A hyperplane Ham Sandwich theorem

We give a direct proof of a result due to Karasev (2008), Karasev-Matschke (2014) and Schnider-Soberón (2023). Given $m+1$ Borel probability measures on the space of affine hyperplanes in a real vector space $V$ of dimension $m+1$, there exist a line $L$ through the origin in $V$ and a point $v\in L$ such that at least half of the hyperplanes, as counted by any of the measures, meet or are parallel to each of the two closed rays in $L$ meeting at $v$.