Proposition 20
Theorem
The angle (∠AOB ) at the center (O) of a circle is double any angle (∠ACB ) at the circumference standing on the same arc.
Commentary
1. Given a circle centered at O, let A, B and C be three points on the circumference.
2. ConnectOA and OB , constructing ∠AOB , and connect CA and CB , constructing ∠ACB .
3. Then∠AOB = 2 ∠ACB . In modern terminology, the central angle ∠AOB is twice the inscribed angle ∠ACB .
4. Note that∠AOB and ∠ACB must stand on the same arc for this proposition to hold. Euclid used the word "circumference" in this proposition to mean an arc or a part of the circumference, whereas in modern terminology "circumference" means the whole perimeter of the circle.
5. The next proposition, Book 3 Proposition 21, covers the case when two angles are both at the circumference.
2. Connect
3. Then
4. Note that
5. The next proposition, Book 3 Proposition 21, covers the case when two angles are both at the circumference.
Original statement
ἐν κύκλῳ ἡ πρὸς τῷ κέντρῳ γωνία διπλασίων ἐστὶ τῆς πρὸς τῇ πϵριϕϵρϵίᾳ, ὅταν τὴν αὐτὴν πϵριϕέρϵιαν βάσιν ἔχωσιν αἱ γωνίαι.
English translation
In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base.
