Two circles touch each other internally at the point A, the smaller one lying inside the larger. A chord BC of the larger circle touches the smaller circle at D. What does the line AD do?

Draw the common tangent at A and call it AT. The alternate-segment theorem applied to the small circle gives ∠TAD = ∠ADB measured in that circle, and applied to the large circle it gives ∠TAB = ∠ACB. Comparing the two, the enlargement centred at A that carries the small circle onto the large one carries D to the point where AD meets the large circle again — and since it carries the tangent line BC to the parallel tangent of the large circle, that image point is the midpoint of arc BC. A line through A and the midpoint of arc BC bisects ∠BAC. The other options fail in general: AD meets BC at a right angle only when the figure is symmetric about the line of centres, AD does not pass through the midpoint of the chord, and AD is a chord of the small circle rather than a diameter unless the same symmetry holds.