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Soddy hyperbolae & ellipses

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  • John Sharp
    Mention of the Soddy hyperbolae made me construct them in Geogebra Does anyone know of the following results ie a reference in the literature: 1) The three
    Message 1 of 3 , Nov 13, 2012
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      Mention of the Soddy hyperbolae made me construct them in Geogebra

      Does anyone know of the following results ie a reference in the literature:

      1) The three hyperbolae all intersect in two points. Using the ETC, these
      are
      X(175) = ISOPERIMETRIC POINT
      X(176) = EQUAL DETOUR POINT

      2) Constructing the corresponding ellipses (two vertices as foci + the
      other as a point)

      These three ellipses intersect in 5 points which are also on the hyperbolae
      These points are not in the ETC

      John S


      [Non-text portions of this message have been removed]
    • rhutson2
      Dear John, I count 6 intersections (at least with the ETC reference triangle). It may depend on the shape of the triangle. I wonder if these 6 points lie on
      Message 2 of 3 , Nov 13, 2012
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        Dear John,

        I count 6 intersections (at least with the ETC reference triangle). It may depend on the shape of the triangle. I wonder if these 6 points lie on a common conic.

        Awhile back, I had found that the lines connecting the intersections of pairs these ellipses concur in X(20).

        Best regards,
        Randy

        --- In Hyacinthos@yahoogroups.com, John Sharp <JS.sliceforms@...> wrote:
        >
        > Mention of the Soddy hyperbolae made me construct them in Geogebra
        >
        > Does anyone know of the following results ie a reference in the literature:
        >
        > 1) The three hyperbolae all intersect in two points. Using the ETC, these
        > are
        > X(175) = ISOPERIMETRIC POINT
        > X(176) = EQUAL DETOUR POINT
        >
        > 2) Constructing the corresponding ellipses (two vertices as foci + the
        > other as a point)
        >
        > These three ellipses intersect in 5 points which are also on the hyperbolae
        > These points are not in the ETC
        >
        > John S
        >
        >
        > [Non-text portions of this message have been removed]
        >
      • Chris Van Tienhoven
        ... Dear John, I know these refences. 1. Both points are briefly mentioned at: http://www.xtec.cat/~qcastell/ttw/ttweng/definicions/d_Soddy_p.html 2. There is
        Message 3 of 3 , Nov 13, 2012
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          >[JS]
          > Does anyone know of the following results ie a reference in the literature:
          >
          > 1) The three hyperbolae all intersect in two points. Using the ETC, these
          > are
          > X(175) = ISOPERIMETRIC POINT
          > X(176) = EQUAL DETOUR POINT

          Dear John,

          I know these refences.
          1. Both points are briefly mentioned at: http://www.xtec.cat/~qcastell/ttw/ttweng/definicions/d_Soddy_p.html
          2. There is a more extensive description at this site:
          http://www.pandd.demon.nl/
          choose: M E E T K U N D E (at the left)
          choose: S
          choose: Soddy cirkels
          The site is in the Dutch language.

          Some years ago I found a most peculiar property of 10 Soddy-related points, including X(175) and X(176).
          These 10 points X(1), X(176), X(1371), X(482), X(1373), X(7), X(1374), X(481), X(1372), X(175) are collinear on the X(1).X(7)-line.
          They always lie in this order.
          They lie in a "perspective row" with vanishing point X(1).
          I made a picture of it in the file "Perspective Fields - part II" page 31.
          See: http://www.chrisvantienhoven.nl/index.php/mathematics/perspective-fields.html
          An explanantion of this feature can be found in the rest of the file.

          Best regards,

          Chris van Tienhoven
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