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Real Cross-Ratios: Complex Analysis Proof

  1. Apr 30, 2010 #1
    1. The problem statement, all variables and given/known data
    Given four complex numbers, z1, z2, z3, and z4, show that (z4-z1)(z3-z2)/(z4-z2)(z3-z1) is real if and only if the four points lie on a circle or a line


    2. Relevant equations

    polar form of complex numbers: z=|z|e^(iarg(z))

    3. The attempt at a solution
    Let r be the cross ratio. Then, r=|(z4-z1)/(z4-z2)|e^(ia) * |(z3-z2)/(z3-z1)|e^(ib)

    Thus, I know that I need to prove that the four points lie on a cirlce or line iff a+b is a multiple of pi, where a and b are opposite angles in the quadrilateral formed by the four complex numbers. I know that if the points lie on a circle, a+b=pi, but I need to prove that if they lie on a line a+b is a multiple of pi. The hard part will be showing that if a+b is a multiple of pi, then the four complex numbers lie on a circle or line
     
  2. jcsd
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