If a diameter is perpendicular to a chord, it bisects the chord

Definition

Substitution

AP + AP = CQ + CQ

AP = CQ

AP + PB = CQ + QD

OP ⟂ AB and OQ ⟂ CD

Equal arcs have equal chords

Algebra

AB = AP + PB and CD = CQ + QD

OP bisects AB and OQ bisects CD

Substitution

AP = PB and CQ = QD

AB = CD

Given

Segment addition theorem