I came across this attractive theorem:

. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that form an obtuse angle with one another.*Theorem*

This was proved^{1} as a corollary of a lemma about irreducible matrices.
I am wondering if anyone knows of an alternative, more geometric proof that somehow
more directly captures the sense that one cannot "pack" more than
$d+1$ obtuse vectors in $\mathbb{R}^d$.

^{1}Lipeng Ning, Tryphon T. Georgiou, Allen Tannenbaum, Stephen P. Boyd.
"Linear models based on noisy data and the Frisch scheme."
*SIAM Review*. 57(2) 2015.
arXiv preprint.