`z_1, z_2a n dz_3` are the vertices of an isosceles triangle in anticlockwise direction with origin as in center , then prove that `z_2, z_1a n d kz_3` are in G.P. where `k in R^+dot`

A(z_1),B(z_2),C(z_3) are the vertices of the triangle A B C (in clockwise). 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

On the Argand plane z_1, z_2a n dz_3 are respectively, the vertices of an isosceles triangle A B C with A C=B C and equal angles are thetadot If z_4 is the incenter of the triangle, then prove that (z_2-z_1)(z_3-z_1)=(1+sectheta)(z_4-z_1)^2dot

Complex numbers z_1 , z_2 , z_3 are the vertices A, B, C respectively of an isosceles right angled trianglewith right angle at C and (z_1- z_2)^2 = k(z_1 - z_3) (z_3 -z_2) , then find k.

If x ,1,a n dz are in A.P. and x ,2,a n dz are in G.P., then prove that x ,a n d4,z are in H.P.

Suppose z_(1),z_(2)andz_(3) are the vertices of an equilateral triangle inscribed in the circle |z| = 2. If z_(1)=1+isqrt3 then find z_(2)andz_(3) .

If z_1, z_2 and z_3 , are the vertices of an equilateral triangle ABC such that |z_1 -i = |z_2 -i| = |z_3 -i| .then |z_1 +z_2+ z_3| equals:

A particle P starts from the point z_0=1+2i , where i=sqrt(-1) . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z_1dot From z_1 the particle moves sqrt(2) units in the direction of the vector hat i+ hat j and then it moves through an angle pi/2 in anticlockwise direction on a circle with centre at origin, to reach a point z_2dot The point z_2 is given by (a) 6+7i (b) -7+6i (c) 7+6i (d) -6+7i

Let z_1, z_2a n dz_3 represent the vertices A ,B ,a n dC of the triangle A B C , respectively, in the Argand plane, such that |z_1|=|z_2|=|z_3|=5. Prove that z_1sin2A+z_2sin2B+z_3sin2C=0.

If z_1, z_2, z_3 be the affixes of the vertices A, B Mand C of a triangle having centroid at G such ;that z = 0 is the mid point of AG then 4z_1 + Z_2 + Z_3 =