Vector through a point with same distance as from this point to a line

Question

I have three points $A,B,C \in \mathbb{R}^3$ (representing a triangle with area $a$). We know the distance from $C$ to $AB = d$. Now I need to find a vector that goes through $C$ (not parallel to $AB$), so that the distance from every point on this vector to $AB$ equals $d$. In other words that a triangle with vertices $A, B$ and any point of this vector will give the area $a$. Any tips will be appreciated!