(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

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 line AB.

Let AC be drawn 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.
[I. 31]

Therefore ADEB is a
parallelogram;

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,
DE,

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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