Proposition 46.
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.
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