Clarification on Barycentric coordinates.

Question

I know how to work on triangles a bit by assigning barycentric coordinates $(1,0,0),\ (0,1,0),$ and $(0,0,1)$ to vertices $A,\ B,$ and $C$ resepectively. Let's say that triangle $ABC$ is equilateral, or $AB=AC$ or some other property, can we assign different coordinates to the vertices so that problem solving becomes easier?