The problem of finding the mean triangle area of a triangle with vertices picked inside a triangle with unit area was proposed by Watson (1865) and solved by Sylvester. It solution is a special case of the general formula for polygon triangle picking.
Since the problem is affine, it can be solved by considering for simplicity an isosceles right triangle with unit leg lengths. Integrating the formula for the area of a triangle over the six coordinates of the vertices (and normalizing to the area of the triangle and region of integration by dividing by the integral of unity over the region) gives
(1)
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(2)
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where
(3)
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is the triangle area of a triangle with vertices , , and .
The integral can be solved using computer algebra by breaking up the integration region using cylindrical algebraic decomposition. This results in 62 regions, 30 of which have distinct integrals, each of which can be directly integrated. Combining the results then gives the result
(4)
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(Pfiefer 1989; Zinani 2003).
The exact distribution function was derived by Philip. and are given by
(5)
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where the subscript 1 denotes the region with and 2 denotes the region with .
The raw moments of for , 2, ... are 1/12, 1/144, 31/9000, 1/450, 1063/617400, 403/264600, ... (OEIS A103474 and A103475).
The central moments of for , 2, ... are 0, 1/144, 61/54000, 343/864000, 9493/66679200, ... (OEIS A130117 and A130118).