---
book: 3
number: 21
id: "III.21"
kind: "theorem"
uses: ["[[book-3/proposition-20]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.21"
license: "CC-BY-SA-4.0"
---

# III.21

*In a circle the angles in the same segment are equal to one another*.

## Proof

Let *ABCD* be a circle, and let the angles *BAD*, *BED* be angles in the same segment *BAED*; I say that the angles *BAD*, *BED* are equal to one another.

For let the centre of the circle *ABCD* be taken, and let it be *F*; let *BF*, *FD* be joined.

Now, since the angle *BFD* is at the centre, and the angle *BAD* at the circumference, and they have the same circumference *BCD* as base, therefore the angle *BFD* is double of the angle *BAD*. [[book-3/proposition-20|III. 20]]

For the same reason the angle *BFD* is also double of the angle *BED*; therefore the angle *BAD* is equal to the angle *BED*.

Therefore etc. Q. E. D.