if `|z_1+z_2|=|z_1|+|z_2|,` then prove that `a r g(z_1)=a r g(z_2)` if `|z_1-z_2|=|z_1|+|z_2|,` then prove that `a r g(z_1)=a r g(z_2)=pi`

Explanation

(i)` |z_(1)+z_(2)|=|z_(1)|+|z_(2)|`
`implies|z_(1)+(-z_(2))|=|z_(1)|+|-z_(2)|`
`implies" "AB=AO+OB` as shown in the figure

`implies" Points "(A(z_(1)),O(0),B(-z_(2))` are collinear
`impliesA(z_(1)),O(0)" and "C(z_(2))` are collinear as shown in figure
`implies" arg"(z_(1))=" arg"(z_(2))`
Alternate method:
`|z_(1)+z_(2)|=|z_(1)|+|z_(2)|`
or `|z_(1)+z_(2)|^(2)=|z_(1)|^(2)+|z_(2)|^(2)+2|z_(1)||z_(2)|`
or `|z_(1)|^(2)+|z_(2)|^(2)+2"Re"(z_(1)bar(z)_(2))=|z_(1)|^(2)+|z_(2)|^(2)+2|z_(1)||z_(2)|`
or `2"Re"(z_(1)bar(z)_(2))=2|z_(1)||z_(2)|`
`impliescos(theta_(1)-theta_(2))=1`
`impliestheta_(1)-theta_(2)=0`
`implies"arg"(z_(1))=arg(z_(2))`
(ii) `|z_(1)-z_(2)|=|z_(1)|+|z_(2)|`
`impliesAB=AO+OB` as shown in the figure.

`implies" Points "z_(1),O("origin"), z_(2)` are collinear as shown in figure If `arg(z_(1))=theta," then "arg(z_(2))=theta-pi`
`impliesarg(z_(1))-arg(z_(2))=pi`
Alternate method:
`|z_(1)-z_(2)|=|z_(1)|+|z_(2)|`
or `|z_(1)-z_(2)|^(2)=|z_(1)|^(2)+|z_(2)|^(2)+2|z_(1)||z_(2)|`
or `|z_(1)|^(2)+|z_(2)|^(2)-2"Re"(z_(1)bar(z)_(2))=|z_(1)|^(2)+|z_(2)|^(2)+2|z_(1)||z_(2)|`1
or `-2"Re"(z_(1)bar(z)_(2))=2|z_(1)||z_(2)|`
`impliescos(theta_(1)-theta_(2))=-1`
`impliestheta_(1)-theta_(2)=pi`