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

# IV.1

*Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle*.

## Proof

Let *ABC* be the given circle, and *D* the given straight line not greater than the diameter of the circle; thus it is required to fit into the circle *ABC* a straight line equal to the straight line *D*.

Let a diameter *BC* of the circle *ABC* be drawn.

Then, if *BC* is equal to *D*, that which was enjoined will have been done; for *BC* has been fitted into the circle *ABC* equal to the straight line *D*.

But, if *BC* is greater than *D*, let *CE* be made equal to *D*, and with centre *C* and distance *CE* let the circle *EAF* be described; let *CA* be joined.

Then, since the point *C* is the centre of the circle *EAF*, *CA* is equal to *CE*.

But *CE* is equal to *D*; therefore *D* is also equal to *CA*.

Therefore into the given circle *ABC* there has been fitted *CA* equal to the given straight line *D*.