---
book: 4
number: 9
id: "IV.9"
kind: "construction"
uses: ["[[book-1/proposition-8]]", "[[book-1/proposition-6]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:4.prop.9"
license: "CC-BY-SA-4.0"
---

# IV.9

*About a given square to circumscribe a circle*.

## Proof

Let *ABCD* be the given square; thus it is required to circumscribe a circle about the square *ABCD*.

For let *AC*, *BD* be joined, and let them cut one another at *E*.

Then, since *DA* is equal to *AB*, and *AC* is common, therefore the two sides *DA*, *AC* are equal to the two sides *BA*, *AC*; and the base *DC* is equal to the base *BC*; therefore the angle *DAC* is equal to the angle *BAC*. [[book-1/proposition-8|I. 8]]

Therefore the angle *DAB* is bisected by *AC*.

Similarly we can prove that each of the angles *ABC*, *BCD*, *CDA* is bisected by the straight lines *AC*, *DB*.

Now, since the angle *DAB* is equal to the angle *ABC*, and the angle *EAB* is half the angle *DAB*, and the angle *EBA* half the angle *ABC*, therefore the angle *EAB* is also equal to the angle *EBA*; so that the side *EA* is also equal to *EB*. [[book-1/proposition-6|I. 6]]

Similarly we can prove that each of the straight lines *EA*, *EB* is equal to each of the straight lines *EC*, *ED*.

Therefore the four straight lines *EA*, *EB*, *EC*, *ED* are equal to one another.

Therefore the circle described with centre *E* and distance one of the straight lines *EA*, *EB*, *EC*, *ED* will pass also through the remaining points; and it will have been circumscribed about the square *ABCD*.

Let it be circumscribed, as *ABCD*.

Therefore about the given square a circle has been circumscribed. Q. E. F.