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?
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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 $(2,0)$.