Stop going in circles

Geometry Level 5

Suppose a regular octagon is inscribed in a unit circle. If \(3\) of the vertices of this octagon are chosen uniformly at random, then the expected area of the triangle formed by these \(3\) vertices is

\(\dfrac{a*(b + \sqrt{c})}{d}\),

where \(a,b,c,d\) are all positive integers with \(a,d\) being coprime and \(c\) square-free.

Find \(a*b*c*d\).

Note: \(a,b,c\) may not necessarily be distinct.


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