---
book: 6
number: 9
id: "VI.9"
kind: "construction"
uses: ["[[book-1/proposition-3]]", "[[book-1/proposition-31]]", "[[book-6/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.9"
license: "CC-BY-SA-4.0"
---

# VI.9

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

## Proof

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*. [[book-1/proposition-3|I. 3]]

Let *BC* be joined, and through *D* let *DF* be drawn parallel to it. [[book-1/proposition-31|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*. [[book-6/proposition-2|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.