1
$\begingroup$

The wiki for regular polygon diagonals says:

"For a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals n."

Can someone provide a basic proof?

https://en.wikipedia.org/wiki/Regular_polygon#Diagonals

$\endgroup$
1
$\begingroup$

Let $f = x-1$ and $g = (x^n-1)/(x-1) = 1+x+x^2+\ldots+x^{n-1}$. Together, $f$ and $g$ have as zeros all the vertices of the unit $n$-gon in the complex plane.

The number we are looking for is now the absolute value of the product $R$ of all terms $a-b$, where $a$ is a zero of $f$ and $b$ is a zero of $g$. This $R$ is known as the resultant of $f$ and $g$.

This resultant is also the determinant of the Sylvester matrix of $f$ and $g$, which is constructed from the coefficients of $f$ and $g$. In this case, the Sylvester matrix has the form

$$ \begin{pmatrix} -1 & & & 1 \\ 1 & \ddots & & \vdots \\ & \ddots & -1 & \vdots \\ & & 1 & 1 \end{pmatrix} $$

and thus the determinant $\pm n$. Hence $|R|= n$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.