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22073Fwd: [EGML] Re: Homothetic Center - Locus [1 Attachment]

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  • Antreas Hatzipolakis
    Mar 26, 2014
    • 0 Attachment

      Cuαrtica lugar de centros de homotecia

      ---------- Forwarded message ----------
      From: Angel Montesdeoca
      Date: Wed, Mar 26, 2014 at 2:44 PM
      Subject: Re: [EGML] Re: Homothetic Center - Locus [1 Attachment]
      To: Anopolis@yahoogroups.com

      [Attachment(s) from amontes included below]

      [Antreas Hatzipolakis Anopolis#1264]:

      In some pairs (P,Q) we have P = Q:
      {2, 2},{15,15}, {16,16},{2574,2574}, {2575,2575}, {2771,2771},
      and the natural question is which is the locus of P such that P = Q?

      *** I excluded the points at infinity. I forgot to remove the centers
      2574, 2575, 2771 and 2772.

      *** The locus of P such that P = Q is the quartic of the barycentric

      b^2 c^2 x^3 y-a^2 c^2 x y^3-b^2 c^2 x^3 z+a^2 b^2 x^2 y z-b^4 x^2 y
      z-a^2 c^2 x^2 y z+c^4 x^2 y z+a^4 x y^2 z-a^2 b^2 x y^2 z+b^2 c^2 x y^2
      z-c^4 x y^2 z+a^2 c^2 y^3 z-a^4 x y z^2+b^4 x y z^2+a^2 c^2 x y z^2-b^2
      c^2 x y z^2+a^2 b^2 x z^3-a^2 b^2 y z^3=0

      Points on the curve:

      X(2), X(15), X(16), X(3413), X(3414)

      The intersections Va, Vb, Vc and Wa, Wb, Wc the interior and exterior
      bisectors with the side lines of ABC.

      Best regards
      Angel Montesdeoca