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Re: [EMHL] Two equivalent theorems (an old and a new(?))

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  • Alexey.A.Zaslavsky
    - ... (resp.) ... Dear Attreas! This is evident. The lines AA , BB , CC are parallel, so their reflections concur in the point isogonally conjugated to
    Message 1 of 2 , Feb 5, 2004
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      ->
      >NEW(?) THEOREM:
      >Let A', B', C' be the orthogonal projections of A,B,C on the line HP.
      >The reflections of AA', BB', CC' in the angle bisectors AA*, BB*, CC*
      (resp.)
      >of ABC concur at P' on the circumcircle of ABC.
      >
      Dear Attreas!
      This is evident. The lines AA', BB', CC' are parallel, so their reflections
      concur in the point isogonally conjugated to infinite.

      Sincerely Alexey
    • xpolakis
      ... HP. ... CC* ... reflections ... Dear Alexey, Yes, this is obvious for the general case (ie for an arbitrary line) But in the special case, ie for a line
      Message 2 of 2 , Feb 6, 2004
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        [APH]:
        > >NEW(?) THEOREM:
        > >Let A', B', C' be the orthogonal projections of A,B,C on the line
        HP.
        > >The reflections of AA', BB', CC' in the angle bisectors AA*, BB*,
        CC*
        > (resp.)
        > >of ABC concur at P' on the circumcircle of ABC.
        > >
        [AZ]:
        > This is evident. The lines AA', BB', CC' are parallel, so their
        reflections
        > concur in the point isogonally conjugated to infinite.

        Dear Alexey,

        Yes, this is obvious for the general case (ie for an arbitrary line)
        But in the special case, ie for a line passing through H,
        the isogonal conjugate is the SAME point we get with the OLD theorem
        in my first message.

        Greetings

        Antreas
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