Proposition 21
Theorem
Angles (∠ACB , ∠ADB ) at the circumference that stand on the same arc of a circle are equal.
Commentary
1. Given a circle centered at O, let A, B, C and D be four points on the circumference.
2. ConnectCA and CB , constructing ∠ACB , and connect DA and DB , constructing ∠ADB .
3. Then∠ACB = ∠ADB .
4. The previous proposition, Book 3 Proposition 20, covers the case when one angle is at the circumference and the other is at the center.
2. Connect
3. Then
4. The previous proposition, Book 3 Proposition 20, covers the case when one angle is at the circumference and the other is at the center.
Original statement
ἐν κύκλῳ αἱ ἐν τῷ αὐτῷ τμήματι γωνίαι ἴσαι ἀλλήλαις ϵἰσίν.
English translation
In a circle the angles in the same segment are equal to one another.
