If complex numbers {z_1}, {z_2}, text{and } { | vedclass

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(D) Given that $	riangle ABC$ is an isosceles right-angled triangle with $\angle C = 90^\circ$ and $AC = BC$.Using the property of rotation in the co

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${(z_1 - z_2)^2} = 2(z_1 - z_3)(z_3 - z_2)$

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(D) Given that $	riangle ABC$ is an isosceles right-angled triangle with $\angle C = 90^\circ$ and $AC = BC$.Using the property of rotation in the complex plane,the vector $\vec{CB}$ is obtained by rotating $\vec{CA}$ by $90^\circ$ (or $\pi/2$ radians).Thus,$(z_2 - z_3) = \pm i(z_1 - z_3)$.Squaring both sides,we get $(z_2 - z_3)^2 = -(z_1 - z_3)^2$.$(z_2 - z_3)^2 + (z_1 - z_3)^2 = 0$.Expanding this: $z_2^2 + z_3^2 - 2z_2z_3 + z_1^2 + z_3^2 - 2z_1z_3 = 0$.$z_1^2 + z_2^2 + 2z_3^2 - 2z_3(z_1 + z_2) = 0$.We know $(z_1 - z_2)^2 = z_1^2 + z_2^2 - 2z_1z_2$.Adding and subtracting $2z_1z_2$ and rearranging terms,we find that $(z_1 - z_2)^2 = 2(z_1 - z_3)(z_3 - z_2)$.

If complex numbers ${z_1}, {z_2}, \text{and } {z_3}$ represent the vertices $A, B, \text{and } C$ respectively of an isosceles triangle $ABC$ of which $\angle C$ is a right angle,then the correct statement is:

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