`|z|<=1,|w|<=1`, then show that `|z- w|^2<=(|z|-|w|)^2+(argz-argw)^2`

Modulus of a Complex Number & its properties If z;z_1;z_2inCC then (i)|z|=0hArrz=0 i.e. Re(z)=Im(z)=0 (ii)|z|=|barz|=|-z| (iii) -|z|leRe(z)le|z|;-|z|leIm(z)le|z| (iv) zbarz=|z|^2 (v)|z_1z_2|=|z_1||z_2| (vi)|(z_1)/(z_2)|=|z_1|/|z_2|; z_2!=0

If z_(1) and z_(2) are two complex numbers,then (A) 2(|z|^(2)+|z_(2)|^(2)) = |z_(1)+z_(2)|^(2)+|z_(1)-z_(2)|^(2) (B) |z_(1)+sqrt(z_(1)^(2)-z_(2)^(2))|+|z_(1)-sqrt(z_(1)^(2)-z_(2)^(2))| = |z_(1)+z_(2)|+|z_(1)-z_(2)| (C) |(z_(1)+z_(2))/(2)+sqrt(z_(1)z_(2))|+|(z_(1)+z_(2))/(2)-sqrt(z_(1)z_(2))|=|z_(1)|+|z_(2)| (D) |z_(1)+z_(2)|^(2)-|z_(1)-z_(2)|^(2) = 2(z_(1)bar(z)_(2)+bar(z)_(1)z_(2))

If z_(1),z_(2),z_(3),z_(4) are the affixes of four point in the Argand plane,z is the affix of a point such that |z-z_(1)|=|z-z_(2)|=|z-z_(3)|=|z-z_(4)| ,then prove that z_(1),z_(2),z_(3),z_(4) are concyclic.

If z_1,z_2,z_3 are vertices of a triangle such that |z_1-z_2|=|z_1-z_3| then arg ((2z_1-z_2-z_3)/(z_3-z_2)) is :