---
book: 2
number: 2
id: "II.2"
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.2"
license: "CC-BY-SA-4.0"
---

# II.2

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

## Proof

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

For let the square *ADEB* be described on *AB* [[book-1/proposition-46|I. 46]], and let *CF* be drawn through *C* parallel to either *AD* or *BE*. [[book-1/proposition-31|I. 31]]

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

Now *AE* is the square on *AB*;

*AF* is the rectangle contained by *BA*, *AC*, for it is contained by *DA*, *AC*, and *AD* is equal to *AB*; and *CE* is the rectangle *AB*, *BC*, for *BE* is equal to *AB*.

Therefore the rectangle *BA*, *AC* together with the rectangle *AB*, *BC* is equal to the square on *AB*.

Therefore etc. Q. E. D.