Say I want to calculate the euclidean distance of all edges of a triangle. I could take the root of the squared distance of each edge and add those. This would give me the right result.
Adding up all the squared distances of the edges and taking the root does not work, I guess because $\sqrt a + \sqrt b \neq \sqrt{a+b}$.
Therefore, is there a way to get "the root of a sum of squared distances"? In other words, is there a way to only use the root once?
Assume that the $3$ vertices of your triangle have integer coordinates and that they are in general position (for instance, "randomly" choose the coordinates). The perimeter of the triangle is in the form $p=\sqrt{a}+\sqrt{b}+\sqrt{c}$ where $a,b,c\in\mathbb{N}^*$ ; $p\in \mathbb{Q}[\sqrt{a},\sqrt{b},\sqrt{c}]$, an algebraic extension of $\mathbb{Q}$ of degree $8$. If you have only one signum $\sqrt{.}$, then you can only obtain elements of an algebraic extension of degree $2$ and certainly not $p$.