---
book: 10
number: 45
id: "X.45"
kind: "theorem"
uses: ["[[book-10/proposition-39]]", "[[book-10/proposition-26]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.45"
license: "CC-BY-SA-4.0"
---

# X.45

*A major straight line is divided at one and the same point only*.

## Proof

Let *AB* be a major straight line divided at *C*, so that *AC*, *CB* are incommensurable in square and make the sum of the squares on *AC*, *CB* rational, but the rectangle *AC*, *CB* medial; [[book-10/proposition-39|X. 39]] 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 incommensurable in square and make the sum of the squares on *AD*, *DB* rational, but the rectangle contained by them medial.

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

Therefore a major straight line is not divided at different points; therefore it is only divided at one and the same point. Q. E. D.