Spoiler:This is Gauss-Lucas Theorem. Highlights of the proof:

If , then is a convex combination of the roots of , otherwise:

write , and take the logarithmic derivative of this:

, which is valid whenever . So if is a root of but not of we get:

, and taking conjugates and dividing we finally get:

, and it's easy now to see these are barycentric coordinates of wrt (please do pay attention

closely to the last expression's indexes).

Tonio