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New centers?

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  • Emelyanov
    Dear friends, I am new in using trilinears. As an exercise in learning trilinears I invented following construction. Let P = x : y : z, A B C be P-cevian
    Message 1 of 2 , Jul 27 1:43 PM
      Dear friends,
      I am new in using trilinears. As an exercise in learning trilinears I
      invented following construction.
      Let P = x : y : z, A'B'C' be P-cevian tringle of ABC. Let LA be the radical
      axis of circles (AC'C) and (AB'B). Similarly we define lines LB and LC. It's
      not difficult to prove that LA, LB and LC concur in P' = a/(by+cz):
      b/(ax+cz): c/(ax+by)= sinA/(ysinB+zsinC): ...
      We have mapping P -> P'. By this mapping, for example, X(1)-> X(58),
      X(2)->X(6), X(3)-> X(54), X(4)-> X(4), X(6)-> X(251), X(8)-> X(1), X(20)->
      X(3), X(63)-> X(284), X(144)-> X(55), X(192)-> X(31), X(193)-> X(25),
      X(329)-> X(9),...
      What are the following centers:
      X{7}-> a^2 - (b - c)^2 : b^2 - (c - a)^2 : c^2 - (a - b)^2 = sinA/[tan(B/2)
      + tan(C/2)] : ...
      X(10) -> a/(2a + b + c) : b/(2b + c+ a) : c/(2c + a + c) ?
      We can obtain more cenetrs (or new construction for known centers).
      Best regards,
      Tatiana
    • yiu@fau.edu
      Dear Tatiana, [TE]: Let P = x : y : z, A B C be P-cevian tringle of ABC. Let LA be the radical axis of circles (AC C) and (AB B). Similarly we define lines
      Message 2 of 2 , Jul 27 5:38 PM
        Dear Tatiana,

        [TE]: Let P = x : y : z, A'B'C' be P-cevian tringle of ABC. Let LA
        be the radical axis of circles (AC'C) and (AB'B). Similarly we define
        lines LB and LC. It's not difficult to prove that LA, LB and LC
        concur in
        P' = a/(by+cz): b/(ax+cz): c/(ax+by)
        = sinA/(ysinB+zsinC): ...
        We have mapping P -> P'.

        *** This is the isogonal conjugate of the inferior of P. The inferor
        of P is the point dividing PG externally in the ratio 3:-1.

        By this mapping, for example, X(1)-> X(58),
        > X(2)->X(6), X(3)-> X(54), X(4)-> X(4), X(6)-> X(251), X(8)-> X(1), X
        (20)->
        > X(3), X(63)-> X(284), X(144)-> X(55), X(192)-> X(31), X(193)-> X
        (25),
        > X(329)-> X(9),...
        > What are the following centers:

        > X{7}-> a^2 - (b - c)^2 : b^2 - (c - a)^2 : c^2 - (a - b)^2
        = sinA/[tan(B/2) + tan(C/2)] : ...

        ***This is X(57). It is a point on the line OI, and is the
        intersection of the the three lines each joining an excenter to the
        point of tangency of the incenter with the corresponding side.

        > X(10) -> a/(2a + b + c) : b/(2b + c+ a) : c/(2c + a + c) ?
        > We can obtain more cenetrs (or new construction for known centers).

        ***This point is not in ETC.

        Best regards,
        Sincerely,
        Paul
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