Given a triangle with the edges $e_1$, $e_2$, $e_3$, it seems (from numerical evidence) that there are coefficients $\alpha_i$ such that $ u^Hv = \sum_{i=1}^3 \alpha_i \, (u^He_i)\, (e_i^H v) $ holds for all vectors $u, v\in\mathbb{C}^2$.
There is of course the brute-force way of taking the edge coefficients in terms of their $x$- and $y$-coordinates to show this is true, but the simplicity of the statement makes me think there should be a more elegant way.
I looked at a bunch of geometry text books and couldn't find it. I'm thinking to maybe take off my geometry hat and put on my linear algebra hat. What's your take on the problem; where do you think are similar results to be found?