---
book: 6
number: 26
id: "VI.26"
kind: "theorem"
uses: ["[[book-1/proposition-31]]", "[[book-6/proposition-24]]", "[[book-5/proposition-11]]", "[[book-5/proposition-9]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.26"
license: "CC-BY-SA-4.0"
---

# VI.26

*If from a parallelogram there be taken away a parallelogram similar and similarly situated to the whole and having a common angle with it*, *it is about the same diameter with the whole*

## Proof

For from the parallelogram *ABCD* let there be taken away the parallelogram *AF* similar and similarly situated to *ABCD*, and having the angle *DAB* common with it; I say that *ABCD* is about the same diameter with *AF*.

For suppose it is not, but, if possible, let *AHC* be the diameter lt of *ABCD* gt, let *GF* be produced and carried through to *H*, and let *HK* be drawn through *H* parallel to either of the straight lines *AD*, *BC*. [[book-1/proposition-31|I. 31]]

Since, then, *ABCD* is about the same diameter with *KG*, therefore, as *DA* is to *AB*, so is *GA* to *AK*. [[book-6/proposition-24|VI. 24]]

But also, because of the similarity of *ABCD*, *EG*, as *DA* is to *AB*, so is *GA* to *AE*; therefore also, as *GA* is to *AK*, so is *GA* to *AE*. [[book-5/proposition-11|V. 11]]

Therefore *GA* has the same ratio to each of the straight lines *AK*, *AE*.

Therefore *AE* is equal to *AK* [[book-5/proposition-9|V. 9]], the less to the greater : which is impossible.

Therefore *ABCD* cannot but be about the same diameter with *AF*; therefore the parallelogram *ABCD* is about the same diameter with the parallelogram *AF*.

Therefore etc. Q. E. D.