---
book: 10
number: 106
id: "X.106"
kind: "theorem"
uses: ["[[book-10/proposition-77]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.106"
license: "CC-BY-SA-4.0"
---

# X.106

*A straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole*.

## Proof

Let *AB* be a straight line which produces with a rational area a medial whole, and *CD* commensurable with *AB*; I say that *CD* is also a straight line which produces with a rational area a medial whole.

For let *BE* be the annex to *AB*; therefore *AE*, *EB* are straight lines incommensurable in square which make the sum of the squares on *AE*, *EB* medial, but the rectangle contained by them rational. [[book-10/proposition-77|X. 77]]

Let the same construction be made.

Then we can prove, in manner similar to the foregoing, that *CF*, *FD* are in the same ratio as *AE*, *EB*, the sum of the squares on *AE*, *EB* is commensurable with the sum of the squares on *CF*, *FD*, and the rectangle *AE*, *EB* with the rectangle *CF*, *FD*; so that *CF*, *FD* are also straight lines incommensurable in square which make the sum of the squares on *CF*, *FD* medial, but the rectangle contained by them rational.

Therefore *CD* is a straight line which produces with a rational area a medial whole. [[book-10/proposition-77|X. 77]] Q. E. D.