Arguments of collinear complex numbers?

Question

This is the exact passage give in my book about collinear complex numbers, I think there is an error somewhere. Would someone please explain the real conditions for arguments of collinear complex numbers? The passage seems rather contradictory as it first says that for collinearity arg (z1/z2) = π implying arg z1 - arg z2 = π or arg z1 = arg z2, and then says that definitely arg z1 = arg z2.

If origin O, z1, z2 are collinear and z1, z2 lie on same side of origin then arg z1 = arg z2 and arg (z1/z2) = 0. If O, z1, z2 are collinear and z1 and z2 are on opposite sides of origin then arg (z1/z2) = π. Hence if z1, O, z2 are collinear then z1/z2 is purely real. This is to say that if |z1+z2| = |z1| + |z2| then arg z1 = arg z2

Answer 1

Nothing is wrong in what is stated. Simply the conclusion,

If $|z1+z2| = |z1| + |z2|$ then arg $(z_1) = $ arg$(z_2)$

does not purely logically follow from what precedes.

One would still have to explain why $$|z1+z2| = |z1| + |z2|$$ occurs if and only if $z_1$ and $z_2$ are collinear and on the same side of the origin. It is true, but not explained here.

Answer 2

If $z_1, z_2, 0$ are col-linear there are two cases

$z_1,z_2$ are on the same side of the origin

$\arg z_1 = \arg z_2\\ \frac {z_1}{z_2} = \frac {|z_1|}{|z_2|}\\ \arg \frac {z_1}{z_2} = \arg 1 = 0\\ $

$\frac {z_1}{z_2}$ is real

Or the origin is between $z_1$ and $z_2$

$\arg z_1 - \arg z_2 = \pm \pi\\ \frac {z_1}{z_2} = -\frac {|z_1|}{|z_2|}\\ \arg \frac {z_1}{z_2} = \arg -1 = \pi\\ $

$\frac {z_1}{z_2}$ is real

Draw an example of each of the two cases....