---
book: 1
number: 10
id: "I.10"
kind: "construction"
uses: ["[[book-1/proposition-1]]", "[[book-1/proposition-9]]", "[[book-1/proposition-4]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.10"
license: "CC-BY-SA-4.0"
---

# I.10

To bisect a given finite straight line.

![I.10](figures/I-10.svg)

## Proof

Let *AB* be the given finite straight line.

Thus it is required to bisect the finite straight line *AB*.

Let the equilateral triangle *ABC* be constructed on it, [[book-1/proposition-1|I. 1]] and let the angle *ACB* be bisected by the straight line *CD*; [[book-1/proposition-9|I. 9]]

I say that the straight line *AB* has been bisected at the point *D*.

For, since *AC* is equal to *CB*, and *CD* is common, the two sides *AC*, *CD* are equal to the two sides *BC*, *CD* respectively; and the angle *ACD* is equal to the angle *BCD*; therefore the base *AD* is equal to the base *BD*. [[book-1/proposition-4|I. 4]]

Therefore the given finite straight line *AB* has been bisected at *D*.

Q. E. F.