Prove that traingle by complex numbers `z_(1),z_(2)` and `z_(3)` is equilateral if `|z_(1)|=|z_(2)| = |z_(3)|` and `z_(1) + z_(2) + z_(3)=0`

Explanation

Triangle is formed by complex numbers `z_(1),z_(2)` and `z_(3)`
Let `|z_(1)| = |z_(2)| = |z_(3)|` and `z_(1) + z_(2)+z_(3)=0`
If `|z_(1)| = |z_(2)| =|z_(3)|`, then `z_(1),z_(2)` and `z_(3)` are equidistant form origin.
So, origin is circumcentre of the triangle.
Also, centroid of the triangle is `(z_(1)+z_(2)+z_(3))/(3) = 0`.
Thus, circumcentre and centroid coincide.
Hence, triangle is equilateral.