Yes, A' will always be a dilation of A. This dilation is unique to a center point P as described below.
Proof: Construct a line containing points A and A'. Measure distance AA'. Construct point P colinear with AA' at a known distance PA on the far side of A from A'. Mark P as center. Dilate A by a scale factor equal to the distance PA' using P as center. The resultant will be the point A'.
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