Construction of Square on Given Straight Line
Theorem
In the words of Euclid:
- On a given straight line to describe a square.
(The Elements: Book $\text{I}$: Proposition $46$)
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, 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 proof is Proposition $46$ of Book $\text{I}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions