Proposition 26a
Theorem
In equal circles, equal angles (∠AOC , ∠DPF ) at the centers stand on equal arcs (ABC , DEF ).
Under Development
Last updated on: 2022-07-28.
Commentary
1. Let circle ABC with center O and circle DEF with center P be equal.
2. Let the angles at the centers∠AOC and ∠DPF be equal. These are called central angles in modern terminology.
3. Then the arcsABC and DEF on which the two central angles stand are equal.
4. This proposition covers the case of angles with vertices at the centers of the circles while the next proposition, Euclid Book 3 Proposition 26b, covers the case of angles with vertices on the circumferences of the circles.
5. This proposition is a partial converse of Euclid Book 3 Proposition 27.
2. Let the angles at the centers
3. Then the arcs
4. This proposition covers the case of angles with vertices at the centers of the circles while the next proposition, Euclid Book 3 Proposition 26b, covers the case of angles with vertices on the circumferences of the circles.
5. This proposition is a partial converse of Euclid Book 3 Proposition 27.
Original statement
ἐν τοῖς ἴσοις κύκλοις αἱ ἴσαι γωνίαι ἐπὶ ἴσων πϵριϕϵρϵιῶν βϵβήκασιν, ἐάν τϵ πρὸς τοῖς κέντροις ἐάν τϵ πρὸς ταῖς πϵριϕϵρϵίαις ὦσι βϵβηκυῖαι.
English translation
In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences.
