Case 1: A falls outside the circle.
OA is the diameter of the new circle. A' is the midpoint of the common chord of the two circles (DE).

Case 2: A falls inside the circle.
A is the midpoint of DE, and OA' the diameter.

Case 3: A is on the circumference of the given circle.
The two circles are tangent: A = A' = D = E.
The transformation is continuous across the boundary of the circle.

Case 4: A = O
The two tangents are at opposite ends of a diameter: they never meet.
However, we can let the line DE cross over the diameter undisturbed. From A's point of view, in the limit the curvature of the second circle passes indistinguishably into a straight line.

That last bit doesn't have to be the hard part. Using limits to decide not-to-tell-the-difference between things which might, in fact, be important is the math that can be made up in the bathroom.
Making something move infinitely far away from me, as a limit procedure, is precisely 75.3%* ridiculous. I just keep pushing until I can't tell the difference between anything but zero and AWAY. Of course, the actual curvature stays at a constant. In the limit, I measure 0% of the arc and I can call it a straight line.
I can also dress myself.
In the first case, with respect to the curvature of the actual circle, my error is infinitely large. It's zero only from my point of view. Depending on the question, that could be exactly the correct description, or it could introduce unbounded (and by procedure, unboundable) error.
In the second case, I just put the clothes on.
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*I am sure of this number because I can't tell the difference between it and any other number. Therefore, they are indistinguishable. hahahah cute.