Hart_Lemma

Hart's lemma

Consider a bundle of conics I(v,w), generated by conics v and w and four points {P,Q,R,S} on conic w. Assume also that a pair of opposite sides of the complete quadrilateral, defined by the four points, PS and QR say, are tangent to v, the chord of contacts being AB. Then every conic of the bundle of conics passing through the four points {P,Q,R,S} is tangent to a conic-member v' of the bundle I(v,w) the chord of contacts being the same line AB.

Selecting the projective base as indicated, the equation of the conic (v) becomes v = xy-z² = 0 (see ProjectiveCoordinates.html ). Consider first the degenerate case of lines (PQ,RS), represented by a product of linear equations mn = 0. This degenerate conic belongs to the conic-bundle generated by w and the degenerate conic xy=0, hence it may be written in the form mn = kw + xy = 0, k being a constant. This however, is mn = kw + v + z² = 0 ==> mn - z² = kw+v. Thus, lines m (representing PQ), n (representing RS) are tangent to the conic v' = kw+v = 0, belonging to the family I(v,w), generated by v and w. Besides they have z = 0 as chord of contacts.
The same argument applies to every conic u of the bundle I(w,xy) i.e. the bundle of conics passing through {P,Q,R,S}. It implies u - z² is equal to a member w' of I(v,w). By considering the intersection points of u and line z and the tangents there, we prove easily the claim. Thus every conic of this bundle is tangent to a conic of the bundle I(v,w), the chord of contacts coinciding with line z = 0.

Remarks
[1] The above figure displays also conic v'' tangent to the pair of lines {QS, PR}. The lemma, in most cases, is applied to prove the existence of conics (like v', v'') tangent to the degenerate members of family I(w,xy).
[2] As seen from the proof, the arguments for the degenerate member-conic mn of I(w,xy) can be restricted to real conics. For more general members of the family I(w,xy) though, one has to work in the complex projective plane.
[3] Hart's lemma is the building block of Hart's proof, of Poncelet's great theorem.

References

Baker, H. F. Plane Geometry New York, Chelsea Publishing Company 1971, p. 276.
Berger, M Geometry II Paris, Springer Verlag, 1987, 16.6.6 p. 205.

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Hart's lemma

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