Is the Riemann Zeta function Is encoded in the triangle inequality?

I had posted this question in MO that has remained unanswered in MO for more than two years now. While working on it, I accidently found an unexpected result.

Let $0<x\leq y\leq z$ be the ordered side lengths of the triangle determined by three independent uniformly distributed points on a circle. Then,$$ \mathbb E\left[ \left(\log\frac{z}{x+y-z}\right)^{s-2} \right] = \frac{12}{\pi^2}\Gamma(s)(1-2^{1-s})\zeta(s), \qquad \operatorname{Re}s>0, $$

Hence the nontrivial zeros of the Riemann zeta function are exactly values of $s$ where the LHS vanishes i.e. of the logarithmic distance from discrepancy in the triangle inequality , and the Riemann hypothesis is equivalent to all such zeros lying on the critical line.

Is this or similar results which connects the Riemann Zeta function the triangle inequality known? Is there any reference in literature? Also I am not an expert in zeta function, so I would appreciate why such a relation should hold?