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excentral triangle as tangents to ellipses

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  • rhutson2
    Dear Hyacinthists, Are these known results? 1) Let Ea be the ellipse with B and C as foci and passing through A. Define Eb, Ec cyclically. Let La be the
    Message 1 of 4 , Jul 29, 2012
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      Dear Hyacinthists,

      Are these known results?

      1) Let Ea be the ellipse with B and C as foci and passing through A. Define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.

      2) Let A', B', C' be points on segments BC, CA, AB respectively. Let Ea be the ellipse with B' and C' as foci and passing through A, and define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.

      3) Let P and P' be isogonal conjugates both inside ABC. Let Ea be the ellipse with P and P' as foci and passing through A, and define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.

      Thank you,
      Randy Hutson
    • Randy Hutson
      Closely related to this:  If hyperbola is substituted for ellipse below, then in each case, the tangents concur in the incenter. Randy ... [Non-text
      Message 2 of 4 , Jul 30, 2012
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        Closely related to this:  If 'hyperbola' is substituted for 'ellipse' below, then in each case, the tangents concur in the incenter.

        Randy




        >________________________________
        > From: rhutson2 <rhutson2@...>
        >To: Hyacinthos@yahoogroups.com
        >Sent: Sunday, July 29, 2012 10:51 PM
        >Subject: [EMHL] excentral triangle as tangents to ellipses
        >
        >

        >Dear Hyacinthists,
        >
        >Are these known results?
        >
        >1) Let Ea be the ellipse with B and C as foci and passing through A. Define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.
        >
        >2) Let A', B', C' be points on segments BC, CA, AB respectively. Let Ea be the ellipse with B' and C' as foci and passing through A, and define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.
        >
        >3) Let P and P' be isogonal conjugates both inside ABC. Let Ea be the ellipse with P and P' as foci and passing through A, and define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.
        >
        >Thank you,
        >Randy Hutson
        >
        >
        >
        >
        >

        [Non-text portions of this message have been removed]
      • Nikolaos Dergiades
        Dear Randy I think that are known because if F, F are the foci of an ellipse and P a point on it then the tangent at P to the ellipse is perpendicular to the
        Message 3 of 4 , Jul 30, 2012
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          Dear Randy
          I think that are known
          because if F, F' are the foci
          of an ellipse and P a point on it
          then the tangent at P to the ellipse
          is perpendicular to the internal bisector PQ
          of triangle PFF' at P.
          In case of an hyperbola the tangent is the line PQ.
          Best regards
          Nikos Dergiades

          > Are these known results?
          >
          > 1) Let Ea be the ellipse with B and C as foci and passing
          > through A.  Define Eb, Ec cyclically.  Let La be
          > the tangent to Ea at A, and define Lb, Lc cyclically. 
          > The lines La, Lb, Lc form the excentral triangle.
          >
          > 2) Let A', B', C' be points on segments BC, CA, AB
          > respectively.  Let Ea be the ellipse with B' and C' as
          > foci and passing through A, and define Eb, Ec
          > cyclically.  Let La be the tangent to Ea at A, and
          > define Lb, Lc cyclically.  The lines La, Lb, Lc form
          > the excentral triangle.
          >
          > 3) Let P and P' be isogonal conjugates both inside
          > ABC.  Let Ea be the ellipse with P and P' as foci and
          > passing through A, and define Eb, Ec cyclically.  Let
          > La be the tangent to Ea at A, and define Lb, Lc
          > cyclically.  The lines La, Lb, Lc form the excentral
          > triangle.
          >
          > Thank you,
          > Randy Hutson
          >
          >
          >
          > ------------------------------------
          >
          > Yahoo! Groups Links
          >
          >
          >     Hyacinthos-fullfeatured@yahoogroups.com
          >
          >
        • Randy Hutson
          Dear Nikos, Of course, this makes perfect sense now.  Thanks! Randy ... [Non-text portions of this message have been removed]
          Message 4 of 4 , Jul 31, 2012
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            Dear Nikos,

            Of course, this makes perfect sense now.  Thanks!

            Randy




            >________________________________
            > From: Nikolaos Dergiades <ndergiades@...>
            >To: Hyacinthos@yahoogroups.com
            >Sent: Tuesday, July 31, 2012 12:42 AM
            >Subject: Re: [EMHL] excentral triangle as tangents to ellipses
            >
            >Dear Randy
            >I think that are known
            >because if F, F' are the foci
            >of an ellipse and P a point on it
            >then the tangent at P to the ellipse
            >is perpendicular to the internal bisector PQ
            >of triangle PFF' at P.
            >In case of an hyperbola the tangent is the line PQ.
            >Best regards
            >Nikos Dergiades
            >
            >> Are these known results?
            >>
            >> 1) Let Ea be the ellipse with B and C as foci and passing
            >> through A.  Define Eb, Ec cyclically.  Let La be
            >> the tangent to Ea at A, and define Lb, Lc cyclically. 
            >> The lines La, Lb, Lc form the excentral triangle.
            >>
            >> 2) Let A', B', C' be points on segments BC, CA, AB
            >> respectively.  Let Ea be the ellipse with B' and C' as
            >> foci and passing through A, and define Eb, Ec
            >> cyclically.  Let La be the tangent to Ea at A, and
            >> define Lb, Lc cyclically.  The lines La, Lb, Lc form
            >> the excentral triangle.
            >>
            >> 3) Let P and P' be isogonal conjugates both inside
            >> ABC.  Let Ea be the ellipse with P and P' as foci and
            >> passing through A, and define Eb, Ec cyclically.  Let
            >> La be the tangent to Ea at A, and define Lb, Lc
            >> cyclically.  The lines La, Lb, Lc form the excentral
            >> triangle.
            >>
            >> Thank you,
            >> Randy Hutson
            >>
            >>
            >>
            >> ------------------------------------
            >>
            >> Yahoo! Groups Links
            >>
            >>
            >>     Hyacinthos-fullfeatured@yahoogroups.com
            >>
            >>
            >
            >
            >------------------------------------
            >
            >Yahoo! Groups Links
            >
            >
            >
            >
            >
            >

            [Non-text portions of this message have been removed]
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