Proposition 14b
Theorem
Alternate name(s): equidistant chords theorem.
In equal circles, chords (AB , CD ) which are equally distant from the center are equal.
Commentary
1. Given a circle centered at O, let AB and CD be two chords of the circle.
2. From O constructOE perpendicular to AB at E and OF perpendicular to CD at F, so that ￜOE ￜ and ￜOF ￜ are the distances from the center to the two chords. Suppose these distances are equal (ￜOE ￜ = ￜOF ￜ ).
3. ThenￜAB ￜ = ￜCD ￜ .
4. We represent the case of the same circle. The case of equal circles follows from this case.
5. This proposition is one part of the equidistant chord theorem.
6. The previous proposition, Book 3 Proposition 14a, is the converse of this proposition.
7. Book 3 Proposition 15a covers the case when the distances from the chords to the center are not equal.
2. From O construct
3. Then
4. We represent the case of the same circle. The case of equal circles follows from this case.
5. This proposition is one part of the equidistant chord theorem.
6. The previous proposition, Book 3 Proposition 14a, is the converse of this proposition.
7. Book 3 Proposition 15a covers the case when the distances from the chords to the center are not equal.
Original statement
ἐν κύκλῳ αἱ ἴσαι ϵὐθϵῖαι ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου, καὶ αἱ ἴσον ἀπέχουσαι ἀπὸ τοῦ κέντρου ἴσαι ἀλλήλαις ϵἰσίν.
English translation
In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal to one another.
