To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.
Let AB be the given straight line, C the given triangle and D the given rectilineal angle;
thus it is required to apply to the given straight line AB, in an angle equal to the angle D, a parallelogram equal to the given triangle C. Let the parallelogram BEFG be constructed equal to the triangle C, in the angle EBG which is equal to D [I. 42];
let it be placed so that BE is in a straight line with AB; letFG be drawn through to H, and let AH be drawn through A parallel to either BG or EF. [I. 31] Let HB be joined. Then, since the straight line HF falls upon the parallels
two right angles meet; [Post. 5]
Then HLKF is a parallelogram, HK is its diameter, and AG, ME are parallelograms. and LB, BF the so-called complements, about HK;
C has been applied to the given straight line AB, in the angle ABM which is equal to D.
Q. E. F. 1