## 9883Re: [EMHL] The Brocard points of a quadrilateral (Geoff Millin)

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• Jun 8, 2004
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Dear Alexey
[AZ]
> >> I have a next hypothesis. Let A_1...A_n is the cyclic polygon.
Then
> >> 1) The Brocard points exist iff the symmedians of triangles
> >> A_{i-1}A_iA_{i+1} from A_i concur in point L.
[JPE]
> >And what do you think of the following conjecture :
> >A cyclic polygon has Brocard points if and only if the polygon is
> >the inverse of a regular polygon?
[AZ]
> I think that this two conditions are equivalent. If the point L
exists then
> the projective transformation T conserving the circumcircle and
such that
> T(L)=O transforms our polygon to regular. In the circle this
transformation
> coincide with some inversion.

Suppose that A_1,...,A_n has Brocard points P,Q; let M be a Poncelet
point of the pencil generated by the circumcircle (O) of the polygon
and the circle OPQ. the line MA_k intersects again (O) at B_k.
Then B_1,...,B_n is regular.
Of course, it is necessary to prove that such a point M is real, ie
that (O) and the circle OPQ cannot intersect.
It is necessary too to prove the reciprocal.
I think that it could be possible to name these two points M the
isodynamic points of the Brocardian polygon because
MA_(i+k)/M_A(i-k) = A_iA_(i+k)/A_iA_(i-k) for all i and k.
Friendly. Jean-Pierre
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