I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook):

Let $(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$ be chosen uniformly at random w.r.t. the Fubini-Study metric, and normalized so that $|z_0|^2 + \cdots |z_n|^2 = 1$. Then the point $(|z_0|^2, \ldots, |z_n|^2)$ is uniformly distributed on the $n$-simplex (the set of $(p_0,\ldots,p_n)\in\mathbb{R}^{n+1}$ such that all $p_i\geq 0$ and $p_0+\cdots+p_n = 1$) w.r.t. its Euclidean metric.

This affords the quantum-mechanical interpretation that if we draw a random quantum entanglement of $n+1$ pure states, and we observe the pure state it is in, we get a uniform probability measure on the pure states. However, I'm not asking for a reference in relation to quantum mechanics.

This is, for example, proposition 1 in this paper but the only reference the authors give after calling the fact “known” is a 600-page book without any specific page number (shame!).

It is also mentioned in this blog post (and attributed to Bill Wootters); and the particular case $n=1$ is mentioned in this other blog post (in relation to the Box-Muller transformation); but I would like something more tangible than a blog post.

As a bonus question, if we take a uniformly random $(n+1)\times(n+1)$ unitary matrix (uniformly w.r.t. the Haar measure) and we look at the square norms of its columns, we get $n+1$ points on the $n$-simplex, each uniformly distributed by the above fact: does this distribution of $n+1$ points on the $n$-simplex have a standard name, and where might I learn more about it?