---
book: 6
number: 22
id: "VI.22"
kind: "theorem"
uses: ["[[book-6/proposition-11]]", "[[book-5/proposition-22]]", "[[book-6/proposition-19]]", "[[book-5/proposition-11]]", "[[book-6/proposition-12]]", "[[book-6/proposition-18]]", "[[book-5/proposition-9]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.22"
license: "CC-BY-SA-4.0"
---

# VI.22

*If four straight lines be proportional*, *the rectilineal figures similar and similarly described upon them will also be proportional*; *and*, *if the rectilineal figures similar and similarly described upon them be proportional*, *the straight lines will themselves also be proportional*.

## Proof

Let the four straight lines *AB*, *CD*, *EF*, *GH* be proportional, so that, as *AB* is to *CD*, so is *EF* to *GH*, and let there be described on *AB*, *CD* the similar and similarly situated rectilineal figures *KAB*, *LCD*, and on *EF*, *GH* the similar and similarly situated rectilineal figures *MF*, *NH*; I say that, as *KAB* is to *LCD*, so is *MF* to *NH*.

For let there be taken a third proportional *O* to *AB*, *CD*, and a third proportional *P* to *EF*, *GH*. [[book-6/proposition-11|VI. 11]]

Then since, as *AB* is to *CD*, so is *EF* to *GH*, and, as *CD* is to *O*, so is *GH* to *P*, therefore, *ex aequali*, as *AB* is to *O*, so is *EF* to *P*. [[book-5/proposition-22|V. 22]]

But, as *AB* is to *O*, so is *KAB* to *LCD*, [[book-6/proposition-19|VI. 19, Por.]] and, as *EF* is to *P*, so is *MF* to *NH*; therefore also, as *KAB* is to *LCD*, so is *MF* to *NH*. [[book-5/proposition-11|V. 11]]

Next, let *MF* be to *NH* as *KAB* is to *LCD*; I say also that, as *AB* is to *CD*, so is *EF* to *GH*.

For, if *EF* is not to *GH* as *AB* to *CD*, let *EF* be to *QR* as *AB* to *CD*, [[book-6/proposition-12|VI. 12]] and on *QR* let the rectilineal figure *SR* be described similar and similarly situated to either of the two *MF*, *NH*. [[book-6/proposition-18|VI. 18]]

Since then, as *AB* is to *CD*, so is *EF* to *QR*, and there have been described on *AB*, *CD* the similar and similarly situated figures *KAB*, *LCD*, and on *EF*, *QR* the similar and similarly situated figures *MF*, *SR*, therefore, as *KAB* is to *LCD*, so is *MF* to *SR*.

But also, by hypothesis, as *KAB* is to *LCD*, so is *MF* to *NH*; therefore also, as *MF* is to *SR*, so is *MF* to *NH*. [[book-5/proposition-11|V. 11]]

Therefore *MF* has the same ratio to each of the figures *NH*, *SR*; therefore *NH* is equal to *SR*. [[book-5/proposition-9|V. 9]]

But it is also similar and similarly situated to it; therefore *GH* is equal to *QR*.

And, since, as *AB* is to *CD*, so is *EF* to *QR*, while *QR* is equal to *GH*, therefore, as *AB* is to *CD*, so is *EF* to *GH*.

Therefore etc. Q. E. D.