**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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