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**DeFacto** I am trying to show that the points 1, a,−(a*), 1/a in the complex plane lie on a circle. Where a* is the conjugate of a.

I know that given four points A,B,C,D then they lie on a circle iff

angle(ADC)+angle(ABC)=pi.

This condition seems wrong. The right condition is: angle(ABC)=angle(ADC).

In term of complex numbers, it means that and have the same argument, which is equivalent to the fact that their ratio is real:

(A,B,C,D are either colinear or cocyclic) iff

.

(it is called the cross-ratio of ) If you substitute with the complex numbers and simplify the ratio, you will see it is real indeed.