Proof: By Euclid
(related to Proposition: 3.21: Angles in Same Segment of Circle are Equal)
- For let the center of circle $ABCD$ have been found [Prop. 3.1], and let it be (at point) $F$.
- And let $BF$ and $FD$ have been joined.
- And since angle $BFD$ is at the center, and $BAD$ at the circumference, and they have the same circumference base $BCD$, angle $BFD$ is thus double $BAD$ [Prop. 3.20].
- So, for the same (reasons), $BFD$ is also double $BED$.
- Thus, $BAD$ (is) equal to $BED$.
- Thus, in a circle, angles in the same segment are equal to one another.
- (Which is) the very thing it was required to show.
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Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"