The perpendicular from the centre to the chord bisects the chord

In the diagram below, AB is the chord of a circle with centre O.

OM is the perpendicular from the centre to the chord.

Look at triangles OAM and OBM.

The hypotenuses (OA and OB) are the same, as they are both the radius of the circle.

OM is common to both triangles.

OMA and OMB are both right angles.

Triangles OAM and OBM are congruent (RHS), so it follows that AM = MB.

Therefore, M is the midpoint of AB, and the chord has been bisected.