Since homographic relations preserve the cross ratio, line bundles X(X',Y',Z',V') and X'(X,Y,Z,V) define the same cross ratio on every line that intersects them. The two bundles having in common the line XX', have, by the well known property of line bundles (see CrossRatioLines.html ), the other pairs of corresponding lines intersect allong a line. Fixing Z, V and their images Z', V' by the homographic relation we get the proof of the above proposition.

Notice that this property implies trivially the theorem of Pappus (see PappusLines.html ), since three points X, Y, Z on line e and three other points X', Y', Z' on line e' uniquely determine a homographic relation mapping X to X' etc..

Notice that this property generalizes to conics, almost verbatim, and defines what is called the

To understain the real story with this line see also its connexion with the conic enveloping lines XX', YY' etc., contained in the file LineHomographyAxis2.html .

CrossRatioLines.html

HomographicRelation.html

HomographyAxis.html

LineHomographyAxis2.html

LineSimilarityAxis.html

PappusLines.html

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