Corollary: 3.01: Bisected Chord of a Circle Passes the Center
(related to Proposition: 3.01: Finding the Center of a given Circle)
(Corollary to Proposition 1 from Book 3 of Euclid's “Elements”)
So, from this, (it is) manifest that if any straight line in a circle cuts any (other) straight line in half, and at right angles, then the center of the circle is on the former (straight line). (Which is) the very thing it was required to do.
Modern Formulation
The straight line which bisects any chord of a circle perpendicularly passes through the center of the circle.
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Calahan
 @Casey
 @Fitzpatrick
References
Adapted from CC BYSA 3.0 Sources:
 Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
Adapted from (Public Domain)
 Casey, John: "The First Six Books of the Elements of Euclid"
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"