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Inconic question

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  • shokoshu2
    Suppose a triangle inconic touches AB in X, BC in Y and CA in Z. Counting parameters implies there is exact one equation in the values of AX,BX,BY,CY,CZ,AZ.
    Message 1 of 4 , Jul 17, 2010
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      Suppose a triangle inconic touches AB in X, BC in Y and CA in Z.
      Counting parameters implies there is exact one equation
      in the values of AX,BX,BY,CY,CZ,AZ. Which is?

      Hauke
    • shokoshu2
      ... Ceva, since Brianchon. OK, the question was lame :-) OK, let s up it a bit. Can any point inside ABC be the Brianchon point of an inconic? (I suspect yes.)
      Message 2 of 4 , Jul 25, 2010
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        --- In Hyacinthos@yahoogroups.com, "shokoshu2" <fc3a501@...> wrote:
        >
        > Suppose a triangle inconic touches AB in X, BC in Y and CA in Z.
        > Counting parameters implies there is exact one equation
        > in the values of AX,BX,BY,CY,CZ,AZ. Which is?
        >
        Ceva, since Brianchon. OK, the question was lame :-)
        OK, let's up it a bit. Can any point inside ABC be the
        Brianchon point of an inconic? (I suspect yes.)
        Which is the locus of all inparabola Brianchon points?
        (This surely is listed on Mathworld...somewhere.)

        Hauke
      • Barry Wolk
        ... So we can take X=(0,y ,z ), Y=(x ,0,z ), Z=(x ,y ,0), abd the conic is (x/x )^2 + (y/y )^2 + (z/z )^2 - 2yz/y z - 2zx/z x - 2xy/x y = 0. ... This conic
        Message 3 of 4 , Jul 27, 2010
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          "shokoshu2" wrote:

          > > Suppose a triangle inconic touches AB in X, BC in Y and CA in Z.
          > > Counting parameters implies there is exact one equation
          > > in the values of AX,BX,BY,CY,CZ,AZ. Which is?
          >
          > Ceva, since Brianchon. OK, the question was lame :-)

          So we can take X=(0,y',z'), Y=(x',0,z'), Z=(x',y',0), abd the conic is

          (x/x')^2 + (y/y')^2 + (z/z')^2 - 2yz/y'z' - 2zx/z'x' - 2xy/x'y' = 0.

          > OK, let's up it a bit. Can any point inside ABC be the
          > Brianchon point of an inconic? (I suspect yes.)
          > Which is the locus of all inparabola Brianchon points?
          > (This surely is listed on Mathworld...somewhere.)
          >
          > Hauke

          This conic is a parabola iff it is tangent to the line at infinity
          iff y'z'+z'x'+x'y'=0, so (x',y',z') is on the Steiner circumellipse.
          --
          Barry Wolk
        • Francois Rideau
          The locus of all inparabola Brianchon points is the Steiner circumellipse. For any inparabola, the join of the focus and the Brianchon point is on a very well
          Message 4 of 4 , Jul 28, 2010
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            The locus of all inparabola Brianchon points is the Steiner circumellipse.
            For any inparabola, the join of the focus and the Brianchon point is on a
            very well known point in ETC.
            Friendly Francois

            On Tue, Jul 27, 2010 at 7:49 PM, Barry Wolk <wolkbarry@...> wrote:

            >
            >
            > "shokoshu2" wrote:
            >
            > > > Suppose a triangle inconic touches AB in X, BC in Y and CA in Z.
            > > > Counting parameters implies there is exact one equation
            > > > in the values of AX,BX,BY,CY,CZ,AZ. Which is?
            > >
            > > Ceva, since Brianchon. OK, the question was lame :-)
            >
            > So we can take X=(0,y',z'), Y=(x',0,z'), Z=(x',y',0), abd the conic is
            >
            > (x/x')^2 + (y/y')^2 + (z/z')^2 - 2yz/y'z' - 2zx/z'x' - 2xy/x'y' = 0.
            >
            > > OK, let's up it a bit. Can any point inside ABC be the
            > > Brianchon point of an inconic? (I suspect yes.)
            > > Which is the locus of all inparabola Brianchon points?
            > > (This surely is listed on Mathworld...somewhere.)
            > >
            > > Hauke
            >
            > This conic is a parabola iff it is tangent to the line at infinity
            > iff y'z'+z'x'+x'y'=0, so (x',y',z') is on the Steiner circumellipse.
            > --
            > Barry Wolk
            >
            >
            >


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