Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/73

Rh 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.

. From the demonstration it is manifest that every parallelogram which has one right angle has all its angles right angles.