POLYA011: Points on a conic (proposed by Jean-Pierre Ehrmann, 11/18/01).

Let C be a conic. A_1,A_2,..., A_n are points (not necessarily distinct) not lying on C. For any point M lying on C,
the line A_1M intersects again C at M_1,
the line A_2M_1 intersects again C at M_2,
..
the line A_nM_(n-1) intersecs again C at M_n.
Under what condition does  the line MM_n go through a fixed point when M moves on C?
For instance, if n = 2, the condition is that A_1 and A_2 are conjugate wrt C; if n = 3 the condition is that A_1, A_2, A_3 lie on a same line.

Background: Hyacinthos messages 4359 (Clark Kimberling) and 4360 (Gilles Boutte) :

Discussion. [JPE, 11/20/01]: Choose projective coordinates such as M(t) = (1,t,t^2) is a parametrization of C. If P = (x,y,z), the line PM(t) intersects C again at M((ty-z)/(tx-y)). Thus, if S(P) is the matrix     | y    -z |
                                                                                 |x     -y | , we get M_k = M((at+b)/(ct+d)) where
    | a    b |
    | c    d |    = S(A_k) S(A_(k-1))....S(A_1)
More over the line M(t) M((at+b)/(ct+d)) usually envelopes a conic bitangent to C at the fixed points of M(t) -> M((at+b)/(ct+d)) and this line goes through a fixed point iff a + d = 0 - the fixed point is (c,a,-b).
Hence we get two equivalent forms for our condition :
(1) trace [S(A_n) S(A_(n-1)).... S(A_1)] = 0
(2) A_n lies on the line joining the fixed points of the homography M ->M_(n-1); this line is the contact chord of C and the envelope of the line
MM_(n-1).
Note that (1) shows that the condition is linear wrt each A_k and invariant under a circular permutation of A_1,..., A_n. But I don't see, except for n = 2,3,  the geometric configuration of A_1,...,A_n, C when the condition is verified.
 

Bibliography.

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