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Re: [EMHL] Proofs of Myakishev's theorems about squares and circumcenters

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  • Barry Wolk
    ... [rest snipped] These results generalize to using similar rectangles instead of squares. If the rectangles have sizes a:af, b:bf, and c:cf, then the
    Message 1 of 3 , Jun 12, 2003
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      --- Darij Grinberg wrote:
      > In the following, I am going to establish some results of
      > Alexei Myakishev, Jean-Pierre Ehrmann and me in Hyacinthos
      > messages #6338, #6339, #6340, #6341, #6344, #6345.
      >
      > Consider a triangle ABC and the squares BBaCaC, CCbAbA and
      > AAcBcB constructed upon its sides BC, CA, AB.
      >
      > _.-'Cb
      > _.-`' \
      > _.-' \
      > Ab \
      > \ \ _Ca
      > \ \ _.-� \
      > \ C-' \
      > \ _.-' \ \
      > \ _.-` \ Ba
      > \ _.-' \ _.-�
      > A----------------B-'
      > | |
      > | |
      > | |
      > | |
      > | |
      > | |
      > | |
      > Ac---------------Bc
      >
      > Let BcBa meet CaCb at A', and analogously define B' and C'.
      > Let XYZ be the antipedal triangle of the centroid G of
      > triangle ABC. Call Oa the circumcenter of triangle BaCaA',
      > and similarly define Ob and Oc.
      >
      > Then,
      >
      > (1) XA' is orthogonal to BC.
      >
      > (2) Triangles A'B'C' and XYZ are homothetic.
      >
      > (3) The homothetic center T is the centroid of both triangles.
      >
      > (4) This T is the reflection of the centroid G of ABC in the
      > circumcenter O of ABC.
      >
      > (5) The symmedian point T' of triangle A'B'C' lies on the
      > Euler line of triangle ABC.
      >
      > (6) Triangles A'B'C' and OaObOc are homothetic.
      >
      > (7) The homothetic center is T'.
      >
      > PROOFS (entirely synthetic).

      [rest snipped]

      These results generalize to using similar rectangles instead of
      squares. If the rectangles have sizes a:af, b:bf, and c:cf, then
      the coordinates of all your points are functions of f, and all 7
      of your results still hold in this general case. Do yuor proofs
      generalize?
      --
      Barry Wolk


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    • Darij Grinberg
      Dear Barry Wolk, ... Many thanks for this remark. The proofs do generalize! In fact, the crucial result that a median of triangle ABC is an altitude of the
      Message 2 of 3 , Jun 12, 2003
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        Dear Barry Wolk,

        You wrote:

        >> These results generalize to using similar rectangles
        >> instead of squares. If the rectangles have sizes a:af,
        >> b:bf, and c:cf, then the coordinates of all your
        >> points are functions of f, and all 7 of your results
        >> still hold in this general case. Do yuor proofs
        >> generalize?

        Many thanks for this remark. The proofs do generalize!
        In fact, the crucial result that a median of triangle
        ABC is an altitude of the corresponding flank is true
        for arbitrary similar rectangles.

        Sincerely,
        Darij Grinberg
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