Kitap 2
2
If a straight line be cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole.
2
For let the straight line AB be cut at random at the point C; I say that the rectangle contained by AB, BC together with the rectangle contained by BA, AC is equal to the square on AB.
2
For let the square ADEB be described on AB [I. 46], and let CF be drawn through C parallel to either AD or BE. [I. 31]
2
Then AE is equal to AF, CE.
2
Now AE is the square on AB;
2
AF is the rectangle contained by BA, AC, for it is contained by DA, AC, and AD is equal to AB; and CE is the rectangle AB, BC, for BE is equal to AB.
2
Therefore the rectangle BA, AC together with the rectangle AB, BC is equal to the square on AB.
2
Therefore etc. Q. E. D.
Proposition 3.
3
If a straight line be cut at random, the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment.
3
For let the straight line AB be cut at random at C; I say that the rectangle contained by AB, BC is equal to the rectangle contained by AC, CB together with the square on BC.
3
For let the square CDEB be described on CB; [I. 46] let ED be drawn through to F, and through A let AF be drawn parallel to either CD or BE. [I. 31]
3
Then AE is equal to AD, CE.
3
Now AE is the rectangle contained by AB, BC, for it is contained by AB, BE, and BE is equal to BC;
3
AD is the rectangle AC, CB, for DC is equal to CB; and DB is the square on CB. Therefore the rectangle contained by AB, BC is equal to the rectangle contained by AC, CB together with the square on BC.
3
Therefore etc. Q. E. D.
Proposition 4.
4
If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.
4
For let the straight line AB be cut at random at C; I say that the square on AB is equal to the squares on AC, CB and twice the rectangle contained by AC, CB.
