Dear community,

I am trying to find the number of diagonals of the regular polygon that are larger than $\displaystyle sqrt(3)$. Assuming that the polygon is circumscribed by a circle of unit radius.

Notice that $\displaystyle sqrt (3)$ is the length of the equilateral triangle edge that is also circumscribed by the same unit circle. The total number of diagonals is $\displaystyle n(n-3)/2$ where $\displaystyle n$ is the number of edges or vertices of regular polygon. The length of the edge of the regular polygon would be $\displaystyle 2sin(pi/n)$ if the circumscribed circle radius is 1. The diagonals of regular polygon that are larger than $\displaystyle sqrt (3)$ are the once that intersect the inscribed circle in that equilateral triangle. So how many are they for the $\displaystyle n$ sided regular polygon?