Euclidean distance vs Squared So I understand that Euclidean distance is valid for all of properties for a metric. But why doesn't the square hold the same way? 

You lose the triangle inequality if you don’t take the square root: the ‘distance’ from the origin to $(2,0)$ would be $4$, which is greater than $2$, the sum of the ‘distances’ from the origin to $(1,0)$ and from $(1,0)$ to $(2,0)$.
