Product of Diagonals from Point of Regular Polygon

Theorem
Let $A_1, A_2, \ldots, A_n$ be the vertices of a regular $n$-gon $P = A_1 A_2 \cdots A_n$ which is circumscribed by a unit circle.

Then:
 * $\displaystyle \prod_{k \mathop = 3}^{n-1} A_1 A_k = \frac {n \csc^2 \frac \pi n} 4$

where $A_1 A_k$ is the length of the line joining $A_1$ to $A_k$.

Proof
First it is worth examining the degenerate case $n = 3$, where there are no such lines.

In this case:

The product $\displaystyle \prod_{k \mathop = 3}^{n-1} A_1 A_k$ is vacuous here.

By convention, such a vacuous product is defined as being equal to $1$.

So the result holds for $n = 3$.


 * RegularPolygonWithDiagonals.png

Let $n > 3$.

The illustration above is a diagram of the case where $n = 7$.

From Roots of Unity lie on Vertices of Regular Polygon, the vertices of $P$ can be identified with the complex $n$th roots of unity.

Thus: