---
book: 2
number: 3
id: "II.3"
kind: "theorem"
uses: ["[[book-1/proposition-46]]", "[[book-1/proposition-31]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:2.prop.3"
license: "CC-BY-SA-4.0"
---

# II.3

*If a straight line be cut at random*, *the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment*.

## Proof

For let the straight line *AB* be cut at random at *C*; I say that the rectangle contained by *AB*, *BC* is equal to the rectangle contained by *AC*, *CB* together with the square on *BC*.

For let the square *CDEB* be described on *CB*; [[book-1/proposition-46|I. 46]] let *ED* be drawn through to *F*, and through *A* let *AF* be drawn parallel to either *CD* or *BE*. [[book-1/proposition-31|I. 31]]

Then *AE* is equal to *AD*, *CE*.

Now *AE* is the rectangle contained by *AB*, *BC*, for it is contained by *AB*, *BE*, and *BE* is equal to *BC*;

*AD* is the rectangle *AC*, *CB*, for *DC* is equal to *CB*; and *DB* is the square on *CB*. Therefore the rectangle contained by *AB*, *BC* is equal to the rectangle contained by *AC*, *CB* together with the square on *BC*.

Therefore etc. Q. E. D.