

In the words of Euclid:

On a given straight line to describe a square.

(The Elements: Book $\text{I}$: Proposition $46$)


Proof

Euclid-I-46.png

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, we have that $AD$ falls on the parallels $AB$ and $DE$.

So from Parallelism 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 theorem is Proposition $46$ of Book $\text{I}$ of Euclid's The Elements.


Sources