From a given straight line to cut off a prescribed part.

Let AB be the given straight line; thus it is required to cut off from AB a prescribed part.

Let the third part be that prescribed.

Let a straight line AC be drawn through from A containing with AB any angle; let a point D be taken at random on AC, and let DE, EC be made equal to AD. [I. 3]

Let BC be joined, and through D let DF be drawn parallel to it. [I. 31]

Then, since FD has been drawn parallel to BC, one of the sides of the triangle ABC, therefore, proportionally, as CD is to DA, so is BF to FA. [VI. 2]

But CD is double of DA; therefore BF is also double of FA; therefore BA is triple of AF.

Therefore from the given straight line AB the prescribed third part AF has been cut off. Q. E. F.