| In a circle the angles in the same segment equal one another. | ||
|
Let ABCD be a circle, and let the angles BAD and BED be angles in the same segment BAED.
I say that the angles BAD and BED equal one another. | ||
| Take the center F of the circle ABCD, and join BF and FD. | III.1 | |
| Now, since the angle BFD is at the center, and the angle BAD at the circumference, and they have the same circumference BCD as base, therefore the angle BFD is double the angle BAD. | III.20 | |
| For the same reason the angle BFD is also double the angle BED. Therefore the angle BAD equals the angle BED. | ||
| Therefore in a circle the angles in the same segment equal one another. | ||
| Q.E.D. | ||
Book III Introduction - Proposition III.20 - Proposition III.22.