Given a triangle with one vertex at the origin and the others at positions and , one might think that a random point inside the triangle
would be given by

(1)

where
and
are uniform variates in the interval . However, as can be seen in the plot above, this samples
the triangle nonuniformly, concentrating points in the corner.

Randomly picking each of the trilinear coordinates from a uniform distribution also does not produce a uniform point spacing on in the
triangle. As illustrated above, the resulting points are concentrated towards the
center.

To pick points uniformly distributed inside the triangle, instead pick

(2)

where
and
are uniform variates in the interval , which gives points uniformly distributed in a quadrilateral
(left figure). The points not in the triangle interior
can then either be discarded, or transformed into the corresponding point inside
the triangle (right figure).

The expected distance of a point picked at random inside an equilateral
triangle of unit side length from the center of the triangle is

(3)

and the expected distance from a fixed vertex is

(4)

The expected distance from the closest vertex is

(5)

while the expected distance from the farthest is

(6)

Picking
points independently and uniformly from a triangle with unit area gives a convex
hull with expected area of

Buchta, C. "Zufallspolygone in konvexen Vielecken." J. reine angew. Math.347, 212-220, 1984.Buchta, C. "A
Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math.30,
653-659, 1986.Sloane, N. J. A. Sequences A093762
and A093763 in "The On-Line Encyclopedia
of Integer Sequences."