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22071Fwd: [EGML] Re: Homothetic Center - Locus

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  • Antreas Hatzipolakis
    Mar 25, 2014


      ---------- Forwarded message ----------
      From:  Angel Montesdeoca
      Date: Tue, Mar 25, 2014 at 7:09 PM
      Subject: [EGML] Re: Homothetic Center - Locus
      To: Anopolis@yahoogroups.com


       


      [Antreas Hatzipolakis, Anopolis#1262]

      Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

      Denote:

      Ab, Ac = the orthogonal projections of A' on PB',PC', resp.
      L1 = the radical axis of the circles (Ab, AbB') and (Ac, AcC')
      M1 = the radical axis of the circle (Ab, AbC') and (Ac, AcB')

      L1 and M1 are parallel (since both are perpendicular to AbAc)
      Similarly L2, M2 and L3, M3

      The triangles bounded by (L1,L2,L3) and (M1,M2,M3) are homothetic.
      Homothetic center ? (in terms of the hom. coordinates of P)


      *** If P=(x:y:z), the first coordinate of the homothetic center Q is:

      a^8xy(y-z)z(x+y+z)+
      a^6(b^2z(x^3y-x^2y(y-2z)-yz^2(y+ z) + 
         x(-2y^3-y^2z-2yz^2+z^3))+ 
        c^2y(-x^3z+x^2z(-2y+z)+y^2z(y+z)+ 
          x(-y^3+2y^2z+yz^2+2z^3)))+
      a^4(c^4y(2x^3z-y^2z(y+z)+x^2(3y^2+yz+z^2)+ 
         x(y^3-3y^2z-2yz^2-z^3))+
        b^4z(-2x^3y+yz^2(y+z)-x^2(y^2+yz+3z^2)+ 
         x(y^3+2y^2z+3yz^2-z^3))+ 
         b^2c^2(y-z)(x^3(y+z)-yz(y+z)^2 - 
         x^2(y^2-yz+z^2)+x(y^3+y^2z+yz^2+z^3)))+
      a^2x(-c^6y(x^2z-yz^2+x(3y^2+z^2))+
         b^6z(x^2y-y^2z+x(y^2+3z^2))+
         b^4c^2(x^3(y+3z)+x^2(-y^2+yz+z^2)-
         yz(2y^2+yz+2z^2)+x(y^3+yz^2-2z^3))- 
         b^2c^4(x^3(3y+z)+x^2(y^2+yz-z^2)- 
         yz(2y^2+yz+2z^2)+x(-2y^3+y^2z+z^3)))
      - b^2c^2(b^2-c^2)x^4(c^2(3y+z)+b^2(y+3z))


      Pairs of points {P,Q}:

      {1, 3},{2, 2}, {4,5}, {5,3628}, {6,3098}, {15,15}, {16,16}, {20,3}, {23,468}, {376,549}, {381,547}, {382,546}, {401,441}, {550,3530}, {631,632}, {858,5159}, {1370,1368},  {1657,548},  {2475,442}, {2574,2574}, {2575,2575}, {2771,2771}, {2772,2772},  {3000,3}, {3091,1656}, {3146,4}, {3151,440}, {3153,2072},  {3522,631}, {3523,3526}, {3529,550}, {3543,381},  {3627,3850},  {3830,5066}, {3832,3090}, {3839,5055},  {4190,474}, {4240,402},  {5046,4187}, {5056,5070}, {5059,20}, {5068,5067}, {5073,3853}, {5189,858}

      Best regards
      Angel Montesdeoca




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      http://anopolis72000.blogspot.com/