[alogo] Complex cross ratio (ABCD) = (( A-C)/(B-C))/((A-D)/(B-D))

Complex cross ratio of four points (ABCD) = ((A-C)/(B-C))/((A-D)/(B-D)).
1) (ABCD) is real if the points are all on a circle or line.
2) Assuming the points on a conic (ellipse), draw the tangents at these points. Let A*, B*, C*, D* be the corresponding intersections of these tangents with a fifth tangent e of the conic. Then set cr = (A*B*C*D*).

Calculate the cross ratio cr, using the User-Tool [ComplexCrossRatio], found and compiled in the file [EUC_Scripts\EUC_User_Tools\ComplexCrossRatio.txt]. The cross ratio is real, hence the point, representing it, lies on the real axis. (cr) remains invariant on the real axis (fixed), as P moves on the conic.
(switch to select-on-contour-tool(CTRL+2), catch and modify P, A, B, C, D).

A picture of another interesting case, related to a conic, is given in the file: Complex_Cross_Ratio2.html .

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By the polarity with respect to the conic, lines correspond to points and points to lines, cross ratio of four points on a line transforms to an equal cross ratio of four lines through a point. In particular the tangents AA*, BB*,... etc. map to A, B, ... etc. whereas points A*, B*, ... map to the lines (not shown) AP, BP, ... etc. Thus, the cross ratio (A*B*C*D*) transforms to the (equal) cross ratio of the four lines AP, BP, ... etc. This, in turn is independent of the position of P on the conic.
An interesting application of this fact can be seen in the file ParabolaProperty.html .

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