---
book: 10
number: 109
id: "X.109"
kind: "theorem"
uses: ["[[book-10/proposition-13]]", "[[book-10/proposition-73]]", "[[book-10/proposition-92]]", "[[book-10/proposition-95]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.109"
license: "CC-BY-SA-4.0"
---

# X.109

*If from a medial area a rational area be subtracted, there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole*.

## Proof

For from the medial area *BC* let the rational area *BD* be subtracted.

I say that the side of the remainder *EC* becomes one of two irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.

For let a rational straight line *FG* be set out, and let the areas be similarly applied.

It follows then that *FH* is rational and incommensurable in length with *FG*, while *KF* is rational and commensurable in length with *FG*; therefore *FH*, *FK* are rational straight lines commensurable in square only; [[book-10/proposition-13|X. 13]] therefore *KH* is an apotome, and *FK* the annex to it. [[book-10/proposition-73|X. 73]]

Now the square on *HF* is greater than the square on *FK* either by the square on a straight line commensurable with *HF* or by the square on a straight line incommensurable with it.

If then the square on *HF* is greater than the square on *FK* by the square on a straight line commensurable with *HF*, while the annex *FK* is commensurable in length with the rational straight line *FG* set out, *KH* is a second apotome. [[book-10/definitions#Definition 2 (part 3)|X. Deff. III. 2]]

But *FG* is rational; so that the side of *LH*, that is, of *EC*, is a first apotome of a medial straight line. [[book-10/proposition-92|X. 92]]

But, if the square on *HF* is greater than the square on *FK* by the square on a straight line incommensurable with *HF*, while the annex *FK* is commensurable in length with the rational straight line *FG* set out, *KH* is a fifth apotome; [[book-10/definitions#Definition 5 (part 3)|X. Deff. III. 5]] so that the side of *EC* is a straight line which produces with a rational area a medial whole. [[book-10/proposition-95|X. 95]]