On a given straight line to describe a square.
Let AB be the given
thus it is required to describe a square on the straight line AB.
Let AC be drawn at right angles
to the straight line AB
from the point A on it,
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. [I. 31]
Therefore ADEB is a
therefore AB is equal to DE, and AD to BE [I. 34] . 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 that it is also right-angled.
For, since the straight line AD
falls upon the parallels AB,
the angles BAD, ADE are equal to two right angles. [I. 29]
But the angle BAD is right;
therefore the angle ADE is also right.
And in parallelogrammic areas the opposite sides and angles are equal to one another; [I. 34]
therefore each of the opposite angles ABE, BED is also right.
Therefore ADEB is right-angled.
And it was also proved equilateral.
Therefore it is a square; and it is described on the straight line AB. Q.E.F.
Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)
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