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22053Fwd: [EGML] Ortholog triangle. Locus

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  • Antreas Hatzipolakis
    Mar 14, 2014
      Keywords:  McCay cubic

      ---------- Forwarded message ----------
      From: César Lozada
      Date: Sat, Mar 15, 2014 at 1:19 AM
      Subject: RE: [EGML] Ortholog triangle. Locus
      To: Anopolis@yahoogroups.com



      Which is the locus of P such that ABC, M1M2M3 are orthologic ?


      M'Cay Cubic – Through: A, B, C, excenters, X(1), X(3), X(4),  X(1075), X(1745), X(3362)


      Orthologic centers (trilinears):

          For P=I

             Qa = 1/((b+c)*a^2+5*a*b*c-(b-c)*(b^2-c^2)) :: = (35,3296) /\ (36,5558)

             Qm = X(5045)


         For P=O

             Qa = 1/(a*((b^2+c^2)*a^6+(-3*b^4+b^2*c^2-3*c^4)*a^4+(b^2+c^2)*(3*c^4-8*b^2*c^2+3*b^4)*a^2-(b^4-b^2*c^2+c^4)*(b^2-c^2)^2)) ::

                   =  (140,578) /\ (235,5063)

             Qm = ((b^2+c^2)*a^6+(-3*b^4+4*b^2*c^2-3*c^4)*a^4+(-3*c^2+b^2)*(-c^2+3*b^2)*(b^2+c^2)*a^2-(b^4+c^4)*(b^2-c^2)^2)*a : :

                   = Midpoint of (3,389)

        For P=H

             Qa = 1/(a*(a^8-4*(b^2+c^2)*a^6+2*(3*c^4+b^2*c^2+3*b^4)*a^4-4*(b^2+c^2)*(b^2-c^2)^2*a^2+(b^4+c^4)*(b^2-c^2)^2)) : :

                  = (3,53) /\ (4,97)     

             Qm = X(389)


      César Lozada




      De: Anopolis@yahoogroups.com [mailto:Anopolis@yahoogroups.com] En nombre de Antreas Hatzipolakis
      Enviado el: Jueves, 13 de Marzo de 2014 09:12 p.m.
      Para: anopolis@yahoogroups.com
      Asunto: [EGML] Ortholog triangle. Locus



      I think there were discussions on the following configuration,

      but I am not sure if the locus was discussed.

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


      Ab, Ac = the reflections of A' in AB,AC, resp.

      A2, A3 = the reflections of A' in PB, PC, resp.

      Mab, Mac = the midpoints of A2Ab, A3Ac, resp.

      Similarly Mbc, Mba and Mca,Mcb

      M1,M2,M3 = the midpoints of MbcMcb, McaMac, MabMba, resp.

      Which is the locus of P such that ABC, M1M2M3 are orthologic ?

      I think H, O are on the locus (orthologic centers?).