Proposition 27
Theorem
In equal circles, angles at the centers (∠AOC , ∠DPF ) or at the circumferences (∠AGC , ∠DHF ), which stand on equal arcs (ABC , DEF ), are equal.
Commentary
1. Let circle ABC with center O and circle DEF with center P be equal, and let arc ABC and arc DEF be equal.
2. The central angles∠AOC and ∠DPF and the inscribed angles ∠AGC and ∠DHF all stand on the equal arcs ABC and DEF .
3. Then∠AOC and ∠DPF are equal, and ∠AGC and ∠DHF are equal.
4. This proposition is the converse of Euclid Book 3 Propositions 26a and 26b.
2. The central angles
3. Then
4. This proposition is the converse of Euclid Book 3 Propositions 26a and 26b.
Original statement
ἐν τοῖς ἴσοις κύκλοις αἱ ἐπὶ ἴσων πϵριϕϵρϵιῶν βϵβηκυῖαι γωνίαι ἴσαι ἀλλήλαις ϵἰσίν, ἐάν τϵ πρὸς τοῖς κέντροις ἐάν τϵ πρὸς ταῖς πϵριϕϵρϵίαις ὦσι βϵβηκυῖαι.
English translation
In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences.
