Length of the space between rays

Question

I am sorry in advance for the awkard phrasing.

Three rays $A, B,$ and $C$ originate from point $(x, y)$. Points $(a, b)$, $(c, d)$, and $(e, f)$ lie on a straight line and lie on rays $A, B,$ and $C$ respectively. If these points are equidistant from each other, does that mean that any three points that lie on a line and lie on those three rays will be equidistant from each other?

Edit: I am asking because of this:

Is this distance $y$ special in the sense that there is only ONE distance $y$ that is the same between rays $1,2$ and $2,3$?