Solution.

From either extremity of the given right line A B draw a line A C perpendicular (XI), and equal to it (III); through C draw C D parallel to A B (XXXI), and through B draw B D parallel to A C; A D is the required square.

Demonstration.

Because A D is a parallelogram (const.), and the angle A a right angle, the angles C, D, and B are also right (153); and because A C is equal to A B (const.), and the sides C D and D B are equal to A B and A C (XXXIV), the four sides A B, A C, C D, D B are equal, therefore A D is a square.