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If $\mathrm{z}_1, \mathrm{z}_2, \mathrm{z}_3$ are the vertices of an equilateral triangle and $\mathrm{z}$ is its circum centre, then
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The correct answer is:
$\frac{\left|z-z_1\right|}{\left|z-z_2\right|}=\frac{\left|z-z_3\right|}{\left|z-z_1\right|}$
Given that $z_1, z_2, z_3$ are the vertices of an equilateral triangle and $\mathrm{z}$ is its circumcentre.
$\begin{aligned} & \therefore\left|z-z_1\right|=\left|z-z_2\right|=\left|z-z_3\right| \\ & \Rightarrow \frac{\left|z-z_1\right|}{\left|z-z_2\right|}=\frac{\left|z-z_3\right|}{\left|z-z_1\right|}\end{aligned}$
$\begin{aligned} & \therefore\left|z-z_1\right|=\left|z-z_2\right|=\left|z-z_3\right| \\ & \Rightarrow \frac{\left|z-z_1\right|}{\left|z-z_2\right|}=\frac{\left|z-z_3\right|}{\left|z-z_1\right|}\end{aligned}$
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