Every pair of points (A, X) defines with the corresponding (F(A), F(X)) the intersection point P of lines [A,F(X)] and [X,F(A)]. The locus of P, is a line the

By considering the bundles of lines A'(A,B,C,X) and A(A',B',C',X') the arguments of LineHomographyAxis.html transfer verbatim to prove this case. The only difference is in the interpretation of the

Notice that the proposition is equivalent to Pascal's theorem, since a hexagon defines by its vertices (several) homographies preserving the conic. For each such homography the homography axis coincides with the line resulting from Pascal's theorem.

Notice that the intersection points (if any) V, W of the homography axis with the conic are the fixed points of F on the conic. The intersection point U of the tangents to the conic (if any) at V and W is a third fixed point of the extension of F to the whole projective plane. See the file ConicHomography.html for a discussion of the extension of F on the whole projective plane (it is always possible and can be done in a unique way).

Consider now the case in which the homography axis intersects the conic and denote the equations of the tangents UW by (v=0), UV by (w=0) and the equation of the chord VW by (u=0). The one-parameter family of conics: v*w - k*u

ConicHomography.html

HomographyConic3by3.html

HomographicRelation.html

InvolutiveHomography.html

LineHomographyAxis.html

LineHomographyAxis2.html

LineSimilarityAxis.html

CrossRatio0.html

CrossRatioLines.html

ParabolaSkew.html

Pascal.html

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