---
book: 10
number: 24
id: "X.24"
kind: "theorem"
uses: ["[[book-6/proposition-1]]", "[[book-10/proposition-11]]", "[[book-10/proposition-23]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.24"
license: "CC-BY-SA-4.0"
---

# X.24

*The rectangle contained by medial straight lines commensurable in length is medial*.

## Proof

For let the rectangle *AC* be contained by the medial straight lines *AB*, *BC* which are commensurable in length; I say that *AC* is medial.

For on *AB* let the square *AD* be described; therefore *AD* is medial.

And, since *AB* is commensurable in length with *BC*, while *AB* is equal to *BD*, therefore *DB* is also commensurable in length with *BC*; so that *DA* is also commensurable with *AC*. [[book-6/proposition-1|VI. 1]], [[book-10/proposition-11|X. 11]]

But *DA* is medial; therefore *AC* is also medial. [[book-10/proposition-23|X. 23, Por.]] Q. E. D.