PROPOSITION 46. PROBLEM.
To describe a square on a given straight line.
Let AB be the given straight line : it is required to describe a square on AB.
From the point A draw AC
at right angles to AB; [I. 11.
and make AD equal to AB; [I. 3.
through D draw DE parallel to
AB ; and through B draw BE
parallel to AD. [I. 31.
ADEB shall be a square.
For ADEB is by construction
a parallelogram ;
therefore AB is equal to DE
and AD to BE. [I. 34.
But AB is equal to AD. [Construction.
Therefore the four straight lines BA,AD, DE, EB are equal to one another, and the parallelogram ADEB is
equilateral. [Axiom 1.
Likewise all its angles are right angles.
For since the straight line AD meets the parallels AB,DE, the angles BAD, ADE are together equal to two
right angles ; [I. 29.
but BAD is a right angle ; [Construction.
therefore also ADE is a right angle. [Axiom 3.
But the opposite angles of parallelograms are equal. [I. 34.
Therefore each of the opposite angles ABE, BED is a
right angle. [Axiom 1.
Therefore the figure ADEB is rectangular; and it has been shewn to be equilateral. Therefore it is a square. [Definition, 30. And it is described on the given straight line AB. q.e.f.
Corollary. From the demonstration it is manifest that every parallelogram which has one right angle has all its angles right angles.
