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Re: [EMHL] Re: A CONJECTURE on NEUBERG (was: Thomson ?)

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  • xpolakis@otenet.gr
    Dear Jean-Pierre, ... So, we have the Theorem: Let AA , BB , CC be the three altitudes of ABC, and Let Ab, Ac be the orth. proj. of A on AB, AC resp. Bc, Ba
    Message 1 of 2 , Sep 1, 2001
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      Dear Jean-Pierre,

      [APH]:
      >> Let ABC be a triangle, and A'B'C' the pedal triangle of P.
      >>
      >>
      >> A
      >> /\
      >> / \
      >> / \
      >> C' B'
      >> / \
      >> / P \
      >> Ab Ac
      >> / \
      >> B-------A'-------C
      >>
      >> Let Ab, Ac be the orth. proj. of A' on AB, AC resp.
      >> Bc, Ba " B' BC, BA
      >> Ca, Cb " C' CA, CB
      >> [...] is it true that the OH/OK lines of the triangles
      >> A'AbAc, B'BcBa, C'CaCb concur [at the OH/OK line of A'B'C']
      >> for P in {H, O, I} ?

      [JPE]:
      >You're right for P = H and the Eulerline : the four Euler lines
      >concur.

      So, we have the Theorem:

      Let AA', BB', CC' be the three altitudes of ABC, and
      Let Ab, Ac be the orth. proj. of A' on AB, AC resp.
      Bc, Ba " B' BC, BA
      Ca, Cb " C' CA, CB

      Then the Euler lines of A'B'C', A'AbAc, B'BcBa, C'CaCb are concurrent.

      Which is the point of concurrence [a point lying on the Euler line of
      the orthic triangle A'B'C' of ABC]? Is it in ETC?

      And another conjecture:
      The Euler lines of the triangles AAbAc, BBcBa, CCaCb are concurrent.
      [are also concurrent for P = I or O ?]

      Thank You!

      Antreas
    • jean-pierre.ehrmann@wanadoo.fr
      Dear Antreas and other Hyacinthists, ... concurrent. ... of ... X-442 of the orthic triangle (not in ETC, I think) ... Very good. Yes, they are, but not on the
      Message 2 of 2 , Sep 1, 2001
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        Dear Antreas and other Hyacinthists,
        > [APH]:

        > So, we have the Theorem:
        >
        > Let AA', BB', CC' be the three altitudes of ABC, and
        > Let Ab, Ac be the orth. proj. of A' on AB, AC resp.
        > Bc, Ba " B' BC, BA
        > Ca, Cb " C' CA, CB
        >
        > Then the Euler lines of A'B'C', A'AbAc, B'BcBa, C'CaCb are
        concurrent.
        >
        > Which is the point of concurrence [a point lying on the Euler line
        of
        > the orthic triangle A'B'C' of ABC]? Is it in ETC?

        X-442 of the orthic triangle (not in ETC, I think)

        > And another conjecture:
        > The Euler lines of the triangles AAbAc, BBcBa, CCaCb are concurrent.

        Very good. Yes, they are, but not on the Euler line of A'B'C'. I
        cannot recognize the common point.

        The locus of P for which The Euler lines of the triangles AAbAc,
        BBcBa, CCaCb are concurrent is again a quartic.

        > Are also concurrent for I and O ?

        I don't think so.

        Friendly. Jean-Pierre
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