[alogo] Proof of Poncelet's theorem

Consider a bundle of conics I(v,w), generated by conics v and w and three points {P,Q,R} on conic w. Assume also that the three sides of triangle ABC are tangent to respectively three members v, v1, v2 of the family. Then one can construct another triangle A'B'C' with vertices on w and sides correspondingly tangent to the same members of the family as the sides of ABC.

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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 v1 and apply Hart's lemma to the pairs lines {BB', CC'} and {BB', CC''}, implying, for each pair, the existence of a conic f', f'' respectively, tangent to both lines of the pair. Since each line (here BB') can be tangent at most to two conics of the bundle I(v,w), one of f', f'' must coincide with f. Assume that this is the case with f', so that BB' and CC' are both tangent to conic f. Then {AA',CC'} are both tangent to f and applying a third time Hart's lemma we see that {AC, A'C'} are both tangent to some conic g, coinciding with one out of the two member-conics tangent to AC. Take v2 = g.
By a continuity argument, we can presrcibe v2 and find A'B'C' near ABC with the stated property.
[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 A1...An inscribed in w and having sides A1A2 tangent to v1, A2A3 tangent to v2 etc.. v1, v2, ... being members of the family I(v,w). The conclusion is the same: one can find infinite many polygons B1...Bn also inscribed in w and having respective sides B1B2, B2B3, ... etc. tangent also to the corresponding conics v1, v2, ...etc..
[4] The theorem remains true if some of the conics v1, v2, ... coincide. In that case gives the proof of theorems similar to the great Poncelet theorem. This great theorem itself follows as a special case of the general theorem, in which v1 = v2 = ... i.e. all these conics coincide.
[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 pn envelope a member-conic of I(v,w).
[6] Starting with a point A1 on w and a series v1, v2, ... vn of member -conics of I(v,w) we can define a tangent from A1 to v1 intersecting again w at a second point A2. From A2 consider a tangent to v2 intersecting again w at A3 ... and continuing that way define a polygonal line A1...An+1, having each side AiAi+1 tangent to vi.
By changing continuously the position of A1, the (generalized) theorem garantees that the corresponding sides will continue to envelope the conics v1, v2, ...etc.. The same will be true for any diagonal AiAi+k of this polygonal line.

See Also



Baker, H. F. Plane Geometry New York, Chelsea Publishing Company 1971, p. 276.

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