Let $z_{1}, z_{2}, z_{3}$ be the vertices of a triangle ABC. Then which of the following statements is correct?

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If $\frac{1}{z - z_{1}} + \frac{1}{z - z_{2}} + \frac{1}{z - z_{3}} = 0,$ where $z = \frac{|z_{1} + z_{2} + z_{3}|}{3},$ then ABC is an equilateral triangle.

If ABC is an equilateral triangle then $\frac{1}{z - z_{1}} + \frac{1}{z - z_{2}} + \frac{1}{z - z_{3}} = 0, w h e r e z = \frac{z_{1} + z_{2} + z_{3}}{3}$

If $| \begin{matrix} 1 & 1 & 1 \\ z_{1} & z_{2} & z_{3} \\ z_{2} & z_{3} & z_{1} \end{matrix} | = 0,$ then the triangle ABC is equilateral

If $| z_{1} | = | z_{2} | = | z_{3} | a n d z_{1} + z_{2} + z_{3} = 0,$ then the triangle ABC is equilateral

answer is B, A, C, D.

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Detailed Solution

A necessary and sufficient condition for a triangle having vertices $z_{1}, z_{2} a n d z_{3}$ to form an equilateral triangle is $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3}$
(A) and (B) will follow by performing some algebraic jugglery on the known condition given above to prove (D) note that $z_{1} + z_{2} + z_{3} = 0$ can be changed to
$\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} = 0 (∵ | z_{1} | = | z_{2} | = | z_{3} |)$