Start with an arbitrary chord A'B' of w tangent to v and apply Hart's lemma, implying that lines {BB', AA'} are tangent to some member conic f of I(v,w). Draw then the tangents {B'C', B'C''} to v

By a continuity argument, we can presrcibe v

[1] The fact that each line L (not passing through the roots of the bundle) has two conics of the family I(v,w) tangent to it, follows from Desargues involution theorem. The two points of tangency with the respective two conics are the fixed points of the involution on L induced by the intersection of L with the members of I(v,w).

[2] By repeating the procedure we can find infinite many triangles inscribed in w and having sides, correspondingly, tangent to the same three members of the family I(v,w) as the sides of ABC.

[3] The theorem generalizes for polygons A

[4] The theorem remains true if some of the conics v

[5] A further application of the (generalized) theorem shows that not only the sides but also the diagonals of such inscribed (in w and circumscribed about v) Poncelet-polygons p

[6] Starting with a point A

By changing continuously the position of A

Hart_Lemma.html

Poncelet.html

ProjectiveBase.html

ProjectiveCoordinates.html

ProjectivePlane.html

Quadratic_Transformation.html

Tangent_Cuts_Circle.html

Tangent_Cuts_Envelope.html

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