---
book: 6
number: 27
id: "VI.27"
kind: "theorem"
uses: ["[[book-6/proposition-26]]", "[[book-1/proposition-43]]", "[[book-1/proposition-36]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.27"
license: "CC-BY-SA-4.0"
---

# VI.27

*Of all the parallelograms applied to the same straight line and deficient by parallelogrammic figures similar and similarly situated to that described on the half of the straight line*, *that parallelogram is greatest which is applied to the half of the straight line and is similar to the defect*.

## Proof

Let *AB* be a straight line and let it be bisected at *C*; let there be applied to the straight line *AB* the parallelogram *AD* deficient by the parallelogrammic figure *DB* described on the half of *AB*, that is, *CB*; I say that, of all the parallelograms applied to *AB* and deficient by parallelogrammic figures similar and similarly situated to *DB*, *AD* is greatest.

For let there be applied to the straight line *AB* the parallelogram *AF* deficient by the parallelogrammic figure *FB* similar and similarly situated to *DB*; I say that *AD* is greater than *AF*.

For, since the parallelogram *DB* is similar to the parallelogram *FB*, they are about the same diameter. [[book-6/proposition-26|VI. 26]]

Let their diameter *DB* be drawn, and let the figure be described.

Then, since *CF* is equal to *FE*, [[book-1/proposition-43|I. 43]] and *FB* is common, therefore the whole *CH* is equal to the whole *KE*.

But *CH* is equal to *CG*, since *AC* is also equal to *CB*. [[book-1/proposition-36|I. 36]]

Therefore *GC* is also equal to *EK*.

Let *CF* be added to each; therefore the whole *AF* is equal to the gnomon *LMN*; so that the parallelogram *DB*, that is, *AD*, is greater than the parallelogram *AF*.

Therefore etc.