A tidbit here some might be interested in is that a version of the accept-flip method works for simplices of arbitrary dimension.

For an n-simplex, start by generating a random point in an n-cube (i.e. n uniform random numbers between 0 and 1).

Next, sort the coordinates. This leaves you with a chain `0 <= c_1 <= ... <= c_n <= 1`.

Taking successive differences from this chain gives you n+1 numbers `d_j` (`0 <= j <= n`) that sum to 1.

Finally, the random point in the n-simplex is just `j_0 v_0 + ... + j_n v_n`, where `v_j` are the vertices of the simplex.

It's not hard to verify that this produces the accept-flip technique in dimension 2. As for why it's uniform: the sorting step is mapping a coordinate in the cube into a fundamental domain for the action of the symmetric group (which breaks the cube into a number of simplices of equal size); the other steps are linear maps, which therefore preserve the uniformity.