# 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?

-
Please pick an answer different than mine to be the correct one, as my answer deals with norms instead of metrics. –  JavaMan Oct 23 '11 at 22:03

The square of the distance does not obey the triangle inequality: $1^2+1^2<(1+1)^2$
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 $(2,0)$.