In the circle through the midpoint O of the chord AC, the chord BD is drawn so that the arcs AB and CD

In the circle through the midpoint O of the chord AC, the chord BD is drawn so that the arcs AB and CD are equal. Prove that O is the midpoint of the chord BD.

Consider the formed triangles AOB and COD.

By condition, AO = OC, since point O is the middle of the chord AC, chord AB = CD.

The inscribed angle ABD rests on the arc AD as well as the inscribed angle ACD, then the angle ADB = ACD.

Then the triangles AOB and COD are equal on two sides and the angle between them, and then OB = OD, and therefore the point O is the midpoint of the chord BD, which was required to prove.