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22000TCS: P16 - P19

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  • Antreas
    Aug 25, 2013
    • 0 Attachment
      [Hatzipolakis - Lozada]

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

      Denote:

      Ab, Ac = the reflections of A' in BB',CC', resp.

      Bc, Ba = the reflections of B' in CC', AA', resp.

      Ca, Cb = the reflections of C' in AA;, BB', resp.

      Na,Nb,Nc = the NPC centers of ABaCa, BCbAb, CAcBc, resp.

      For P = H,I the triangles ABC, NaNbNc are orthologic.


      ORTHOLOGIC CENTERS. [in trilinears]

      O1 stands for the orthologic center (ABC, NaNbNc)
      O2 stands for the orthologic center (NaNbNc,ABC)


      FOR P=I:

      P16:

      O1=1/(a*(a^5-(b+c)*a^4-(2*(b^2+b*c+c^2))*a^3+(2*(b+c))*(b^2-b*c+c^2)*a^2+(b^2+b*c+c^2)^2*a-(b+c)*(c^4+b^4-b*c*(2*b^2-b*c+2*c^2)))) ::

      O1=(71,5341)/\(72,3585)

      O1= -0.780302077498124, -0.98098196223250, 4.679945260759860

      O1 on Jerabek hyperbola

      P17:

      O2=(2*a^7-(2*b*c+3*c^2+3*b^2)*a^5-(b+c)^3*a^4-b*c*(b^2+c^2)*a^3+2*(b^2-c^2)^2*(b+c)*a^2+(b^2+3*b*c+c^2)*(b^2-c^2)^2*a-(b-c)*(b^2-c^2)^3)/a::

      O2=(1,30)/\(3,229)

      O2= -0.720017732347441, -0.71365923086162, 4.467052133587395

      FOR P=H:

      P18:

      O1=1/(a*((b^2+c^2)*a^6+(4*b^2*c^2-3*c^4-3*b^4)*a^4+(b^2+c^2)*(3*c^4-7*b^2*c^2+3*b^4)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2))::

      No relations with ETC centers 1-5543

      P19:

      O2=((b^2+c^2)*a^6+(-3*b^4+8*b^2*c^2-3*c^4)*a^4+(3*(b^2+c^2))*(b^2-c^2)^2*a^2-(b^4+8*b^2*c^2+c^4)*(b^2-c^2)^2)*a::

      O2=(4,51)/\(25,3357)

      O2=-14.372486901377580, -16.09367038759592, 21.415891781648130


      Reference:
      Sun Aug 25, 2013
      http://tech.groups.yahoo.com/group/Anopolis/message/877