To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

Let *AB* be the given straight line.

It is required to cut *AB* so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

Describe the square *ABDC* on *AB.* Bisect *AC* at the point *E,* and join *BE.*

Draw *CA* through to *F,* and make *EF* equal to *BE.* Describe the square *FH* on *AF,* and draw *GH* through to *K.*

I say that *AB* has been cut at *H* so that the rectangle *AB* by *BH* equals the square on *AH.*

Since the straight line *AC* has been bisected at *E,* and *FA* is added to it, the rectangle *CF* by *FA* together with the square on *AE* equals the square on *EF.*

But *EF* equals *EB,* therefore the rectangle *CF* by *FA* together with the square on *AE* equals the square on *EB.*

But the sum of the squares on *BA* and *AE* equals the square on *EB,* for the angle at *A* is right, therefore the rectangle *CF* by *FA* together with the square on *AE* equals the sum of the squares on *BA* and *AE.*

Subtract the square on *AE* from each. Therefore the remaining rectangle *CF* by *FA* equals the square on *AB.*

Now the rectangle *CF* by *FA* is *FK,* for *AF* equals *FG,* and the square on *AB* is *AD,* therefore *FK* equals *AD.*

Subtract *AK* from each. Therefore *FH,* which remains, equals *HD.*

And *HD* is the rectangle *AB* by *BH,* for *AB* equals *BD,* and *FH* is the square on *AH,* therefore the rectangle *AB* by *BH* equals the square on *HA.*

Therefore the given straight line *AB* has been cut at *H* so that the rectangle *AB* by *BH* equals the square on *HA.*

Q.E.F.

This construction is used in the proof of IV.10, which is later used to construct a regular pentagon. It accomplishes the same thing as the construction of proposition VI.30, which cuts a line into extreme and mean ratio, defined in VI.Def.3, and that construction is used later in XIII.17.

The difference between this proposition and VI.30 is a matter of terminology. Propositions dealing with ratios of lines are postponed until Book VI, but any ratio concerning lines can be converted into a statement about areas of rectangles. Proposition VI.16 states that the line The construction of this proposition cuts a line into two parts ## Construction stepsFor the purposes of cutting the line Altogether, there are six circles to be drawn, two lines connected, and one line extended. In order, they are as follows. Extend |