z1, z2 r 2 complex numbers on z-plane, |z1| and |z2| represent modulus (magnitude) of z1 n z2.
|z1 + z2| is again a complex number on same plane.
z1, z2 and z1+z2 will make a triangle,
in triangle, difference of any two side is greater than third side.
=> |z1| - |z2| > |z1 + z2|
but |z1| - |z2| = |z1 + z2| is given => z1, z2, z1+z2 don't make a triangle. they are in one line.
Now if they are on the same side of origin, obviously |z1 + z2| = |z1| + |z2|
but we have, |z1 + z2| = |z1| - |z2|
=> z1 and z2 must be on the different side of origin.
=> if we rotate line of z1 by pi-angle we get line of z2.
=> arg z1 - arg z2 = pi
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