---
book: 6
number: 23
id: "VI.23"
kind: "theorem"
uses: ["[[book-6/proposition-12]]", "[[book-6/proposition-1]]", "[[book-5/proposition-11]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.23"
license: "CC-BY-SA-4.0"
---

# VI.23

*Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides*.

## Proof

Let *AC*, *CF* be equiangular parallelograms having the angle *BCD* equal to the angle *ECG*; I say that the parallelogram *AC* has to the parallelogram *CF* the ratio compounded of the ratios of the sides.

For let them be placed so that *BC* is in a straight line with *CG*; therefore *DC* is also in a straight line with *CE*.

Let the parallelogram *DG* be completed; let a straight line *K* be set out, and let it be contrived that, as *BC* is to *CG*, so is *K* to *L*, and, as *DC* is to *CE*, so is *L* to *M*. [[book-6/proposition-12|VI. 12]]

Then the ratios of *K* to *L* and of *L* to *M* are the same as the ratios of the sides, namely of *BC* to *CG* and of *DC* to *CE*.

But the ratio of *K* to *M* is compounded of the ratio of *K* to *L* and of that of *L* to *M*; so that *K* has also to *M* the ratio compounded of the ratios of the sides.

Now since, as *BC* is to *CG*, so is the parallelogram *AC* to the parallelogram *CH*, [[book-6/proposition-1|VI. 1]] while, as *BC* is to *CG*, so is *K* to *L*, therefore also, as *K* is to *L*, so is *AC* to *CH*. [[book-5/proposition-11|V. 11]]

Again, since, as *DC* is to *CE*, so is the parallelogram *CH* to *CF*, [[book-6/proposition-1|VI. 1]] while, as *DC* is to *CE*, so is *L* to *M*, therefore also, as *L* is to *M*, so is the parallelogram *CH* to the parallelogram *CF*. [[book-5/proposition-11|V. 11]]

Since then it was proved that, as *K* is to *L*, so is the parallelogram *AC* to the parallelogram *CH*, and, as *L* is to *M*, so is the parallelogram *CH* to the parallelogram *CF*, therefore, ex aequali, as *K* is to *M*, so is *AC* to the parallelogram *CF*.

But *K* has to *M* the ratio compounded of the ratios of the sides; therefore *AC* also has to *CF* the ratio compounded of the ratios of the sides.

Therefore etc. Q. E. D.