Proposition 26b
Theorem
In equal circles, equal angles (∠AGC , ∠DHF ) at the circumferences stand on equal arcs (ABC , DEF ).
Commentary
1. Let circle ABC and circle DEF be equal.
2. Let the angles at the circumferences∠AGC and ∠DHF be equal. These are called inscribed angles in modern terminology.
3. Then the arcsABC and DEF on which the two inscribed angles stand are equal.
4. This proposition covers the case of angles with vertices on the circumferences of the circles while the previous proposition, Euclid Book 3 Proposition 26a, covers the case of angles with vertices at the centers of the circles.
5. This proposition is a partial converse of Euclid Book 3 Proposition 27.
2. Let the angles at the circumferences
3. Then the arcs
4. This proposition covers the case of angles with vertices on the circumferences of the circles while the previous proposition, Euclid Book 3 Proposition 26a, covers the case of angles with vertices at the centers 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.
