Sample uniformly random points within a triangle

I would like to know how to get random vectors inside a triangle, in python, but it seemed more difficult than I expected. The plane would have coordinates like [a, b], [x, y], [u, v] (three points):

Let u and v be vectors defining a triangle centered at the origin. By this triangle point picking method, one can generate random points within a parallelogram defined by u and v. If points are outside the triangle, simply reject, or invert the points about the diagonal between u and v.

import random

def uniform_triangle(u, v):
    while True:
        s = random.random()
        t = random.random()
        in_triangle = s + t <= 1
        p = s * u + t * v if in_triangle else (1 - s) * u + (1 - t) * v
        yield p

Figure generated via:

from itertools import islice
import matplotlib.pyplot as plt
import numpy as np

triangle = np.array([
    [1, 2],
    [3, 8],
    [7, 5],
])

it = uniform_triangle(
    triangle[1] - triangle[0],
    triangle[2] - triangle[0],
)

points = np.array(list(islice(it, 0, 1000)))
points += triangle[0]

fig, ax = plt.subplots()
ax.scatter(points[:, 0], points[:, 1], s=1)
fig.savefig("triangle.png", dpi=200)