---
book: 1
number: 9
id: "I.9"
kind: "construction"
uses: ["[[book-1/proposition-3]]", "[[book-1/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.9"
license: "CC-BY-SA-4.0"
---

# I.9

To bisect a given rectilineal angle.

![I.9](figures/I-9.svg)

## Proof

Let the angle *BAC* be the given rectilineal angle.

Thus it is required to bisect it.

Let a point *D* be taken at random on *AB*; let *AE* be cut off from *AC* equal to *AD*; [[book-1/proposition-3|I. 3]] let *DE* be joined, and on *DE* let the equilateral triangle *DEF* be constructed; let *AF* be joined.

I say that the angle *BAC* has been bisected by the straight line *AF*.

For, since *AD* is equal to *AE*, and *AF* is common, the two sides *DA*, *AF* are equal to the two sides *EA*, *AF* respectively.

And the base *DF* is equal to the base *EF*; therefore the angle *DAF* is equal to the angle *EAF*. [[book-1/proposition-8|I. 8]]

Therefore the given rectilineal angle *BAC* has been bisected by the straight line *AF*.

Q. E. F.