Proposition 4
Theorem
If two chords (AB , CD ) of a circle bisect each other, then they are both diameters.
Commentary
1. Given a circle centered at O, let AB and CD be two chords bisecting each other.
2. ThenAB and CD are both diameters and they intersect at the center of the circle.
3. Euclid stated the proposition in a negative way, while this site uses the contrapositive of his statement.
2. Then
3. Euclid stated the proposition in a negative way, while this site uses the contrapositive of his statement.
Original statement
ἐὰν ἐν κύκλῳ δύο ϵὐθϵῖαι τέμνωσιν ἀλλήλας μὴ διὰ τοῦ κέντρου οὖσαι, οὐ τέμνουσιν ἀλλήλας δίχα.
English translation
If in a circle two straight lines cut one another which are not through the centre, they do not bisect one another.
