First, let us move to the complex plane. The $n$ vertices form the $n$ roots of unity. However, if we were to take the unit circle centered at the origin, we'll have a product in the form of $\prod_{1 = 0}^{n-1} |z_k - z_0|$, where $z_k$ is a $n$th root of unity. Instead, we would want $z_0$ to be $0$, so we can now consider the equation $(z+1)^n = 1$, making for a unit circle centered at $-1$. The LHS expands as $z^n + nz^{n-1} + \cdots + nz + 1 = 1 \implies z_0 z_1 \cdots z_{n-1} = (-1)^{n} n$. Then, the product of the distances from a fixed vertex, WLOG $0$, is simply the magnitude of the product of the vertices, using the fact that the magnitude is multiplicative, is $|(-1)^{n} n| = \boxed{n}$.