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AF_a, BF_b, CF_c are concurrent

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  • Jean-Pierre.EHRMANN
    Antreas wrote in message 1 of Jan 1 : ? Theorem The lines AF_a, BF_b, CF_c are concurrent where the nine-point circle touches the excircles at F_a, F_b, F_c.
    Message 1 of 5 , Jan 7, 2000
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      Antreas wrote in message 1 of Jan 1 :
      ? Theorem
      The lines AF_a, BF_b, CF_c are concurrent where the nine-point circle
      touches the excircles at F_a, F_b, F_c.
      This is true. The intersection of the three lines is barycenter of A, B, C
      with coefficients :
      (b + c)^2 (a + b - c) (a + c - b)
      (c + a)^2 (b + c - a) (b + a - c)
      (a + b)^2 (c + a - b) (c + b - a)
      where a, b, c are the legths of the sides BC, CA, AB.
      May be there is a special name for this point. Why not the Hatzipolakis
      point or the first Hatzipolakis point - many other ones will perhaps
      follow -?
      Proof by brute force :
      A barycentric equation for the nine-point circle is
      (b^2 + c^2 - a^2) u^2 + (c^2 + a^2 - b^2) v^2 + (a^2 + b^2 - c^2) w^2 - 2
      c^2 u v - 2 a^2 v w - 2 b^2 w u = 0
      and for the A-excircle
      (a + b + c)^2 u^2 + (a + b - c)^2 v^2 + (a + c - b)^2 w^2 + 2 (a + b + c) (a
      + b - c) u v - 2 (a + b - c) (a + c - b) v w + 2 (a + b +c) (a + c - b) w u
      = 0
      Hence F_a is barycenter of A, B, C with coefficients
      - (b - c)^2 (a + b + c)
      (a + c)^2 (a + b - c)
      (a + b)^2 (a + c - b)
      By symetry, we have, for F_b
      (b + c)^2 (b + a - c)
      - (c - a)^2 (b + c + a)
      (b + a)^2 (b + c - a)
      and, for F_c,
      (c + b)^2 (c + a - b)
      (c + a)^2 (c + b - a)
      - (a - b)^2 (c + a + b)
      and the result with an easy computation.
      Friendly from France.
      Jean-Pierre Ehrmann
    • xpolakis@xxxxxx.xxxxxxxxxxxxx.xxxxxxxxxx
      Dear Jean-Pierre ... This Theorem is well-known. The (?) means that I don t know who first discovered it. See:
      Message 2 of 5 , Jan 7, 2000
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        Dear Jean-Pierre

        You wrote:

        >Antreas wrote in message 1 of Jan 1 :
        >? Theorem
        >The lines AF_a, BF_b, CF_c are concurrent where the nine-point circle
        >touches the excircles at F_a, F_b, F_c.

        This Theorem is well-known. The (?) means that I don't know who first
        discovered it.
        See:
        http://cedar.evansville.edu/~ck6/tcenters/class/feuer.html

        >This is true. The intersection of the three lines is barycenter of A, B, C
        >with coefficients :
        >(b + c)^2 (a + b - c) (a + c - b)
        >(c + a)^2 (b + c - a) (b + a - c)
        >(a + b)^2 (c + a - b) (c + b - a)
        >where a, b, c are the legths of the sides BC, CA, AB.
        >May be there is a special name for this point. Why not the Hatzipolakis
        >point or the first Hatzipolakis point - many other ones will perhaps
        >follow -?

        :-))

        >Proof by brute force :
        >A barycentric equation for the nine-point circle is
        >(b^2 + c^2 - a^2) u^2 + (c^2 + a^2 - b^2) v^2 + (a^2 + b^2 - c^2) w^2 - 2
        >c^2 u v - 2 a^2 v w - 2 b^2 w u = 0
        >and for the A-excircle
        >(a + b + c)^2 u^2 + (a + b - c)^2 v^2 + (a + c - b)^2 w^2 + 2 (a + b + c) (a
        >+ b - c) u v - 2 (a + b - c) (a + c - b) v w + 2 (a + b +c) (a + c - b) w u
        >= 0
        >Hence F_a is barycenter of A, B, C with coefficients
        >- (b - c)^2 (a + b + c)
        >(a + c)^2 (a + b - c)
        >(a + b)^2 (a + c - b)
        >By symetry, we have, for F_b
        >(b + c)^2 (b + a - c)
        >- (c - a)^2 (b + c + a)
        >(b + a)^2 (b + c - a)
        >and, for F_c,
        >(c + b)^2 (c + a - b)
        >(c + a)^2 (c + b - a)
        >- (a - b)^2 (c + a + b)
        >and the result with an easy computation.
        >Friendly from France.
        >Jean-Pierre Ehrmann

        Greetings from Athens

        Antreas
      • John Conway
        ... Antreas - this is in the Greek book on triangle geometry that you kindly sent me. It is in fact part of the desmic theory. There are four points
        Message 3 of 5 , Jan 9, 2000
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          On Sat, 8 Jan 2000, Jean-Pierre.EHRMANN wrote:

          > Antreas wrote in message 1 of Jan 1 :
          > ? Theorem
          > The lines AF_a, BF_b, CF_c are concurrent where the nine-point circle
          > touches the excircles at F_a, F_b, F_c.

          Antreas - this is in the Greek book on triangle geometry that you
          kindly sent me. It is in fact part of the "desmic" theory. There are
          four points Fa,Fb,Fc,Fo where the NPC touches the four incircles,
          and four perspectors of ABC with the triangles formed by 3 of them,
          namely

          Fo' of Fa,Fb,Fc
          Fa' of Fo,Fc,Fb
          Fb' of Fc,Fo,Fa
          Fc' of Fb,Fa,Fo

          Then symmetrically, Fo,Fa,Fb,Fc are the perspectors of ABC with
          the triangles formed by taking triples from Fo',Fa',Fb',Fc', and these
          two quartets form a desmic triple with A,B,C,D, where D (the "desmon")
          is where the lines FoFo',FaFa',FbFb',FcFc' meet.

          John Conway
        • Steve Sigur
          on 1/7/00 8:16 PM Jean-Pierre.EHRMANN wrote ... I call this point the Feuerbach mate but perhaps it deserves a better name. It is the second center of
          Message 4 of 5 , Jan 10, 2000
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            on 1/7/00 8:16 PM Jean-Pierre.EHRMANN wrote

            >The lines AF_a, BF_b, CF_c are concurrent where the nine-point circle
            >touches the excircles at F_a, F_b, F_c.
            >This is true. The intersection of the three lines is barycenter of A, B, C
            >with coefficients :
            >(b + c)^2 (a + b - c) (a + c - b)
            >(c + a)^2 (b + c - a) (b + a - c)
            >(a + b)^2 (c + a - b) (c + b - a)


            I call this point the Feuerbach mate but perhaps it deserves a better
            name. It is the second center of similarity of the incircle and the 9pt
            circle. This point, along with Fo, is on the line connecting the incenter
            to the 9pt circle.

            The Feuerbach points are the first centers of similarity between the 9pt
            circles and the 4 in/excircles.

            Since it is weak and the above formulas depend on the sides, this point
            has 3 extra versions, which are themselves the second centers of
            similarity of the 9PC and the excircles.

            I denote these 4 points Fmo, Fma, Fmb, and Fmc.

            The points Fx, Fmx, and ABCN form a desmic triple of quadrangles in the
            following way


            o a b c

            I N A B C
            II Fo Fa Fb Fc
            III Fmo Fma Fmb Fmc

            N, the 9 pt center is the strong point that hold them together (the
            "desmon," in Conway's terminology). The four incenter Ix are the harmonic
            "halo" accompanying these points.

            Steve
          • darij_grinberg <darij_grinberg@web.de>
            I am going to resurrect a discussion which ended two years ago. Hereby, I refer to #61, #93, #94, #110, #125. The problem is to prove that the lines AFa, BFb,
            Message 5 of 5 , Dec 10, 2002
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              I am going to resurrect a discussion which ended two years ago.
              Hereby, I refer to #61, #93, #94, #110, #125.

              The problem is to prove that the lines AFa, BFb, CFc are concurrent,
              where Fa, Fb, Fc are the points of touch of the 9pt circle with the
              excircles. I am going to show that all three lines pass through the
              point F', the internal center of similtude of the 9pt circle with the
              incircle.

              In fact, note that the Feuerbach point F is the EXTERNAL CENTER OF
              SIMILTUDE of 9pt and incircle, and Fa, Fb, Fc are the INTERNAL
              CENTERS OF SIMILTUDE of 9pt and respective excircles. Now,

              * F' is the internal center of similtude of 9pt and incircle;
              * Fa is the internal center of similtude of 9pt and a-excircle;
              * A is the external center of similtude of incircle and a-excircle
              (really easy to prove).

              By Monge's theorem, the point A, F' and Fa are collinear, i. e. AFa
              passes through F'. Similarly, BFb and CFc pass through F', qed.

              The point F' is Clark Kimberling's X(12); I call it HARMONICAL
              FEUERBACH POINT. Similar points can be defined for the 9pt circle and
              the excircles.

              Sincerely,
              Darij Grinberg
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