The diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.

As long as a straight line has any two of the following five conditions, the other three conclusions can be deduced. It is called knowing two makes three (knowing two pushes three).

The best arc to bisect the chord.

The lower arc bisecting the chord (the first two together are: the two arcs bisecting the chord)

Bisect a chord (not a diameter)

Perpendicular to the chord

Pass through the center (or diameter)

Mathematical proof

As shown in the figure, in ⊙O, DC is the diameter, AB is the chord, AB⊥DC is at point E, AB and CD intersect at point E, and it is proved that AE=BE, arc AC= arc BC, arc AD= arc BD.

Diagram of vertical diameter theorem

Proof: Connect OA and OB to ⊙O at point A and point B respectively.

∵OA and OB are the radii of∵ O.

∴OA=OB

△ OAB is an isosceles triangle.

∵AB⊥DC

∴AE=BE, ∠AOE=∠BOE (isosceles triangle with three lines in one)

∴ arc AD= arc BD, ∠AOC=∠BOC.

∴ arc AC= arc BC