---
book: 6
number: 11
id: "VI.11"
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.11"
license: "CC-BY-SA-4.0"
---

# VI.11

*To two given straight lines to find a third proportional*.

## Proof

Let *BA*, *AC* be the two given straight lines, and let them be placed so as to contain any angle; thus it is required to find a third proportional to *BA*, *AC*.

For let them be produced to the points *D*, *E*, and let *BD* be made equal to *AC*; [[book-1/proposition-3|I. 3]] let *BC* be joined, and through *D* let *DE* be drawn parallel to it. [[book-1/proposition-31|I. 31]]

Since, then, *BC* has been drawn parallel to *DE*, one of the sides of the triangle *ADE*, proportionally, as *AB* is to *BD*, so is *AC* to *CE*. [[book-6/proposition-2|VI. 2]]

But *BD* is equal to *AC*; therefore, as *AB* is to *AC*, so is *AC* to *CE*.

Therefore to two given straight lines *AB*, *AC* a third proportional to them, *CE*, has been found. Q. E. F.