Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/78

54 PROPOSITION 2. THEOREM.

If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square on the whole line.

Let the straight line AB be divided into any two parts at the point C: the rectangle contained by AB, BC, together with the rectangle AB, AC, shall be equal to the square on AB.

[ Note. To avoid repeating the word contained too frequently, the rectangle contained by two straight lines AB, AC is sometimes simply called the rectangle AB, AC. ]

On AB describe the square ADEB; [I.46. and through C draw CF parallel to AD or BE. [I.31.

Then AE is equal to the rectangles AF, CE. But AE is the square on AB. And AF is the rectangle contained by BA, AC, for it is contained by DA, AC, of which DA is equal to BA; and CE is contained by AB, BC, for BE is equal to AB. Therefore the rectangle AB, AC, together with the rectangle AB, BC, is equal to the square on AB.

Wherefore, if a straight line &c.

PROPOSITION 3. THEOREM.

If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square on the aforesaid part.

Let the straight line AB be divided into any two parts at the point C: the rectangle AB, BC shall be equal to the rectangle AC, CB, together with the square on BC.