The tangent drawn to a circle at the point A is parallel to the chord BC of the same circle. Which statement must be true?

Call the tangent line XAY, with X on the same side as B. The alternate-segment theorem gives ∠XAB = ∠ACB. Because XY ∥ BC, the chord AB acts as a transversal and the alternate angles give ∠XAB = ∠ABC. Therefore ∠ACB = ∠ABC, and sides opposite equal angles are equal, so AB = AC — A is the midpoint of arc BC. Nothing in the data forces BC to pass through the centre or the angle at A to be 90°.