KKM meets colorful Carathéodory

The colorful Carathéodory theorem states that if $V_1,\dots ,V_{d+1}\subset {\mathbb{R}}^d$ satisfy $0\in \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}{V}_i$ for every $i$, then there exist points $v_i\in V_i$ such that $0\in \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\{v_1,\dots ,v_{d+1}\}$. We give a new proof based on triangulations and the classical KKM lemma. We lift the sets $V_i$ to parallel affine copies of ${\mathbb{R}}^d$ in ${\mathbb{R}}^d\times {\mathbb{R}}^d$ and use a triangulation of the resulting point configuration to construct a KKM cover. Using the same approach, we prove the very colorful Carathéodory theorem.