Pick three points P = (x_1, y_1), Q = (x_2, y_2), and R = (x_3, y_3) distributed independently and uniformly in a unit disk K (i.e., in the interior of the unit circle). Then the average area of the triangle determined by these points is A^_ = ( integral integral_(P element K) integral integral_(Q element K) integral integral_(R element K) 1/2 left bracketing bar x_1 | y_1 | 1 x_2 | y_2 | 1 x_3 | y_3 | 1 right bracketing bar d y_3 d y_2 d y_1 d x_3 d x_2 d x_1)/( integral integral_(P element K) integral integral_(Q element K) integral integral_(R element K) d y_3 d y_2 d y_1 d x_3 d x_2 d x_1).

ball triangle picking | circle triangle picking | disk line picking | Gaussian triangle picking | Heilbronn triangle problem | hexagon triangle picking | obtuse triangle | simplex simplex picking | square triangle picking | Sylvester's four-point problem | triangle triangle picking