Proposition 3a
Theorem
Alternate name(s): perpendicular chord bisector theorem.
If a line (AB ) passing through the center of a circle bisects a chord (CD ), which does not pass through the center, then these two lines are perpendicular.
Commentary
1. Given a circle centered at O, let CD be a chord that does not pass through the center.
2. LetAB be a diameter (a chord passing through the center O) that bisects CD .
3. ThenAB and CD are perpendicular.
4. This proposition is known as the perpendicular chord bisector theorem, and Book 3 Proposition 3b is the converse of this proposition.
2. Let
3. Then
4. This proposition is known as the perpendicular chord bisector theorem, and Book 3 Proposition 3b is the converse of this proposition.
Original statement
ἐὰν ἐν κύκλῳ ϵὐθϵῖά τις διὰ τοῦ κέντρου ϵὐθϵῖάν τινα μὴ διὰ τοῦ κέντρου δίχα τέμνῃ, καὶ πρὸς ὀρθὰς αὐτὴν τέμνϵι: καὶ ἐὰν πρὸς ὀρθὰς αὐτὴν τέμνῃ, καὶ δίχα αὐτὴν τέμνϵι.
English translation
If in a circle a straight line through the centre bisects a straight line not through the centre, it also cuts it at right angles; and if it cuts it at right angles, it also bisects it.
