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reflections in cevians

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  • Antreas P. Hatzipolakis
    Let ABC be a triangle and A B C the cevian triangle of I. In an old discussion we have seen that the reflections of AA in B C , BB in C A and CC in A B
    Message 1 of 2 , Sep 1, 2005
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      Let ABC be a triangle and A'B'C' the cevian triangle of I.
      In an old discussion we have seen that the reflections of
      AA' in B'C', BB' in C'A' and CC' in A'B' are concurrent.

      How about the reflections of B'C' in AA', C'A' in BB',
      A'B' in CC'?

      Is the triangle formed by these reflections in perspective
      with ABC?

      In general:

      Let ABC be a triangle, P a point, A'B'C' its cevian triangle
      and P* its isogonal conjugate.

      Let La, Lb, Lc be the reflections of B'C' in AA*, C'A' in BB*,
      A'B' in CC*.

      Which is the locus of P such that ABC, (La,Lb,Lc) are perspective?

      [If La, Lb, Lc are the reflections of AA* in B'C', BB* in C'A',
      CC* in A'B', then the locus is the whole plane]

      APH


      --
    • Paul Yiu
      Dear Antreas, [APH]: Let ABC be a triangle and A B C the cevian triangle of I. ... *** Yes, the perspector is X(81) with barycentric coordinates
      Message 2 of 2 , Sep 1, 2005
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        Dear Antreas,

        [APH]: Let ABC be a triangle and A'B'C' the cevian triangle of I.
        >In an old discussion we have seen that the reflections of
        >AA' in B'C', BB' in C'A' and CC' in A'B' are concurrent.
        >
        >How about the reflections of B'C' in AA', C'A' in BB',
        >A'B' in CC'?
        >
        >Is the triangle formed by these reflections in perspective
        >with ABC?

        *** Yes, the perspector is X(81) with barycentric coordinates
        (a/(b+c):b/(c+a):c/(a+b)).

        Best regards
        Sincerely
        Paul



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