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A Generalized Feuerbach Theorem

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  • iloveprimenumbers
    We have recently finished writing up a several year project in geometry titled Synthetic Cevian Geometry . Starting from work of Grinberg, Ehrmann, and Yiu
    Message 1 of 4 , Mar 1, 2009
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      We have recently finished writing up a several year project in
      geometry titled "Synthetic Cevian Geometry". Starting from work of
      Grinberg, Ehrmann, and Yiu in these Hyacinthos messages (Hyacinthos
      #6423, #959, #1790) we have been able to find synthetic proofs of many
      new results in the theory of cevian triangles. One important result
      that we have obtained is a generalized Feuerbach theorem.

      Let P be a point not on the sides of a triangle ABC or its
      anti-complementary triangle, and let Q denote its isotomcomplement =
      the complement of the isotomic P' of P. We define a generalized
      orthocenter H depending on P, as follows. Let DEF be the cevian
      triangle of P with respect to ABC. Then H is defined as the
      intersection of the lines AH, BH, CH through the vertices A, B, C
      which are parallel, respectively, to the lines QD, QE, QF. We have
      proved that H lies on the conic ABCPQ. We have also proved the
      following affine formula for the point H. Let the cevian triangle of
      P' be D'E'F' and let the mapping T_2 be the unique affine map taking
      triangle ABC to D'E'F'. If K denotes the complement mapping, then

      H = K^-1 * (T_2)^-1 * K(Q).

      The related point K(H) = O = (T_2)^-1 * K(Q) is called the generalized
      circumcenter of P, and is the center of a circumconic C_O of ABC which
      can be given as

      C_O = (T_2)^-1(N_P'),

      where N_P' is the 9-point conic of the quadrangle ABCP'. This formula
      is proved by showing that C_O is the 9-point conic of the quadrangle
      A_2B_2C_2Q, where A_2B_2C_2 =
      (T_2)^-1(ABC) is the anticevian triangle of Q with respect to ABC. We
      also show that the complement of the conic C_O is the 9-point conic
      N_H of the quadrangle ABCH. Let I_P be the inconic of ABC which is
      tangent to the sides of ABC at D, E, F, and let T_1 be the unique
      affine map taking ABC to DEF. We have proved the following theorem.

      Generalized Feuerbach Theorem. If P does not lie on a median of ABC,
      the 9-point conic N_H is tangent to the conic I_P at the point Z which
      is the center of the conic C = ABCPQ.

      We prove this theorem by showing that the map M = T_1 * K^-1 * T_2 *
      K^-1 takes the conic N_H to I_P and fixes the point Z. This map M can
      be broken down as follows:

      1) The anti-complement map K^-1 takes the conic N_H to the
      circumconic C_O, whose center is of course O.

      2) The map T_2 takes the circumconic C_O to the 9-point conic
      N_P', whose center is K(Q).

      3) The map K^-1 takes the conic N_P' to a circumconic of ABC with
      center Q, which coincides with the inconic I_Q' of the anticevian
      triangle of the point Q', defined as the isotomcomplement of the point
      P' = complement of P. The anticevian triangle of Q' with respect to
      ABC is the triangle (T_1)^-1(ABC), and the conic
      K^-1(N_P') is tangent to the sides of this triangle at A, B, C because
      of the general fact that Q is the isotomcomplement of Q' with respect
      to triangle (T_1)^-1(ABC).

      4) Grinberg (Hyacinthos #6423) has shown that the center of the
      conic I_P is Q, and Ehrmann (Hyacinthos #959) noticed that Q is a
      fixed point of the map T_1. Further, we have proved that T_1(Q') = P.
      Thus, the map T_1 takes the conic I_Q' of step 3 to the inconic I_P.

      This shows that M takes the 9-point conic N_H to the conic I_P. Now
      the maps T_1 and T_2 are inverse maps on the line at infinity, so M is
      a homothety or translation. We use the representation Z = GV.T_1(GV),
      where V is the reflection of P in the point Q and G is the centroid,
      to show that Z is the center of the mapping M and is therefore a fixed
      point. Finally, Z lies on the 9-point conic N_H because Z is the
      center of one of the conics lying on the quadrangle ABCH. Since M
      fixes lines through Z, the tangent to N_H at Z coincides with the
      tangent to I_P at Z, and this implies the theorem as stated above.

      This gives a new proof of the Feuerbach's theorem in the special case
      that P is the Gergonne point of triangle ABC, since in that case Q is
      the incenter of ABC and H is the ordinary orthocenter. It turns out
      that four different points P have the same generalized orthocenter, so
      the theorem implies that four different inscribed conics are tangent
      to the 9-point conic of ABCH. Thus, we get the full content of
      Feuerbach's theorem in the classical case.

      The full (synthetic) proofs can be found in our PDF manuscript in the
      IUPUI Math Department Preprint Series at the website
      http://www.math.iupui.edu/preprint/2009/ .

      Igor Minevich and Patrick Morton
      Dept. of Mathematical Sciences
      Indiana University - Purdue University at Indianapolis (IUPUI)
      pmorton@...
    • Francois Rideau
      Dear Igor If I understand you, you give the affine plane a metric structure, in order to P would be the Gergonne point, P the Nagel point and Q the center
      Message 2 of 4 , Mar 2, 2009
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        Dear Igor
        If I understand you, you give the affine plane a metric structure, in
        order to P would be the Gergonne point, P' the Nagel point and Q the center
        of the incircle.
        So with this new metrix , you have just to write again the old theory?
        Friendly
        Francois


        [Non-text portions of this message have been removed]
      • iloveprimenumbers
        Dear Francois, We work in the extended Euclidean plane, and use Euclidean, affine, and projective arguments. If we understand your question correctly, you are
        Message 3 of 4 , Mar 3, 2009
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          Dear Francois,

          We work in the extended Euclidean plane, and use Euclidean, affine,
          and projective arguments. If we understand your question correctly,
          you are suggesting that we could derive our whole theory by applying
          affine maps to the classical theory. This is not the case, because
          for some points P the inconic is a hyperbola, as P is allowed to lie
          outside of the triangle ABC, and there is no affine map that will take
          such an inconic to the incircle of some other triangle.

          Whenever the inconic is an ellipse, the induced involution of
          conjugate points on the line at infinity is elliptic. In this case
          the theory does follow from classical Euclidean results a la Coxeter
          (in his "The Real Projective Plane") , except that along the way we
          prove some results we believe to be new. When the inconic is a
          hyperbola, the induced involution on the line at infinity is
          hyperbolic, and the results for this case cannot be proven directly
          from the old Euclidean results as far as we know.

          The case P = X(7) = Gergonne point is one interesting case, but there
          are many others. If P = X(4) = the orthocenter, the point Z is the
          center of the Jerabek hyperbola, and our results show that a Feuerbach
          theorem exists for the inconic of triangle ABC whose center is the
          symmedian point X(6). This inconic is tangent at Z to the nine-point
          conic of ABCH, where the generalized orthocenter H = X(66), the
          isogonal conjugate of the Exeter point X(22). We do not know if this
          result was noticed before.

          We would be interested in knowing if anyone has used an affine map
          like our map M = T_1*K^-1*T_2*K^-1 to prove the classical Feuerbach
          theorem.

          With friendly regards,
          Igor Minevich and Patrick Morton


          --- In Hyacinthos@yahoogroups.com, Francois Rideau
          <francois.rideau@...> wrote:
          >
          > Dear Igor
          > If I understand you, you give the affine plane a metric structure, in
          > order to P would be the Gergonne point, P' the Nagel point and Q the
          center
          > of the incircle.
          > So with this new metrix , you have just to write again the old theory?
          > Friendly
          > Francois
          >
          >
          > [Non-text portions of this message have been removed]
          >
        • Francois Rideau
          Dear Igor and Dear Patrick I was only joking for I knew that point P was allowed to go everywhere it could inside as well outside the reference triangle ABC.
          Message 4 of 4 , Mar 3, 2009
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            Dear Igor and Dear Patrick
            I was only joking for I knew that point P was allowed to go everywhere it
            could inside as well outside the reference triangle ABC.
            Whenever you start with an affine plane , a reference triangle ABC in it
            and some point P as a parameter, we can always ask if there exists an
            euclidian structure on the plane in order to P becomes the orthocenter, or
            the circumcenter, or the Gergonne point or any other point in the Kimberling
            list.
            Doing so we get an affine partition of the plane in two sets, in the first
            there exists such an euclidian metric given by some quadratic form with a
            (2,0) signature and in the other there exists also such a metric but
            associated to a quadratic form with a (1, 1) signature and we have to work
            in a hyperbolic plane.
            For example, in the case of the orthocenter, the partition is given by the
            side lines, for the circumcenter it is given by the sides of the medial
            triangle, for the Lemoine point by the inner Steiner ellipse and so on...
            I notice that in some cases, euclidian properties of the point P are still
            valid provided they are translated properly, for example euclidian circles
            becoming "circles" in the hyperbolic geometry, that is to say hyperbolas of
            which asymptots are isotropic lines, and so on..
            Maybe Poncelet should have been happy to see his continuity principle in
            action!
            I think doing geometry in a hyperbolic plane is very funny and interesting
            and I will look at your paper with great pleasure.
            Very friendly regards
            Francois Rideau


            [Non-text portions of this message have been removed]
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