Proposition 15a
Theorem
The diameter (AB ) is the longest chord in a circle; and of the others, a chord (CD ) which is closer to the center is longer than one (EF ) more remote.
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 closer to the center than EF , that is, ￜOG ￜ < ￜOH ￜ .
4. Then the diameterAB is the longest and ￜCD ￜ > ￜEF ￜ .
5. The next proposition, Book 3 Proposition 15b, is the converse of this proposition.
6. Book 3 Proposition 14b covers the case when the distances from chords to the center are equal.
2. From O construct
3. Let
4. Then the diameter
5. The next proposition, Book 3 Proposition 15b, is the converse of this proposition.
6. Book 3 Proposition 14b covers the case when the distances from chords to the center are equal.
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.
