Triangle inscribed-triangle picking is the selection of three independent uniformly distributed points, one on each polygon
edge of a fixed nondegenerate triangle. Let 
be the ratio of the area of the
resulting inscribed triangle to the area
of the fixed triangle. The distribution
function of 
is independent of the shape and size of the fixed triangle,
since an invertible affine transformation
preserves uniform sampling along each polygon edge
and multiplies both areas by the same factor.
For a nonnegative integer 
, the 
th raw moment is
![<C^n>=1/((n+1)[(n+1)!]^2)sum_(k=0)^n[k!(n-k)!]^2,](/images/equations/TriangleInscribed-TrianglePicking/NumberedEquation1.svg) |
(1)
|
where brackets denote the expectation value. The integer sums in the numerator, starting at 
, are 1, 2, 9, 80, 1240, 30240, ... (OEIS A279055).
In particular, the mean and variance
are
 |  |  |
(2)
|
 |  |  |
(3)
|
The mean area is therefore one quarter of the fixed triangle's area (Maesumi 2019, pp. 365-366).
In triangle triangle picking, the three points are instead chosen independently from the interior of the fixed triangle.
