Question on intersecting lines. [closed]

Solution 1:

The points in the plane which are equidistant to $AB$ and $CD$ are the points on the angle bisector of $\angle AOC,\angle DOB$ and the angle bisector of $\angle COB,\angle DOA$ (blue lines in the figure).

Because the line $XY$ generally intersects both angle bisectors, then it passes through two points equidistant to $AB$ and $CD$.

The line $XY$ passes through only one such point if and only if it is parallel to, but does not coincide with, one of the angle bisectors, or it passes through O without coinciding with either angle bisector.

Added note: the line $XY$ passes through infinitely many such points if and only if it coincides with one of the angle bisectors.