If complex numbers `z_(1)z_(2)` and `z_(3)` are such that `|z_(1)| = |z_(2)| = |z_(3)|`, then prove that `arg((z_(2))/(z_(1))) = arg ((z_(2) - z_(3))/(z_(1) - z_(3)))^(2)`.

Explanation

Given that `|z_(1)|=|z_(2)|=|z_(3)|`
So, `z_(1),z_(2)` and `z_(3)` are equidistant from origin.
Thus, origin is circumcenter of the triangle formed by these three complex numbers.

Now, from one of the properties of the chord of the circle,
`angleAOB=2angleACB`
`:." ""arg"(z_(2)-0)/(z_(1)-0)=2"arg"(z_(2)-z_(3))/(z_(1)-z_(3))`
`implies" "arg((z_(2))/(z_(1)))=arg((z_(2)-z_(3))/(z_(1)-z_(3)))^(2)`