The previous calculation shows that the cross ratio (A,B,G,H) is a ratio of the sines of the angles between the lines (see also ProjectiveLine.html ). Hence it is the same for every line intersecting these four lines. Thus, if a line crossing a bundle defines four harmonically conjugate points (i.e. (A,B,G,H)=-1), the same must be true for every other line intersecting this bundle.

This simple fact has many implications, some of them discussed below.

In fact, in this case

A well known variation of this rule is that two adjacent sides of the parallelogram and its diagonals (drawn parallel from the common vertex) build a harmonic bundle.

To prove this it suffices to apply (2) to the bundle of lines at D: (DE, DF, DH, DI), which form a harmonic bundle of lines (see Harmonic.html ).

There is another aspect of the invariance of the cross ratio of four lines intercepted by a fifth. This is the aspect of

The proof follows immediately by completing the figure to the complete quadrilateral defined by the four points {A,B,C,D}. Line FG is the polar of point E (see Polar_Construction.html ) and as such it is orthogonal to OE as is also KL. Since the bundle of lines F(A,B,E,G) is harmonic and KL is parallel to line FG the result of section 3 applies.

For another application in this context see Harmonic_Bundle2.html .

Butterfly.html

CrossRatio0.html

CrossRatioLines.html

Complex_Cross_Ratio.html

Complete_Quadrangle.html

Harmonic.html

Harmonic_Bundle2.html

Newton.html

Polar.html

Polar_Construction.html

ProjectiveLine.html

RectHypeRelation.html

TrilinearPolar.html

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