Motivated by a [geometric proof of the Fundamental Theorem of Algebra](https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra#Geometric_proofs) we ask:


Is there a geometric proof for the [Gauss-Lucas theorem](https://en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_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?



A geometric proof of the Gauss-Lucas theorem
Good question! I think the following article may qualify as a "yes":


Arnaud Ch´eritat, Yan Gao, Yafei Ou, Lei Tan. "A refinement of the Gauss-Lucas theorem (after
W. P. Thurston)". 2015. 
<https://hal.archives-ouvertes.fr/hal-01157602/document>


(An essential observation here is that you **can** make the Gauss-Lucas theorem explicitly about the behavior half-planes in $\mathbb{C}$. )


