Question

Let A, B and C represent the complex numbers $z_1,z_2,z_3$ respectively on the complex plane. If the circumcentre of the triangle ABC lies at the origin, then the nine point centre is represented by the complex number

• A. $\frac{z_1+z_2}{2}-z_3$
• B. $\frac{z_1+z_2-z_3}{2}$
• C. $\frac{z_1+z_2+z_3}{4}$
• D. $\frac{z_1-z_2-z_3}{2}$

Answer 1

The correct option is C $\frac{z_1+z_2+z_3}{4}$

Let $O,N,G,C$ be the orthocenter, ninepoint centre, centroid & circumcentre of the triangle ABC respectively.
As given circumcentre of the triangle $ABC$ is origin.
We know that, centroid divide orthocenter and circumcentre in the ratio $2:1$
$\therefore G=\frac{2O+C}{3}$
$\Rightarrow \frac{z_1+z_2{+z}_3}{3}=\frac{2O}{3}$
$\Rightarrow O=\frac{z_1+z_2{+z}_3}{2}$
Ninepoint centre is the mid-point of orthocenter and circumcentre
$\Rightarrow N=\frac{O+C}{2}$
$\therefore N=\frac{z_1+z_2{+z}_3}{4}$