---
book: 10
number: 43
id: "X.43"
kind: "theorem"
uses: ["[[book-10/proposition-26]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.43"
license: "CC-BY-SA-4.0"
---

# X.43

*A first bimedial straight line is divided at one point only*.

## Proof

Let *AB* be a first bimedial straight line divided at *C*, so that *AC*, *CB* are medial straight lines commensurable in square only and containing a rational rectangle; I say that *AB* is not so divided at another point.

For, if possible, let it be divided at *D* also, so that *AD*, *DB* are also medial straight lines commensurable in square only and containing a rational rectangle.

Since, then, that by which twice the rectangle *AD*, *DB* differs from twice the rectangle *AC*, *CB* is that by which the squares on *AC*, *CB* differ from the squares on *AD*, *DB*, while twice the rectangle *AD*, *DB* differs from twice the rectangle *AC*, *CB* by a rational area—for both are rational— therefore the squares on *AC*, *CB* also differ from the squares on *AD*, *DB* by a rational area, though they are medial: which is absurd. [[book-10/proposition-26|x. 26]]

Therefore a first bimedial straight line is not divided into its terms at different points; therefore it is so divided at one point only.