Let f and g be analytic functions on the open unit disk
${\mathbb D}$
such that
$|f|=|g|$
on a set A. We give an alternative proof of the result of Perez that there exists c in the unit circle
${\mathbb T}$
such that
$f=cg$
when A is the union of two lines in
${\mathbb D}$
intersecting at an angle that is an irrational multiple of
$\pi $
, and from this, deduce a sequential generalization of the result. Similarly, the same conclusion is valid when f and g are in the Nevanlinna class and A is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of this result and analyze the case
$A=r{\mathbb T}$
. Finally, we examine the most general situation when there is equality on two distinct circles in the disk, proving a result or counterexample for each possible configuration.