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Re: [EMHL] Concyclic Circumcenters

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  • Barry Wolk
    Here is a variation that is symmetric in ABC. Given D, for what P do the circumcenters of the six triangles PBC, PCA, PAB, PDA, PDB, and PDC all lie on a
    Message 1 of 10 , May 18, 2010
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      Here is a variation that is symmetric in ABC.

      Given D, for what P do the circumcenters of the six triangles PBC, PCA, PAB, PDA, PDB, and PDC all lie on a conic?

      This locus will contain the circumcircles of triangles BCD, ACD, ABD, and ABC. The rest of the locus should be of small degree.

      Francisco Javier wrote:

      > Dear Antreas,
      >
      > Let D=(u:v:w) be the coordinates of $D$ with respect $ABC$
      >
      > If P is on some of the circumcircles of DAB, ABC, BCD, CDA
      > then we have only three circumcenters and they will be
      > concyclic. On the remaining cases I get that P=(x:y:z)
      > should be on the cubic
      >
      > c^2 v x^2 y - c^2 u x y^2 - a^2 v x^2 z + b^2 v x^2 z + c^2
      > v x^2 z +
      > b^2 w x^2 z + a^2 u x y z - b^2 u x y z - c^2 u x y z +
      > a^2 w x y z +
      >   b^2 w x y z - c^2 w x y z + a^2 w y^2 z - b^2 u x
      > z^2 -
      > a^2 v x z^2 - b^2 v x z^2 + c^2 v x z^2 - a^2 v y z^2 =
      > 0.
      >
      > Perhaps Bernard Gibert can explain us the meaning of this cubic.
      >
      > I am able to check that this cubic passes through A, B, C,
      > D, but I have not more information on it.
      >
      > Best regards,
      >
      > Francisco Javier.
      >
      > "Antreas" <anopolis72@...> wrote:
      > >
      > > Let ABCD be a quadrilateral.
      > >
      > > For which points P, the circumcenters of PAB,
      > > PBC, PCD, PDA are concyclic?
      > >
      > > APH
      --
      Barry Wolk
    • Bernard Gibert
      Dear Barry, ... I far prefer this more symmetrical approach but I m just wondering how can you predict this small degree ? Anything to do with Lemoyne s
      Message 2 of 10 , May 18, 2010
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        Dear Barry,

        > Given D, for what P do the circumcenters of the six triangles PBC, PCA, PAB, PDA, PDB, and PDC all lie on a conic?
        >
        > This locus will contain the circumcircles of triangles BCD, ACD, ABD, and ABC. The rest of the locus should be of small degree.

        I far prefer this more symmetrical approach but I'm just wondering how can you predict this "small" degree ?

        Anything to do with Lemoyne's theories ?

        Best regards

        Bernard

        [Non-text portions of this message have been removed]
      • Barry Wolk
        ... Because I was able to keep reducing the degree by canceling some common factors while attempting (unsuccessfully) to do the calculation. And symmetry meant
        Message 3 of 10 , May 19, 2010
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          > Dear Barry,
          >
          > > Given D, for what P do the circumcenters of the six
          > triangles PBC, PCA, PAB, PDA, PDB, and PDC all lie on a conic?
          > >
          > > This locus will contain the circumcircles of triangles
          > BCD, ACD, ABD, and ABC. The rest of the locus should be of
          > small degree.
          >
          > I far prefer this more symmetrical approach but I'm just
          > wondering how can you predict this "small" degree ?

          Because I was able to keep reducing the degree by canceling some common factors while attempting (unsuccessfully) to do the calculation. And symmetry meant there would be other common factors which hadn't occurred yet.

          This turned out to be trivial. The perpendicular bisectors of the 4 segments PA, PB, PC, PD each contain 3 of those 6 circumcenters. And the 6 intersection points of 4 lines can lie on a conic only when they partially collapse, where 2 of its points or lines coincide.

          So the expected "small degree" turned out to be degree 0.

          >
          > Anything to do with Lemoyne's theories ?

          I don't know what that theory is.

          >
          > Best regards
          >
          > Bernard

          --
          Barry Wolk
        • Antreas
          ... Variations: We can replace circumcenters with orthocenters, or even concyclicity with orthocentricity. That is: For which points P, the orthocenters of of
          Message 4 of 10 , May 19, 2010
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            --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
            >
            > Let ABCD be a quadrilateral.
            >
            > For which points P, the circumcenters of PAB,
            > PBC, PCD, PDA are concyclic?

            Variations:

            We can replace circumcenters with orthocenters, or
            even concyclicity with orthocentricity.

            That is:

            For which points P, the orthocenters of
            of PAB, PBC, PCD, PDA form an orthocentric
            tetrad?

            APH
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