---
book: 2
number: 1
id: "II.1"
kind: "theorem"
uses: ["[[book-1/proposition-11]]", "[[book-1/proposition-3]]", "[[book-1/proposition-31]]", "[[book-1/proposition-34]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:2.prop.1"
license: "CC-BY-SA-4.0"
---

# II.1

*If there be two straight lines*, *and one of them be cut into any number of segments whatever*, *the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments*.

## Proof

Let *A*, *BC* be two straight lines, and let *BC* be cut at random at the points *D*, *E*; I say that the rectangle contained by *A*, *BC* is equal to the rectangle contained by *A*, *BD*, that contained by *A*, *DE* and that contained by *A*, *EC*.

For let *BF* be drawn from *B* at right angles to *BC*; [[book-1/proposition-11|I. 11]] let *BG* be made equal to *A*, [[book-1/proposition-3|I. 3]] through *G* let *GH* be drawn parallel to *BC*, [[book-1/proposition-31|I. 31]] and through *D*, *E*, *C* let *DK*, *EL*, *CH* be drawn parallel to *BG*.

Then *BH* is equal to *BK*, *DL*, *EH*.

Now *BH* is the rectangle *A*, *BC*, for it is contained by *GB*, *BC*, and *BG* is equal to *A*;

*BK* is the rectangle *A*, *BD*, for it is contained by *GB*, *BD*, and *BG* is equal to *A*; and *DL* is the rectangle *A*, *DE*, for *DK*, that is *BG* [[book-1/proposition-34|I. 34]], is equal to *A*.

Similarly also *EH* is the rectangle *A*, *EC*.

Therefore the rectangle *A*, *BC* is equal to the rectangle *A*, *BD*, the rectangle *A*, *DE* and the rectangle *A*, *EC*.

Therefore etc. Q. E. D.