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If the complex number z1,z2 ,z3 represent the vertices of an equilateral triangle such that |z1|=|z2|=|z3|, then find z1+z2+z3

If the complex number z1,z2 ,z3 represent the vertices of an equilateral triangle such that |z1|=|z2|=|z3|, then find z1+z2+z3

Grade:11

2 Answers

mycroft holmes
272 Points
7 years ago
Here circumcentre is the origin O. Let the vertices be A,B,C. We have angles AOB = BOC = COA = 120o and OA = OB = OC since its given that |z1| =  |z2| =  |z3|
So we have z_2 = z_1 \omega, z_3 = z_1 \omega^2 where \omega = e^{\frac{2\pi i}{3} } is the primitive cube root of infinity.
 
Soz_1+z_2+z_3 = z_1(1+\omega + \omega^2 ) = 0
mycroft holmes
272 Points
7 years ago
Shorter way to go about it is: Invoke the result that if the circumcenter is at the origin then the complex number corresponding to the orthocenter is z1+z2+z3.
 
Since the triangle is equilateral, orthocenter and circumcenter coincide. So we have z1+z2+z3 = 0

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