On a given straight line to describe a square.
Let AB be the given straight line; thus it is required to describe a square on
the straight AB.
Let AC be dawn at right angles to the straight line AB from the point A on it,
[I.11],
and let AD be made equal to AB;
through the point D let DE be drawn parallel to AB,
and through the point B let BE be drawn parallel to AD.
Therefore ADEB is a parallelogram;
therefore AB is equal to DE, and AD to BE.
But AB is equal to AD;
therefore the four straight lines BA, AD, DE, EB are equal to one another;
therefore the parallelogram ADEB is equilateral.
I say next it is also right-angled.
For, since the straight line AD falls upon the parallels AB, DE,
the angles BAD, ADE are equal to two right angles.
But the angle BAD is right;
therefore the angle ADE is also right.
And in parallelogramic areas the opposite sides and angles are equal to one
another;
therefore each of the opposite angles ABE, BED is also right.
Therefore ADEB is right-angled.
And it was proved equilateral..
Therefore it is a square; and it is described on the straight line AB.
Q.E.F