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