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PRIZE (Re: ORTHOCENTER - REFLECTIONS - CONCURRENT CIRCLES)

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  • Antreas Hatzipolakis
    [APH]: In fact we can take any point P (instead of H) and any points O1,O2,O3 on the circumcircles of PBC,PCA,PAB, resp. Then the circumcircles of the
    Message 1 of 2 , Sep 14 3:04 AM
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    [APH]:

    In fact we can take any point P (instead of H) and any points
    O1,O2,O3 on the circumcircles of PBC,PCA,PAB, resp.

    Then the circumcircles of the triangles

    AO2O3, BO3O1, CO1O2

    are concurrent.

    Anopolis #850
    http://groups.yahoo.com/neo/groups/Anopolis/conversations/messages/850

    For a proof I offer the book:

    R. G. SANGER: SYNTHETIC PROJECTIVE GEOMETRY (1939)

    APH

  • Antreas Hatzipolakis
    The problem is equivalent to this: Let ABC, A B C be two triangles. If the circumcircles of A BC, B CA, C AB are concurrent, then the circumcircles of AB C ,
    Message 2 of 2 , Sep 14 10:13 AM
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      The problem is equivalent to this:

      Let ABC, A'B'C' be two triangles.

      If the circumcircles of A'BC, B'CA, C'AB are concurrent, then
      the circumcircles of AB'C', BC'A', CA'B' are also concurrent .

      It sounds old! And for a reference the offer holds as well !!.

      Note: From the other version ie
      "If P is a fixed point and O1,O2,O3 three variable points on
      the circumcircles of PBC,PCA,PAB, resp  then the circumcircles
      of AO2O3, BO3O1, CO1O2 are concurrent" we may ask:
      As O1,O2,O3 move on the respective  circles, in what plane region
      the points of concurrence are located on?
      (I think it is not the entire plane)

      APH


      On Sat, Sep 14, 2013 at 1:04 PM, Antreas Hatzipolakis <anopolis72@...> wrote:
       
      [Attachment(s) from Antreas Hatzipolakis included below]

      [APH]:

      In fact we can take any point P (instead of H) and any points
      O1,O2,O3 on the circumcircles of PBC,PCA,PAB, resp.

      Then the circumcircles of the triangles

      AO2O3, BO3O1, CO1O2

      are concurrent.

      Anopolis #850
      http://groups.yahoo.com/neo/groups/Anopolis/conversations/messages/850

      For a proof I offer the book:

      R. G. SANGER: SYNTHETIC PROJECTIVE GEOMETRY (1939)

      APH




      --
      http://anopolis72000.blogspot.com/
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