Quote:
Originally Posted by Laurent  Quote:
Originally Posted by DeFacto 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 fact, the correct condition is a combination of those two. If B and D are on the same side of AC then angle(ABC)=angle(ADC). If they are on opposite sides then angle(ADC)+angle(ABC)=π. In either case, the cross-ratio is real. In one case it is positive and in the other case it is negative.