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Radical axes, orhologic

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  • Antreas
    Let ABC be an acute triangle, (Ia), (Ib), (Ic) its excircles and (A),(B),(C) the excircles of the orthic triangle (ie centered at A,B,C). Denote: Ra = the
    Message 1 of 3 , Apr 11, 2013
      Let ABC be an acute triangle, (Ia), (Ib), (Ic) its
      excircles and (A),(B),(C) the excircles of the orthic triangle
      (ie centered at A,B,C).

      Denote:

      Ra = the radical axis of (Ia), (A)
      Rb = the radical axis of (Ib), (B)
      Rc = the radical axis of (Ic), (C)

      The triangles ABC, Triangle bounded by (Ra,Rb,Rc)
      are orthologic. The one orthologic center is the I of ABC.
      The other one?

      Variation:
      The excentral triangle IaIbIc as reference triangle
      and the ABC as orthic triangle:

      ie:

      Let ABC be an acute triangle, and IaIbIc its excentral triangle.
      Let (Ja), (Jb), (Jc) be excircles of IaIbIc.

      Denote:

      Ra = the radical axis of (Ia),(Ja)
      Rb = the radical axis of (Ib),(Jb)
      Rc = the radical axis of (Ic),(Jc)

      The triangles ABC, Triangle bounded by (Ra,Rb, Rc) are
      orthologic. The one orhologic center is the incenter of IaIbIc.
      The other one?

      APH
    • rhutson2
      Antreas, For your variation, I assume you meant The triangles IaIbIc, Triangle bounded by (Ra,Rb, Rc) are orthologic. In this case, the 2nd orthology center
      Message 2 of 3 , Apr 11, 2013
        Antreas,

        For your variation, I assume you meant 'The triangles IaIbIc, Triangle bounded by (Ra,Rb, Rc) are orthologic.' In this case, the 2nd orthology center is the circumcenter of the triangle bounded by (Ra, Rb, Rc). It is not in ETC (search 4.811597276294553), nor are its isogonal/isotomic/excentral isogonal/excentral isotomic conjugates or (anti)complements (wrt either ABC or IaIbIc).

        Randy

        --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
        >
        > Let ABC be an acute triangle, (Ia), (Ib), (Ic) its
        > excircles and (A),(B),(C) the excircles of the orthic triangle
        > (ie centered at A,B,C).
        >
        > Denote:
        >
        > Ra = the radical axis of (Ia), (A)
        > Rb = the radical axis of (Ib), (B)
        > Rc = the radical axis of (Ic), (C)
        >
        > The triangles ABC, Triangle bounded by (Ra,Rb,Rc)
        > are orthologic. The one orthologic center is the I of ABC.
        > The other one?
        >
        > Variation:
        > The excentral triangle IaIbIc as reference triangle
        > and the ABC as orthic triangle:
        >
        > ie:
        >
        > Let ABC be an acute triangle, and IaIbIc its excentral triangle.
        > Let (Ja), (Jb), (Jc) be excircles of IaIbIc.
        >
        > Denote:
        >
        > Ra = the radical axis of (Ia),(Ja)
        > Rb = the radical axis of (Ib),(Jb)
        > Rc = the radical axis of (Ic),(Jc)
        >
        > The triangles ABC, Triangle bounded by (Ra,Rb, Rc) are
        > orthologic. The one orhologic center is the incenter of IaIbIc.
        > The other one?
        >
        > APH
        >
      • Antreas Hatzipolakis
        Yes, Randy, IaIbIc....., which plays the role of the reference triangle! Thanks ... -- http://anopolis72000.blogspot.com/ [Non-text portions of this message
        Message 3 of 3 , Apr 11, 2013
          Yes, Randy, IaIbIc....., which plays the role of the reference triangle!

          Thanks


          On Fri, Apr 12, 2013 at 12:34 AM, rhutson2 <rhutson2@...> wrote:

          > **
          >
          >
          > Antreas,
          >
          > For your variation, I assume you meant 'The triangles IaIbIc, Triangle
          > bounded by (Ra,Rb, Rc) are orthologic.' In this case, the 2nd orthology
          > center is the circumcenter of the triangle bounded by (Ra, Rb, Rc). It is
          > not in ETC (search 4.811597276294553), nor are its
          > isogonal/isotomic/excentral isogonal/excentral isotomic conjugates or
          > (anti)complements (wrt either ABC or IaIbIc).
          >
          > Randy
          >
          >
          > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
          > >
          > > Let ABC be an acute triangle, (Ia), (Ib), (Ic) its
          > > excircles and (A),(B),(C) the excircles of the orthic triangle
          > > (ie centered at A,B,C).
          > >
          > > Denote:
          > >
          > > Ra = the radical axis of (Ia), (A)
          > > Rb = the radical axis of (Ib), (B)
          > > Rc = the radical axis of (Ic), (C)
          > >
          > > The triangles ABC, Triangle bounded by (Ra,Rb,Rc)
          > > are orthologic. The one orthologic center is the I of ABC.
          > > The other one?
          > >
          > > Variation:
          > > The excentral triangle IaIbIc as reference triangle
          > > and the ABC as orthic triangle:
          > >
          > > ie:
          > >
          > > Let ABC be an acute triangle, and IaIbIc its excentral triangle.
          > > Let (Ja), (Jb), (Jc) be excircles of IaIbIc.
          > >
          > > Denote:
          > >
          > > Ra = the radical axis of (Ia),(Ja)
          > > Rb = the radical axis of (Ib),(Jb)
          > > Rc = the radical axis of (Ic),(Jc)
          > >
          > > The triangles ABC, Triangle bounded by (Ra,Rb, Rc) are
          > > orthologic. The one orhologic center is the incenter of IaIbIc.
          > > The other one?
          > >
          > > APH
          > >
          >
          >
          >



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


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