`A(z_1),B(z_2),C(z_3)` are the vertices of he triangle `A B C` (in anticlockwise). If `/_A B C=pi//4` and `A B=sqrt(2)(B C)` , then prove that `z_2=z_3+i(z_1-z_3)dot`

A(z_(1)),B(z_(2)),C(z_(3)) are the vertices of he triangle ABC (in anticlockwise).If /_ABC=pi/4 and AB=sqrt(2)(BC), then prove that z_(2)=z_(3)+i(z_(1)-z_(3))

A(z_(1));B(z_(2)) and C(z_(3)) are the vertices of the triangle ABC( in anticlockwise order).If /_ABC=(pi)/(4) and AB=sqrt(2)(BC); then prove that z_(2)=z_(3)+i(z_(1)-z_(3))

If z_(1),z_(2),z_(3) are the vertices A,B,C of a right angled triangle taken in counte-clockwise direction with right angle at B and (AC)/(BC)=sqrt(5) ,then (z_(3)-z_(2))=

If the points A(z), B(-z) and C(1-z) are the vertices of an equilateral triangle, then value of z is

If A(z_1), B(z_2). C(z_3) are the vertices of an equilateral triangle ABC, then arg((z_2+z_2-2z_1)/(z_3-z_2))=pi/4 Reason(R): If /_B=alpha, then

Let A(z_(1)),B(z_(2)),C(z_(3)) be the vertices of an equilateral triangle ABC in the Argand plane, then the number (z_(2)-z_(3))/(2z_(1)-z_(2)-z_(3)) , is

Let z_(1),z_(2),z_(3),z_(4) be the vertices A,B,C,D respectively of a square on the Argand diagramtaken in anticlockwise direction then prove that: 2z_(2)=(1+i)z_(1)+(1-i)z_(3)

If A(z_(1)),B(z_(2)), C(z_(3)) are the vertices of an equilateral triangle ABC, then arg (2z_(1)-z_(2)-z_(3))/(z_(3)_z_(2))=

Let A(z_1),B(z_2) and C(z_3) be the vertices of an equilateral triangle in the Argand plane such that |z_1|=|z_2|=|z_3|. Then (A) (z_2+z_3)/(2z_1-z_2-z_3) is purely real (B) (z_2-z_3)/(2z_1-z_2-z_3) is purely imaginary (C) |arg(z_1/z_2)|=2 arg((z_3-z_2)/(z_1-z_2))| (D) none of these

If the complex number z_1,z_2 and z_3 represent the vertices of an equilateral triangle inscribed in the circle |z|=2 and z_1=1+isqrt(3) then (A) z_2=1,z_3=1-isqrt(3) (B) z_2=1-isqrt(3),z_3=-isqrt(3) (C) z_2=1-isqrt(3), z_3=-1+isqrt(3) (D) z_2=,z_3=1-isqrt(3)