As shown in the figure below, it is known that the tangent circle O of the straight line CD is at point D, and BC and AC are the chords of the circle O..

The included angle between the tangent DE of point C on the chord BC and the no-load AC is ∠ECA, which is clamped by the tail arc AC, so ∠ECA= 1/2∠AOC=∠ABC.

So put it in this question:

When the chord AB is the chord of the circle O, AD is the tangent of the circle O, and the arc sandwiched between them is the arc AB, so ∠DAB=∠ACB.

Similarly, when the chord AB is the chord of the circle O', AC is the tangent of the circle O', and the arc sandwiched between them is the arc AB, so ∠CAB=∠ADB.