Triangle Point Picking -- from Wolfram MathWorld

Given a triangle with one vertex at the origin and the others at positions v_1 and v_2, one might think that a random point inside the triangle would be given by x=a_1v_1+(1-a_1)a_2v_2, (1) where A_1 and A_2 are uniform variates in the interval [0,1]. However, as can be seen in the plot above, this samples the triangle nonuniformly, concentrating points in the v_1 corner. Randomly picking each of the trilinear coordinates from a uniform distribution [0,1] also does not produce a uniform

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