Let ABCD be a non-collinear parallelogram, E be the midpoint of AB, and F be the midpoint of BC. Prove that D, E, F form an affine basis, and find the barycentric coordinates of the centroid of A, B, C, D with respect to D, E, F.
I am able to show that D, E, F form an affine basis, but I do not understand how to find the barycentric coordinates. I tried starting with since D, E, F form an affine basis then their exists $a,b,c \in \mathbb{R}$ such that $a + b + c =1$ and $ aD + bE + cF = (A+B+C+D)/4$. Where $(A+B+C+D)/4$ is the centroid of ABCD. But I keep getting stuck in simplifing.