Book 1 – Proposition 46

On a given straight line to describe a square.
[Desmos graphs can be found here]

Let AB be the given straight line. We will construct a square on AB.

Use Proposition 11 (constructing a straight line at a right angle to a given line at a point) to draw AC at a right angle to AB at the point A.

Select point D on AC such that AD is equal to AB.
Using Proposition 31 (covers drawing a straight line through a point that is parallel to a given line), through point D draw DE that is parallel to AB.
Also, through point B draw BE that is parallel to AD.

ADEB is a parallelogram, and by Proposition 34 (in a parallelogram opposite sides are equal and opposite angles are equal) AB is equal to DE and AD is equal to BE.

Since AB is equal to AD, all four sides of the parallelogram are equal to each other. Therefore the parallelogram ADEB is equilateral.

Now we will show that the parallelogram is also right-angled, making it a square.
Since AD falls upon the parallel lines AB and DE, Proposition 29 (if a line falls on two parallel lines, then the interior angles on the same side are equal to two right angles) tells us that the two angles BAD and ADE are together equal to two right angles.
But angle BAD is a right angle, so angle ADE is also a right angle.
Using Proposition 34 again, the two angles opposite BAD and ADE are equal to those two angles. So, angles ABE and BED are also each right angles.

Since ADEB is right-angles and equilateral, it is a square described on the straight line AB.

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Next proposition … the Pythagorean Theorem!