Question

Let the complex numbers `z_(1),z_(2)` and `z_(3)` be the vertices of an equailateral triangle. If `z_(0)` is the circumcentre of the triangle , then prove that ` z_(1)^(2) + z_(2)^(2) + z_(3)^(2) = 3z_(0)^(2)`.

Answer 1

Let `A(z_(1)),B(z_(2))" and "C(z_(3))` be the vertices of an equilateral triangle.
`:." "z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(1)z_(3)" "(1)`
Now, `(z_(1)+z_(2)+z_(3))^(2)=z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+2(z_(1)z_(2)+z_(2)z_(3)+z_(1)z_(3))`
`=3(z_(1)^(2)+z_(2)^(2)+z_(3)^(2))" "("Using (1)")`
Also, we have `z_(0)=(z_(1)+z_(2)+z_(3))/(3)` (as centroid will coincide with circumcentre)
`(z_(1)+z_(2)+z_(3))^(2)=9z_(0)^(2)`
So, `z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=3z_(0)^(2)`