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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 $\displaystyle \frac{z_B-z_A}{z_B-z_C}$ and $\displaystyle \frac{z_D-z_A}{z_D-z_C}$ 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 $\displaystyle \frac{\tfrac{z_B-z_A}{z_B-z_C}}{\tfrac{z_D-z_A}{z_D-z_C}}\in\mathbb{R}$.

(it is called the cross-ratio of $\displaystyle z_A,z_B,z_C,z_D$) If you substitute with the complex numbers $\displaystyle a,-\bar{a},1,\frac{1}{a}$ and simplify the ratio, you will see it is real indeed.