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

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  • jpehrmfr
    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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