TITLE: A geometric proof of the Gauss-Lucas theorem
QUESTION [13 upvotes]: Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask:
Is there a geometric proof for the Gauss-Lucas theorem? Since we are working on a half plane, can one imagine a possible proof of the above fact  with some methods in hyperbolic geometry?

REPLY [4 votes]: Charles R. Johnson and I gave a proof of the Gauss–Lucas theorem that utilizes the field of values (or numerical range) of a matrix, which can be viewed as geometric in nature (the field itself is an uncountably infinite intersection of half-planes). The proof can be found in the article Matricial Proofs of Some Classical Results about Critical Point Location, which recently appeared in The American Mathematical Monthly.
Our methods also yield a proof of Marden's theorem and a weaker version of Siebeck's theorem.