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11964回覆: [EMHL] Another way to look at X(1276) and X(1277).

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  • 黃嘉麟
    Jan 1, 2006
      Dear all,

      Sorry, my mistake, there is no typo in the description of X(1276) and X(1277) in ETC.

      For the concyclic properties of X(1), X(484), X(1276), X(1277), here is another proof.
      X(1277) = Xd( 15| 1), X(1276) = Xd( 16| 1),
      X( 1) = Xd( 4| 1), X( 484) = Xd(186| 1). d = "anticevian triangle"
      Since X(15)=inverse-in-circumcircle of X(16) and X(4)=inverse-in-circumcircle of X(186), it is easy to show X(4),X(186),X(15),X(16) are concyclic. Then it follows that X(1), X(484), X(1276), X(1277) are also concyclic.

      Note: For a circle with center O, radius R, if (P1,P2) and (Q1,Q2) are inverse pairs in circle O, then OP1*OP2=OQ1*OQ2=R^2. So triangle OP1Q2 is similar to triangle OP2Q1 and P1,P2,Q1,Q2 are concyclic.

      Happy new year
      Best regards
      Chia-Lin Hwang

      space <hwangd2000@...> 說:
      Dear all,

      In ETC, X(1276) and X(1277)(2nd and 3rd EVANS PERSPECTOR)
      are described as perspectors of excentral triangle and special
      triangles T, T'(some typoes in the description I believe).
      Moses(8/9/2004),further gave a relationship of these two points,
      X(1276) = inverse-in-Bevan-circle of X(1277).

      Here give another way to look at X(1276) and X(1277).
      Actually they are ISODYNAMIC POINTs X(15), X(16) wrt excentral

      Since X(15)=inverse-in-circumcircle of X(16) and the Bevan circle is
      the circumcircle of the excentral triangle,then the result of Moses
      follows obviously.

      I would like to suggest "double-index notation," Xd( p| q),
      to describe those notable points wrt special triangle.
      Xd( p| q) = X(p) wrt X(q)-special triangle,
      d denotes some kind of special triangle.
      I believe it is helpful to build new concept and extend Kimberling's
      points. For example:
      X(1277) = Xd(15| 1) = X(15)-of-excentral triangle
      X(1276) = Xd(16| 1) = X(16)-of-excentral triangle
      where d = "anticevian triangle" and
      excentral triangle= anticevian triangle of X(1).

      The properties of X(1276) and X(1277) in ETC is not many,
      now we know how to add more(with the help of X(15),X(16)). For
      example the pedal triangle of X(1276),X(1277) wrt excentral triangle
      is equilateral.

      Best regards
      Chia-Lin Hwang
      Taipei, Taiwan

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