What Inequality holds true if and only if when $|x|+|y|+|z|+|x+y+z|=|x+y|+|y+z|+|z+x|$ is true?

Question

What inequality holds true if and only if when $$|x|+|y|+|z|+|x+y+z|=|x+y|+|y+z|+|z+x|$$ is true?

Here, the variables $x,y,z$ are all real numbers.
I tried to find out the answer by dividing the cases:
when
$x\geq0, y\geq0, z\geq0$
$x\geq0, y\geq0, z\leq0$
and so on... but can anyone simplify this procedure?

Answer 1

After squaring of the both sides we obtain: $$\sum_{cyc}(x^2+2|xy|+x^2+2xy+2|x(x+y+z))=\sum_{cyc}((x+y)^2+2|(x+y)(x+z)|)$$ or $$\sum_{cyc}(|xy|+|x^2+xy+xz|)=\sum_{cyc}|x^2+xy+xz+yz|.$$ But by the triangle inequality $$|xy|+|x^2+xy+xz|\geq|xy+x^2+xy+xz+yz|=|(x+y)(x+z)|,$$ which says that it's enough to understand, when does the equality occur.

Can you end it now?