Proposition 28
Theorem
In equal circles, equal chords (AB , EF ) divide the circumferences into equal major arcs and minor arcs.
Commentary
1. Let circle ACBD and circle EGFH be equal and let AB and EF be two equal chords.
2. The chordAB divides circle ACBD into two parts: arc ACB and arc BDA . The chord EF divides circle EGFH into two parts: arc EGF and arc FHE .
3. In the figure above,ACB and EGF are the shorter arcs in the circles ACBD and EGFH while BDA and FHE are the longer arcs.
4. In this case, arcACB and arc EGF are equal and arc BDA and arc FHE are equal.
5. The next proposition, Euclid Book 3 Proposition 29, is a partial converse of this proposition.
2. The chord
3. In the figure above,
4. In this case, arc
5. The next proposition, Euclid Book 3 Proposition 29, is a partial converse of this proposition.
Original statement
ἐν τοῖς ἴσοις κύκλοις αἱ ἴσαι ϵὐθϵῖαι ἴσας πϵριϕϵρϵίας ἀϕαιροῦσι τὴν μὲν μϵίζονα τῇ μϵίζονι τὴν δὲ ἐλάττονα τῇ ἐλάττονι.
English translation
In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less.
