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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` |
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Answer» 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. |
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