Where does this formula for the product of Gamma Functions come from?

Question

$$\prod_{r=1}^{n} \Gamma \Big(\frac{r}{n+1}\Big) = \sqrt{\frac{(2\pi)^{n}}{n+1}}$$

I came across the formula at the bottom of the Wikipedia page on specific values of the Gamma function but it was mentioned in passing and a proof is not referenced.

Answer 1

A lot of relations like this can be derived using the Euler Form of the Gamma Function, which re-writes the gamma function as a limit. See here.

In this case this is the Gauss Multiplication Formula, whose proof can be found here.