A chord BD is drawn in a circle in the middle O of the chord AC so that the arcs AB and CD are equal

A chord BD is drawn in a circle in the middle O of the chord AC so that the arcs AB and CD are equal, prove that O is the midpoint of the chord BD.

Let’s connect points B and C, as well as A and D.
Angle СAD = SВD as they rely on the same arc of the СAD.
Angle АСВ = АDВ, since they are based on the same arc AB.
By condition, the degree measure of the arc AB is equal to the degree measure of the arc of the СD, then in the triangles OAD and OBC the angles at the bases of the BC and AD are equal.
The ВOD angle is equal to the AOD angle as vertical angles.
By condition, OA = OС, then the triangles OAD and OBC are equal in side and two adjacent angles, and therefore BO = DO, then point O is the middle of the BD, which was required to prove.