---
book: 6
number: 12
id: "VI.12"
kind: "construction"
uses: ["[[book-1/proposition-31]]", "[[book-6/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.12"
license: "CC-BY-SA-4.0"
---

# VI.12

*To three given straight lines to find a fourth proportional*.

## Proof

Let *A*, *B*, *C* be the three given straight lines; thus it is required to find a fourth proportional to *A*, *B*, *C*.

Let two straight lines *DE*, *DF* be set out containing any angle *EDF*; let *DG* be made equal to *A*, *GE* equal to *B*, and further *DH* equal to *C*; let *GH* be joined, and let *EF* be drawn through *E* parallel to it. [[book-1/proposition-31|I. 31]]

Since, then, *GH* has been drawn parallel to *EF*, one of the sides of the triangle *DEF*, therefore, as *DG* is to *GE*, so is *DH* to *HF*. [[book-6/proposition-2|VI. 2]]

But *DG* is equal to *A*, *GE* to *B*, and *DH* to *C*; therefore, as *A* is to *B*, so is *C* to *HF*.

Therefore to the three given straight lines *A*, *B*, *C* a fourth proportional *HF* has been found. Q. E. F.