Expert: Paul Klarreich - 7/10/2011

Question
Hi i would like to ask for help about this problem:
Prove that if z1+z2+z3=0 and norm of z1=norm of z2=norm of z3=1, then z1,z2,z3 are the vertices of an equilateral triangle inscribed in a unit circle.

thank you very much

Answer
Questioner:keithbads
Country:Rizal, Philippines
Category:Advanced Math
Private:No

Subject:complex analysis

Question:

Hi i would like to ask for help about this problem:
Prove that if z1+z2+z3=0 and |z1| = |z2| = |z3| = 1, then z1,z2,z3
are the vertices of an equilateral triangle inscribed in a unit
circle.

thank you very much
......................................................
Assume, wlog, that z1 lies along the positive x-axis. (Rotate the axes if necessary.)

So  z1 = 1 + 0i = 1    << norms are 1

Now  z2 = x2 + i y2  

and  z3 = x3 + i y3

Then if z1 + z2 + z3 = 0, or:

1 + 0i + x2 + i y2 + x3 + i y3 = 0 + 0i

We have these equations:

y2 + y3 = 0  -->   y3 = - y2, which we can just call y.

x2 + x3 = - 1

x2^2 + y^2 = 1  << norms are 1
x3^2 + y^2 = 1  << norms are 1

x2^2 = 1 - y^2
x3^2 = 1 - y^2

So x2^2 = x3^2 and  x2 = +- x3.

If x2 = - x3, then x2 + x3 = 0, but x2 + x3 = - 1

So x2 = x3, which means:

x2 + x2 = -1 -->  x2 = -1/2.

Now x2^2 = 1 - y^2, so

(-1/2)^2 = 1 - y^2

1/4 = 1 - y^2

y^2 = 3/4,

y = +- sqrt(3)/2

I think you can finish this up now.