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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. 1

1 any angle. The expression here and in the two following propositions is τυχοῦσα γωνία, corresponding exactly to τυχὸν σημεῖον which I have translated as “a point (taken) at random” ; but “an angle (taken) at random” would not be so appropriate where it is a question, not of taking any angle at all, but of drawing a straight line casually so as to make any angle with another straight line.

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