On a given straight line to describe a square.
be the given straight line; thus it is required to describe a square on the straight line AB
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
parallel to AB
, and through the point B
be drawn parallel to AD
. [I. 31
is a parallelogram; therefore AB is equal to DE, and AD to BE. [I. 34]
is equal to AD
; therefore the four straight lines BA, AD, DE, EB are equal to one another;
therefore the parallelogram ADEB
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.