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Re: [EMHL] Three Triangle Circles

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  • Nikolaos Dergiades
    Dear Andreas, Do you have a construction of circles (Kabb), (Kacc)? If A1 is the mid point of (Kab), (Kac) then the triangles ABC, A1B1C1 are perspective at
    Message 1 of 3 , Feb 23, 2013
      Dear Andreas,

      Do you have a construction of circles
      (Kabb), (Kacc)?

      If A1 is the mid point of (Kab), (Kac)
      then the triangles ABC, A1B1C1 are perspective
      at X(1123) isotomic conjugate of (bc+S:ca+S:ab+S)

      If the circles
      (Kabb) is tangent externally to (Kab) and to BC, and (Kacc)to
      (Kac) and to BC then the triangles ABC, K1K2K3
      are perspective at the isotomic conjugate of
      (SA + 2S + 2bc : SB + 2S +2ca : . . .)

      But your problem seems to be difficult.

      Best regards
      Nikos Dergiades


      > Let ABC be a triangle and (Kab),
      > (Kac) the two congruent
      > and tangent circles such that (Kab) is tangent to the
      > sides of the internal angle B and (Kac) to the sides of the
      >
      > internal angle C.
      > Let (Kabb), (Kacc) be the two congruent tangent circles such
      > that
      > (Kabb) is tangent externally to (Kab) and to AB, and (Kacc)
      > to
      > (Kac) and to AC.
      >
      > The quadrilateral KabKabbKaccKac is inscriptible. Let (K1)
      > be its incircle. Similarly define (K2), (K3).
      >
      > (For external angles we get another three circles.)
      >
      > K1K2K3 is perspective with triangles????
      > Is it with ABC?
      >
      > APH
      >
      >
      >
      > ------------------------------------
      >
      > Yahoo! Groups Links
      >
      >
      >     Hyacinthos-fullfeatured@yahoogroups.com
      >
      >
    • Antreas Hatzipolakis
      Dear Nikos, A pure geometric construction of (Kabb), (Kacc) would be quite difficult, since even the construction for lines instead of circles (Kab), (Kac) is
      Message 2 of 3 , Feb 23, 2013
        Dear Nikos,

        A pure geometric construction of (Kabb), (Kacc) would be quite difficult,
        since even the construction for lines instead of circles (Kab), (Kac)
        is difficult !!
        (it is the problem of constucting two mutually tangent congruent
        circlles, each one of whose is tangent to the sides of opposite angles
        of quadrilateral)

        But algebraically/trigonometrically we can compute the radius of the first
        circles (Kab), (Kac), and after that the radius of (Kabb), (Kacc), more or less
        easily I guess!!

        APH

        On Sat, Feb 23, 2013 at 4:18 PM, Nikolaos Dergiades <ndergiades@...> wrote:
        > Dear Andreas,
        >
        > Do you have a construction of circles
        > (Kabb), (Kacc)?
        >
        > If A1 is the mid point of (Kab), (Kac)
        > then the triangles ABC, A1B1C1 are perspective
        > at X(1123) isotomic conjugate of (bc+S:ca+S:ab+S)
        >
        > If the circles
        > (Kabb) is tangent externally to (Kab) and to BC, and (Kacc)to
        > (Kac) and to BC then the triangles ABC, K1K2K3
        > are perspective at the isotomic conjugate of
        > (SA + 2S + 2bc : SB + 2S +2ca : . . .)
        >
        > But your problem seems to be difficult.
        >
        > Best regards
        > Nikos Dergiades
        >
        >
        >> Let ABC be a triangle and (Kab),
        >> (Kac) the two congruent
        >> and tangent circles such that (Kab) is tangent to the
        >> sides of the internal angle B and (Kac) to the sides of the
        >>
        >> internal angle C.
        >> Let (Kabb), (Kacc) be the two congruent tangent circles such
        >> that
        >> (Kabb) is tangent externally to (Kab) and to AB, and (Kacc)
        >> to
        >> (Kac) and to AC.
        >>
        >> The quadrilateral KabKabbKaccKac is inscriptible. Let (K1)
        >> be its incircle. Similarly define (K2), (K3).
        >>
        >> (For external angles we get another three circles.)
        >>
        >> K1K2K3 is perspective with triangles????
        >> Is it with ABC?
        >>
        >> APH
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