This is Gauss-Lucas Theorem. Highlights of the proof:
If
=p'(z)=0)
, then
is a convex combination of the roots of
)
, otherwise:
write
=a\prod\limits^n_{i=1}(z-a_i) \,,\,\,a,a_i\in\mathbb{C})
, and take the logarithmic derivative of this:
}{p(z)}=\sum^n_{i=1}\frac{1}{z-a_i}=\sum^n_{i=1}\frac{\overline{z}-\overline{a_i}}{|z-a_i|^2})
, which is valid whenever
\neq 0)
. So if
is a root of
)
but not of
we get:
, and taking conjugates and dividing we finally get:
a_i}}{\displaystyle{\sum^n_{i=1}\fra c{1}{|z-a_i|^2}}})
, 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
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Originally Posted by Bruno J. 
