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Re: [EMHL] Gossard again

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  • Antreas P. Hatzipolakis
    ... No. It is ZEEMAN. Julio s message in HM (Fri, 03 Dec 1999) on Z. reads: _____________________________________ Dear John, Antreas, & all, 1. If a,
    Message 1 of 84 , Aug 14 5:08 PM
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      Richard Guy wrote:

      >Perhaps `Zeman' ??? R.

      No. It is ZEEMAN.

      Julio's message in HM (Fri, 03 Dec 1999) on Z. reads:

      _____________________________________

      <quote>
      Dear John, Antreas, & all,

      1. If a, b, c, d denote four straight lines (of the same plane) such
      that $d$ be parallel to the Euler line of the triangle $abc$, then
      the three Euler lines

      a' of bcd, b' of cda, c' of dab,

      are parallel to a, b, c, respectively.

      This theorem of professor P. Zeeman Gz [from Delft, renowned all over
      the world as the city of Delft blue earthenware, in the Netherlands] was
      published early in 1903 in the Dutch journal _Wiskundige Opgaven_ [vol
      VIII (1899-1902), p 305]. I first heard of this theorem thanks to a brief
      note [_Mathesis_ (3) 3 (1903) p 60] written by the famous triangle guru
      Joseph Neuberg (1840-1926) of Luxembourg:

      "Si quatre droites a, b, c, d d'un plan sont telles
      que l'une d'elles est parall\ele \a la droite d'Euler
      du triangle des trois autres, cette propri/et/e a
      encore lieu pour chacune des autres droites."

      Neuberg noticed he could easily prove this proposition by using a formula
      he had discussed in an earlier volume of _Wiskundige Opgaven_ [vol III
      (1886-1889), p 372].


      2. An interesting complement to Zeeman's theorem is that of professor
      C.A. Cikot [from Bois-le-Duc (s'Hertogenbosch) capital of North Brabant
      province, south central Netherlands], a proposition first published in
      an article written also by Neuberg [_Mathesis_ (3) 8 (1908) p 233]:

      Let a, b, c, d denote four straight lines (of the same plane). If $d$
      is the Euler line of the triangle $abc$, then the three Euler lines a',
      b', c' respectively of the triangles $bcd$, $cda$, $dab$ form a triangle
      congruent to $abc$, and the Euler line d' of $a'b'c'$ coincides with $d$.

      What about "The Cikot Point"? ...

      John, this theorem implicitly includes the congruence-by-reflection
      statement you've independently discovered. Do you agree? ...

      By the way, many thanks for your kindest words in your previous letter
      to me on this thread!

      With best regards,
      Julio Gonzalez Cabillon

      </quote>

      I don't know what the initial P. stands for
      but the name appears as:

      ZEEMAN GZ., P.
      ZEEMAN GZN, P.

      Probably the P. stands for PIETER (must
      not be confused with the 1902 Physics nobelist
      PIETER ZEEMAN)

      P. ZEEMAN GZN wrote several geometrical articles
      in Dutch periodicals.

      In my next message I will give the German abstract
      of an article of his about triangle cubics.

      Antreas


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    • Antreas Hatzipolakis
      The following is an obvious consequence of Gossard theorem, so probably was already studied. Gossard Theorem: Let ABC be a triangle and L its Euler linr. The
      Message 84 of 84 , Jun 27, 2014
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        The following is an obvious consequence of Gossard theorem,
        so probably was already studied.

        Gossard Theorem:

        Let ABC be a triangle and L its Euler linr.
        The Euler line La of the triangle bounded by (AB,AC, L)
        is parallel to BC

        Similarly Lb, Lc. So the triangle bounded by (La,Lb,Lc) is
        homothetic to ABC. To homothetic center is called
        Zeeman - Gossard perspector X(402)

        Now, let ABC be a triangle and P a point.

        Let L1 be the Euler line of PBC and L11 the Euler line
        of the triangle bounded by (PB,PC,L1).

        Similarly:

        Let L2 be the Euler line of PCA and L22 the Euler line
        of the triangle bounded by (L2, PC, PA)

        and

        Let L3 be the Euler line of PAB and L33 the Euler line
        of the triangle bounded by (L3, PA, PB).

        We have that ABC, triangle bounded by (L11, L22, L33) are
        homothetic.

        Homothetic center in terms of the coordinates of P?

        Locus: For which P's the L11, L22, L33 are concurrent?

        Antreas





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