Proposition 15b
Theorem
Given two chords, the longer chord (CD ) is closer to the center than the shorter chord (EF ).
Commentary
1. Given a circle centered at O, let AB be the diameter of the circle.
2. From O constructOG perpendicular to a chord CD at G and OH perpendicular to a chord EF at H, so that ￜOG ￜ and ￜOH ￜ are the distances from the center to the two chords.
3. LetCD be longer than EF .
4. ThenￜOG ￜ < ￜOH ￜ .
5. The previous proposition, Book 3 Proposition 15a, is the converse of this proposition.
6. Book 3 Proposition 14a covers the case when the chords are equal in length.
2. From O construct
3. Let
4. Then
5. The previous proposition, Book 3 Proposition 15a, is the converse of this proposition.
6. Book 3 Proposition 14a covers the case when the chords are equal in length.
Original statement
ἐν κύκλῳ μϵγίστη μὲν ἡ διάμϵτρος τῶν δὲ ἄλλων ἀϵὶ ἡ ἔγγιον τοῦ κέντρου τῆς ἀπώτϵρον μϵίζων ἐστίν.
English translation
Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the centre is always greater than the more remote.
