Complex_Cross

1. Complex cross ratio

The complex cross ratio of four points in the (complex) plane is defined to be:
(ABCD) = ((A-C)/(B-C))/((A-D)/(B-D)),
where points are identified with complex numbers.

[1] (ABCD) is real if and only if the points are all four on a circle or line.
[2] Assuming the points are on a circle, project them on a line (e), from a point X of the circle. Let A', B', C', D' the corresponding projections. Then (ABCD) = (A'B'C'D').

Where (ACB), (ADB) denotes the oriented angle-measures. This quotient is real if and only if these angles are equal or complementary and inverse oriented, which shows the first claim.
The second claim is proved in [3] of CrossRatio0.html .

2. Remarks

[1] The cross ratio of four points on a circle generalizes to the cross ratio of four points on a conic. In the case of a circle we can express this ratio with complex numbers. This is not so with the generalization, which uses instead property [2] of the previous section.
[2] The cross ratio of four points on tangential conics transfers this idea to the cross ratio of four tangents of a conic. This is considered in FourTangentsCrossRatio.html .
[3] Harmonic quadrangles are defined by four points on a circle which are harmonic i.e. (A,B,C,D) = -1. This is discussed in HarmonicQuad.html .

1. Complex cross ratio

2. Remarks

See Also