AP = CQ

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

AP + PB = CQ + QD

Definition

Substitution

OP is perpendicular to AB and OQ is perpendicular to CD

AP + AP = CQ + CQ

Substitution

AB = AP + PB and CD = CQ + QD

Equal arcs have equal chords

OP bisects AB and OQ bisects CD

Algebra

AP = PB and CQ = QD

AB = CD

Given

Segment addition theorem