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July 22

Geometric significance of cross-ratio values

In the article Cross-ratio, specifically Cross-ratio#Exceptional orbits, it says a cross ratio of 1, 1/2 or 2 indicates that two of the four points are projective harmonic conjugates of each other w.r.t. the other two. Similarly a cross ratio of 0, 1 or ∞ indicates that two of the four points are equal. It mentions one other orbit, e^±π/3, but does not give a similar geometric interpretation. Perhaps this isn't surprising since this value is impossible on the real line, but it is possible for geometries over other fields. The projective line over the complex numbers is essentially the Riemann sphere, so there ought to be a geometric condition on four points in the plane equivalent to saying that when the four points are interpreted as complex numbers their cross-ratio is e^±π/3. This property would be preserved under Circle inversions. Does anyone know if this condition can be stated in simple geometrical terms? --RDBury (talk) 15:31, 22 July 2019 (UTC)[reply]

I think I have the answer for this now. Draw circles through the four points three at a time to get four circles, any three of which intersect at one of the four points. Then the condition is that each pair of circles intersect each other at an angle of π/3. Circle inversion preserves angles and circles so that checks, and using a Lorenz transformation to map three of the points to three specific points you only need to check one case. I tried {0, 1, e^±π/3} which gives the desired cross-ratio and has the geometric property described, so everything checks. The case where three of the points are collinear yields a proposition that may be interesting in it's own right: Let A, B, and C be collinear with B between A and C. Draw equilateral triangles ABX and BCZ on the same side of the line, and ACY on the opposite side. Then the circles circumscribing the three triangles have a common point of intersection. --RDBury (talk) 03:45, 28 July 2019 (UTC)[reply]