Proposition 3

If a straight line is cut at random, then the rectangle contained by the whole and one of the segments equals the sum of the rectangle contained by the segments and the square on the aforesaid segment.

Let the straight line AB be cut at random at C.

java applet or image

I say that the rectangle AB by BC equals the sum of the rectangle AC by CB and the square on BC.

I.46
I.31

Describe the square CDEB on CB. Draw ED through to F, and draw AF through A parallel to either CD or BE.

Then AE equals AD plus CE.

Now AE is the rectangle AB by BC, for it is contained by AB and BE, and BE equals BC; AD is the rectangle AC by CB, for DC equals CB; and DB is the square on CB.

Therefore the rectangle AB by BC equals the sum of the rectangle AC by CB and the square on BC.

Therefore if a straight line is cut at random, then the rectangle contained by the whole and one of the segments equals the sum of the rectangle contained by the segments and the square on the aforesaid segment.

Q.E.D.

Guide

 This proposition is another special case of II.1. In modern algebraic notation it says that if x = y + z, then xy = y2 + yz. Identities that are logically equivalent to this implication can be found by eliminating one of the three variables x, y, or z. Here are two of them.

(y + z) y = y2 + yz,
and
xy = y2 + y (x – y).

Use of this proposition

This proposition refers to lines and rectangles, but the analogous statement for numbers is used in a proposition in one of the Euclid’s books on number theory, namely, of proposition IX.15.