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.
SI recta linea secta sit vtcunque: Rectangula, quæ sub tota, & quolibet segmentorum comprehenduntur, æqualia sunt ei, quod à tota sit, quadrato.
一直線。任兩分之。其元線上直角方形。與元線偕兩分線、兩矩內直角形幷、等。
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.
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]
Then AE is equal to AF, CE.
Now AE is the square on AB;
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.
Therefore the rectangle BA, AC together with the rectangle AB, BC is equal to the square on AB.
Complete text
Book I