Real Cross-Ratios: Complex Analysis Proof

[g1990]

Homework Statement

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

Homework Equations

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

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

 

[mighty2000]

. This can be done by showing that the cross ratio is real, and therefore the four points are collinear or concyclic. One way to show that the cross ratio is real is by using the fact that the argument of the cross ratio is equal to the sum of the arguments of the four points. If a+b is a multiple of pi, then the sum of the arguments of the four points is also a multiple of pi. This means that the cross ratio is real, and therefore the four points are collinear or concyclic. This completes the proof.