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how to solve below complex number problem .

The points $A,B,C$ represent the complex numbers $z_1,z_2,z_3$ respectively, and $G$ is the centroid of the triangle $ABC$ . If $4z_1+z_2+z_3=0$, show that the origin is the mid point of $AG$ ?

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The centroid G is defined to be $\frac{z_1+z_2+z_3}{3}$. So divide the constraint by 3 to get $(3/3) z_1 + (1/3) (z_1 + z_2 +z_3) = 0$. This implies that $z_1 = -G$. Thus the midpoint of $z_1$ and G must be 0.

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  • $\begingroup$ I got it . Thank you so much. $\endgroup$ – Suraj Apr 8 '14 at 1:46

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