Proposition 14a
Theorem
Alternate name(s): equidistant chords theorem.
In equal circles, equal chords (ￜAB ￜ = ￜCD ￜ ) are equally distant from the center.
Commentary
1. Given a circle centered at O, let AB and CD be two equal 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.
3. Then these distances are equal (ￜOE ￜ = ￜOF 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 next proposition, Book 3 Proposition 14b, is the converse of this proposition.
7. Book 3 Proposition 15b covers the case when the chords are not the same length.
2. From O construct
3. Then these distances are equal (
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 next proposition, Book 3 Proposition 14b, is the converse of this proposition.
7. Book 3 Proposition 15b covers the case when the chords are not the same length.
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.
