If `|z|=2a n d(z_1-z_3)/(z_2-z_3)=(z-2)/(z+2)` , then prove that `z_1, z_2, z_3` are vertices of a right angled triangle.

If z_1=i z_2 and z_1-z_3=i(z_3-z_2) ,then prove that |z_3|=sqrt(2)|z_1|.

The complex numbers z_1, z_2 and z_3 satisfying (z_1-z_3)/(z_2-z_3) =(1- i sqrt(3))/2 are the vertices of triangle which is (1) of area zero (2) right angled isosceles(3) equilateral (4) obtuse angled isosceles

Find the condition in order that z_(1),z_(2),z_(3) are vertices of an.isosceles triangle right angled at z_(2) .

If |z_1|=|z_2|=1, then prove that |z_1+z_2| = |1/z_1+1/z_2∣

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0

If z_(1),z_(2),z_(3) are the vertices of an isoscles triangle right angled at z_(2) , then

If 2z_1-3z_2 + z_3=0 , then z_1, z_2 and z_3 are represented by

Let z_1, z_2, z_3 be the three nonzero complex numbers such that z_2!=1,a=|z_1|,b=|z_2|a n d c=|z_3|dot Let |a b c b c a c a b|=0 a r g(z_3)/(z_2)=a r g((z_3-z_1)/(z_2-z_1))^2 orthocentre of triangle formed by z_1, z_2, z_3, i sz_1+z_2+z_3 if triangle formed by z_1, z_2, z_3 is equilateral, then its area is (3sqrt(3))/2|z_1|^2 if triangle formed by z_1, z_2, z_3 is equilateral, then z_1+z_2+z_3=0

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|=1,|z_2|=2,|z_3|=3,a n d|9z_1z_2+4z_1z_3+z_2z_3|=12 , then find the value of |z_1+z_2+z_3|dot