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

Let a straight line *AB* be cut at random at the point *C.*

I say that the sum of the squares on *AB* and *BC* equals twice the rectangle *AB* by *BC* plus the square on *CA.*

Describe the square *ADEB* on *AB,* and let the figure be drawn.

Then, since *AG* equals *GE,* add *CF* to each, therefore the whole *AF* equals the whole *CE.*

Therefore the sum of *AF* and *CE* is double *AF.*

But the sum of *AF* and *CE* equals the gnomon *KLM* plus the square *CF,* therefore the gnomon *KLM* plus the square *CF* is double *AF.*

But twice the rectangle *AB* by *BC* is also double *AF,* for *BF* equals *BC,* therefore the gnomon *KLM* plus the square *CF* equal twice the rectangle *AB* by *BC.*

Add *DG,* which is the square on *AC,* to each. Therefore the gnomon *KLM* plus the sum of the squares *BG* and *GD* equals twice the rectangle *AB* by *BC* plus the square on *AC.*

But the gnomon *KLM* plus the sum of the squares *BG* and *GD* equals the whole *ADEB* plus *CF,* which are squares described on *AB* and *BC.*

Therefore the sum of the squares on *AB* and *BC* equals twice the rectangle *AB* by *BC* plus the square on *CA.*

Therefore *if a straight line is cut at random, then the sum of the square on the whole and that on one of the segments equals twice the rectangle contained by the whole and the said segment plus the square on the remaining segment.*

Q.E.D.

This proposition is used later in Book II to prove proposition II.13, and it is used repeatedly in Book X.