Quote:
Originally Posted by 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.