---
book: 6
number: 19
id: "VI.19"
kind: "theorem"
uses: ["[[book-6/proposition-11]]", "[[book-5/proposition-16]]", "[[book-5/proposition-11]]", "[[book-6/proposition-15]]", "[[book-6/proposition-1]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.19"
license: "CC-BY-SA-4.0"
---

# VI.19

*Similar triangles are to one another in the duplicate ratio of the corresponding sides*.

## Proof

Let *ABC*, *DEF* be similar triangles having the angle at *B* equal to the angle at *E*, and such that, as *AB* is to *BC*, so is *DE* to *EF*, so that *BC* corresponds to *EF*; [[book-5/definitions#Definition 11|V. Def. 11]] I say that the triangle *ABC* has to the triangle *DEF* a ratio duplicate of that which *BC* has to *EF*.

For let a third proportional *BG* be taken to *BC*, *EF*, so that, as *BC* is to *EF*, so is *EF* to *BG*; [[book-6/proposition-11|VI. 11]] and let *AG* be joined.

Since then, as *AB* is to *BC*, so is *DE* to *EF*, therefore, alternately, as *AB* is to *DE*, so is *BC* to *EF*. [[book-5/proposition-16|V. 16]]

But, as *BC* is to *EF*, so is *EF* to *BG*; therefore also, as *AB* is to *DE*, so is *EF* to *BG*. [[book-5/proposition-11|V. 11]]

Therefore in the triangles *ABG*, *DEF* the sides about the equal angles are reciprocally proportional.

But those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal; [[book-6/proposition-15|VI. 15]] therefore the triangle *ABG* is equal to the triangle *DEF*.

Now since, as *BC* is to *EF*, so is *EF* to *BG*, and, if three straight lines be proportional, the first has to the third a ratio duplicate of that which it has to the second, [[book-5/definitions#Definition 9|V. Def. 9]] therefore *BC* has to *BG* a ratio duplicate of that which *CB* has to *EF*.

But, as *CB* is to *BG*, so is the triangle *ABC* to the triangle *ABG*; [[book-6/proposition-1|VI. 1]] therefore the triangle *ABC* also has to the triangle *ABG* a ratio duplicate of that which *BC* has to *EF*.

But the triangle *ABG* is equal to the triangle *DEF*; therefore the triangle *ABC* also has to the triangle *DEF* a ratio duplicate of that which *BC* has to *EF*.

Therefore etc.

Porism. From this it is manifest that, if three straight lines be proportional, then, as the first is to the third, so is the figure described on the first to that which is similar and similarly described on the second. Q. E. D.