Euclid's Elements Book II Proposition 2

If a straight line is cut at random, then the sum of the rectangles contained by the whole and each of the segments equals the square on the whole.

Let the straight line AB be cut at

random at the point C.

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

Describe the square ADEB on AB, and draw CF through C parallel to either AD or BE.

Then AE equals AF plus CE.

Now AE is the square on AB;

AF is the rectangle BA by AC, for it is contained by DA and AC, and AD equals AB;

and CE is the rectangle AB by BC, for BE equals AB.

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

Therefore if a straight line is cut at random, then the sum of the rectangles contained by the whole and each of the segments equals the square on the whole.

Q.E.D.

Proposition 1 <-- Proposition 2 --> Proposition 3