Build a positive Cartesian coordinate system in the space so that the $Oz$ axis is orthogonal to the triangle defined by the vertices $(1,0,0),(0,1,0),(0,0,1)$ and goes through the centre of mass.
The centroid here is $(1/3, 1/3, 1/3)$. The normal $\vec{n} = (1,1,1)$ is orthogonal to the triangle.
Am I just meant to multiply $(a,b,c)$ by a rotation matrix so that $Oz$ changes to $\vec{n}$?