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What I don't like about the Droz-Farny theorem

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  • Floor en Lyanne van Lamoen
    Dear all, In recent times there have been some discussions on proofs of the following theorem by Droz-Farny: Let L1 and L2 be perpendicular and passing through
    Message 1 of 19 , Oct 17, 2004
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      Dear all,

      In recent times there have been some discussions on proofs of the following
      theorem by Droz-Farny:

      Let L1 and L2 be perpendicular and passing through H. The intersections of
      Li with the sides of ABC are Ai, Bi and Ci respectively. The midpoints of
      A1A2, B1B2 and C1C2 are collinear.

      In some versions of this theorem the collinearity of the midpoints is
      replaced by the coaxiality of the circles with diameters A1A2, B1B2 and
      C1C2.

      There is something I don't like about this theorem. My main objection is
      that it is not necessary to take midpoints. Any points A3, B3 and C3
      dividing the segments A1A2, B1B2 and C1C2 in the same ratio will do. Of
      course it turns out that A1B1C1, A2B2C2 and A3B3C3 are 'similar' inscribed
      degenerate triangles. In stead of these lines L1 and L2 we could have taken
      any two (directly) similar inscribed triangles A1B1C1 and A2B2C2 and we
      would find A3B3C3 similar to these.

      Going back to the A1B1C1 and A2B2C2 perpendicular lines through H:
      The more crucial fact of the DF theorem is IMHO (and here I repeat myself)
      that A1B1C1 and A2B2C2 are 'similar' degenerate triangles. Or, stated in a
      more classical way: the circles (AB1C1), (AB2C2), (A1BC1), (A2BC2), (A1B1C),
      (A2B2C) intersect in one point on (ABC). This implies the stronger form of
      DF, not only the midpoints.

      Kind regards,
      Floor.
    • Darij Grinberg
      Dear Floor, ... This really opened my eyes! The degenerate triangles A1B1C1 and A2B2C2 are similar; in other words, we have B1C1 / C1A1 = B2C2 / C2A2.
      Message 2 of 19 , Oct 17, 2004
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        Dear Floor,

        In Hyacinthos message #10716, you wrote:

        >> Any points A3, B3 and C3 dividing the segments A1A2,
        >> B1B2 and C1C2 in the same ratio will do. Of course
        >> it turns out that A1B1C1, A2B2C2 and A3B3C3 are
        >> 'similar' inscribed degenerate triangles.

        This really opened my eyes! The degenerate triangles
        A1B1C1 and A2B2C2 are similar; in other words, we have
        B1C1 / C1A1 = B2C2 / C2A2. Therefore, we can find two
        reals k and l with k + l = 1 such that
        C1 = k A1 + l B1 and C2 = k A2 + l B2, where we
        identify points with vectors from an (arbitrarily
        chosen) origin. Now, if the points A3, B3, C3 divide
        the segments A1A2, B1B2, C1C2 in the same ratio, we
        can find two reals u and v with u + v = 1 such that
        A3 = u A1 + v A2, B3 = u B1 + v B2 and
        C3 = u C1 + v C2. Thus,

        C3 = u C1 + v C2 = u (k A1 + l B1) + v (k A2 + l B2)
        = k (u A1 + v A2) + l (u B1 + v B2)
        = k A3 + l B3,

        and since k + l = 1, it follows that the point C3
        lies on the line A3B3, i. e. the points A3, B3, C3
        are collinear. Also, it is now clear that

        B1C1 : C1A1 : A1B1 = B2C2 : C2A2 : A2B2
        = B3C3 : C3A3 : A3B3.

        Now this explains really a lot. The Droz-Farny
        theorem turns out to be even simpler than I thought.
        I must say I still like the Droz-Farny theorem, but
        now I see the most crucial part of it is the
        similarity of the degenerate triangles A1B1C1 and
        A2B2C2 rather than the coaxality of the circles.

        Sincerely,
        Darij Grinberg
      • Floor en Lyanne van Lamoen
        Dear Darij, [DG] ... Of course I still like the DF-theorem. I only think it needs be restated, to see the complete figure, the reason why DF is true. It seems
        Message 3 of 19 , Oct 17, 2004
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          Dear Darij,

          [DG]
          > Now this explains really a lot. The Droz-Farny
          > theorem turns out to be even simpler than I thought.
          > I must say I still like the Droz-Farny theorem, but
          > now I see the most crucial part of it is the
          > similarity of the degenerate triangles A1B1C1 and
          > A2B2C2 rather than the coaxality of the circles.

          Of course I still like the DF-theorem. I only think it needs be restated, to
          see the complete figure, the reason why DF is true. It seems that you agree.
          Your use of vectors to show this is really clear and simple.
          Using Miquel theory is as simple - when M is the Miquel point, then
          triangles A1MA2, B1MB2, C1MC2 are similar, and hence so are A1MA3, B1MB3 and
          C1MC3, but that means that A3B3C3 is a result of pivoting as well, and thus
          similar.

          Kind regards,
          Floor.
        • Paul Yiu
          Dear Floor, [FvL]: In recent times there have been some discussions on proofs of the following theorem by Droz-Farny: Let L1 and L2 be perpendicular and
          Message 4 of 19 , Oct 29, 2004
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            Dear Floor,

            [FvL]: In recent times there have been some discussions on proofs of
            the following theorem by Droz-Farny:

            Let L1 and L2 be perpendicular and passing through H. The
            intersections of Li with the sides of ABC are Ai, Bi and Ci
            respectively. The midpoints of A1A2, B1B2 and C1C2 are collinear.
            ... Tere is something I don't like about this theorem. My main
            objection is that it is not necessary to take midpoints. Any points
            A3, B3 and C3 dividing the segments A1A2, B1B2 and C1C2 in the same
            ratio will do.

            *** It is very interesting. The envelope of the line containing A3,
            B3, C3, as the ratio varies, is the inscribed parabola tangent to L1
            and L2.

            Best regards
            Sincerely
            Paul
          • Paul Yiu
            Dear Floor, [FvL]: In recent times there have been some discussions on proofs of the following theorem by Droz-Farny: Let L1 and L2 be perpendicular and
            Message 5 of 19 , Oct 29, 2004
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              Dear Floor,

              [FvL]: In recent times there have been some discussions on proofs of
              the following theorem by Droz-Farny:

              Let L1 and L2 be perpendicular and passing through H. The
              intersections of Li with the sides of ABC are Ai, Bi and Ci
              respectively. The midpoints of A1A2, B1B2 and C1C2 are collinear.

              *** As the perpendicular lines rotate about H, the envelope of the
              Droz-Farny line is the inscribed ellipse with foci O and H (hence
              center N).

              Best regards
              Sincerely
              Paul
            • Paul Yiu
              Dear Floor, [FvL]: In recent times there have been some discussions on proofs of the following theorem by Droz-Farny: Let L1 and L2 be perpendicular and
              Message 6 of 19 , Oct 29, 2004
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                Dear Floor,

                [FvL]: In recent times there have been some discussions on proofs of
                the following theorem by Droz-Farny:

                Let L1 and L2 be perpendicular and passing through H. The
                intersections of Li with the sides of ABC are Ai, Bi and Ci
                respectively. The midpoints of A1A2, B1B2 and C1C2 are collinear.

                *** For two fixed perpendicular directions, consider the lines L1 and
                L2 through a point P with these directions, intersecting the
                sidelines of ABC at Ai, Bi, Ci (for i = 1, 2). The locus of P for
                which the midpoints of A1A2, B1B2, C1C2 are collinear is the
                rectangular circum-hyperbola with asymptotes along the given
                directions.

                Best regards
                Sincerely
                Paul
              • Floor en Lyanne van Lamoen
                Dear Paul, ... [PY] ... Let P be a point on the circumcircle and A B C be its Simson-Wallace line. Isn t for each P the collection of pivoted triangles
                Message 7 of 19 , Oct 30, 2004
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                  Dear Paul,

                  > [FvL]: In recent times there have been some discussions on proofs of
                  > the following theorem by Droz-Farny:
                  >
                  > Let L1 and L2 be perpendicular and passing through H. The
                  > intersections of Li with the sides of ABC are Ai, Bi and Ci
                  > respectively. The midpoints of A1A2, B1B2 and C1C2 are collinear.
                  > ... There is something I don't like about this theorem. My main
                  > objection is that it is not necessary to take midpoints. Any points
                  > A3, B3 and C3 dividing the segments A1A2, B1B2 and C1C2 in the same
                  > ratio will do.

                  [PY]
                  > *** It is very interesting. The envelope of the line containing A3,
                  > B3, C3, as the ratio varies, is the inscribed parabola tangent to L1
                  > and L2.

                  Let P be a point on the circumcircle and A'B'C' be its Simson-Wallace line.
                  Isn't for each P the collection of pivoted "triangles" A'B'C' a parabola?

                  Kind regards,
                  Floor.
                • Floor en Lyanne van Lamoen
                  Dear Paul, ... [PY] For two fixed perpendicular directions, consider the lines L1 and ... For two general fixed directions, the locus of P as above is the
                  Message 8 of 19 , Oct 30, 2004
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                    Dear Paul,

                    > [FvL]: In recent times there have been some discussions on proofs of
                    > the following theorem by Droz-Farny:
                    >
                    > Let L1 and L2 be perpendicular and passing through H. The
                    > intersections of Li with the sides of ABC are Ai, Bi and Ci
                    > respectively. The midpoints of A1A2, B1B2 and C1C2 are collinear.

                    [PY] For two fixed perpendicular directions, consider the lines L1 and
                    > L2 through a point P with these directions, intersecting the
                    > sidelines of ABC at Ai, Bi, Ci (for i = 1, 2). The locus of P for
                    > which the midpoints of A1A2, B1B2, C1C2 are collinear is the
                    > rectangular circum-hyperbola with asymptotes along the given
                    > directions.

                    For two general fixed directions, the locus of P as above is the
                    circumhyperbola with asymptotes along the given directions.

                    In stead of midpoints of A1A2, B1B2 and C1C2 we may again take any point
                    dividing these segments in the same ratio, and the locus stays the same.

                    Kind regards,
                    Floor.
                  • Milorad Stevanovic
                    Dear Floor and Paul, ... [PY] For two fixed perpendicular directions, consider the lines L1 and ... For any point P different from orthocenter H there is
                    Message 9 of 19 , Oct 30, 2004
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                      Dear Floor and Paul,

                      > [FvL] In recent times there have been some discussions on proofs of
                      > the following theorem by Droz-Farny:
                      >
                      > Let L1 and L2 be perpendicular and passing through H. The
                      > intersections of Li with the sides of ABC are Ai, Bi and Ci
                      > respectively. The midpoints of A1A2, B1B2 and C1C2 are collinear.

                      [PY] For two fixed perpendicular directions, consider the lines L1 and
                      > L2 through a point P with these directions, intersecting the
                      > sidelines of ABC at Ai, Bi, Ci (for i = 1, 2). The locus of P for
                      > which the midpoints of A1A2, B1B2, C1C2 are collinear is the
                      > rectangular circum-hyperbola with asymptotes along the given
                      > directions.

                      > [FvL]
                      >For two general fixed directions, the locus of P as above is the
                      >circumhyperbola with asymptotes along the given directions.

                      For any point P different from orthocenter H there is unique
                      pair of two perpendiculars through P with Droz-Farny
                      property(midpoints of A1A2,B1B2,C1C2 are collinear).
                      I have mentioned this result earlier when we have tried to
                      find good proof of Droz-Farny theorem..

                      I have two questions.

                      1.What is the reason that for orthocenter any pair of
                      two perpendiculars through H has D-F property,but
                      for any other point there is only one pair of perpendiculars
                      through it with D-F property?

                      2.What is the meaning of this unique pair of perpendiculars
                      through point P different form orthocenter H,with D-F property?
                      Are they asymptotes of some hyperbola?

                      Best regards

                      Milorad R.Stevanovic



                      [Non-text portions of this message have been removed]
                    • Bernard Gibert
                      Dear Floor, Milorad and Paul, ... The locus of P for which the midpoints of A1A2, B1B2, C1C2 form a triangle perspective to ABC is a central pK - whose pole is
                      Message 10 of 19 , Oct 31, 2004
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                        Dear Floor, Milorad and Paul,

                        > [FvL] In recent times there have been some discussions on proofs of
                        > > the following theorem by Droz-Farny:
                        > >
                        > > Let L1 and L2 be perpendicular and passing through H. The
                        > > intersections of Li with the sides of ABC are Ai, Bi and Ci
                        > > respectively. The midpoints of A1A2, B1B2 and C1C2 are collinear.
                        >
                        > [PY] For two fixed perpendicular directions, consider the lines L1 and
                        > > L2 through a point P with these directions, intersecting the
                        > > sidelines of ABC at Ai, Bi, Ci (for i = 1, 2). The locus of P for
                        > > which the midpoints of A1A2, B1B2, C1C2 are collinear is the
                        > > rectangular circum-hyperbola with asymptotes along the given
                        > > directions.
                        >
                        > > [FvL] For two general fixed directions, the locus of P as above is
                        > the
                        > >circumhyperbola with asymptotes along the given directions.

                        The locus of P for which the midpoints of A1A2, B1B2, C1C2 form a
                        triangle perspective to ABC is a central pK
                        - whose pole is the perspector of the circumhyperbola above,
                        - whose center is the center of the circumhyperbola,
                        - whose pivot is the anticomplement of the isoconjugate of the center.

                        This gives another characterization of the central pKs as seen in
                        Special Isocubics §3.1.3.

                        Best regards

                        Bernard

                        [Non-text portions of this message have been removed]
                      • Floor en Lyanne van Lamoen
                        Dear Milorad, ... I think the following is also true: When we consider pairs of lines making a fixed acute angle, then for any point P different from H,A,B,C
                        Message 11 of 19 , Oct 31, 2004
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                          Dear Milorad,

                          [MS]:
                          > For any point P different from orthocenter H there is unique
                          > pair of two perpendiculars through P with Droz-Farny
                          > property(midpoints of A1A2,B1B2,C1C2 are collinear).
                          > I have mentioned this result earlier when we have tried to
                          > find good proof of Droz-Farny theorem..

                          I think the following is also true:
                          When we consider pairs of lines making a fixed acute angle, then for any
                          point P different from H,A,B,C there are two pairs with the DF-property. For
                          H there are none.

                          Can someone confirm?

                          Kind regards,
                          Floor.
                        • jpehrmfr
                          Dear Floor and Milorad ... Two lines L, L have the Droz-Farny property if and only if they touch the same inscribed parabola (in which case the line joining
                          Message 12 of 19 , Nov 1, 2004
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                            Dear Floor and Milorad
                            > [MS]:
                            > > For any point P different from orthocenter H there is unique
                            > > pair of two perpendiculars through P with Droz-Farny
                            > > property(midpoints of A1A2,B1B2,C1C2 are collinear).
                            > > I have mentioned this result earlier when we have tried to
                            > > find good proof of Droz-Farny theorem..
                            Two lines L, L' have the Droz-Farny property if and only if they
                            touch the same inscribed parabola (in which case the line joining
                            the midpoints touch the parabola too)
                            Hence, if P<>H, there is an unique pair of perpendiculars through P
                            with Droz-Farny property : the tangents from P to the inscribed
                            parabola with directrix HP (this gives an easy construction of these
                            lines)
                            [FvL]
                            > I think the following is also true:
                            > When we consider pairs of lines making a fixed acute angle, then
                            for any
                            > point P different from H,A,B,C there are two pairs with the DF-
                            property. For
                            > H there are none.
                            I don't think so. If t is the measure in [-Pi/2,Pi/2] of the
                            required line angle, I think that there exist two pairs of solutions
                            if and only if OP*>R cos(t) where P* = isogonal conjugate of P.
                            Of course, if P* is outside the circumcircle, there are two pairs of
                            solutions for any t.
                            If Ta is the part of the plane limited by the half-lines AB, AC and
                            the segment BC and opposite to A, P* is outside the circumcircle if
                            and only if P lies in Ta union Tb union Tc.
                            Friendly. Jean-Pierre
                            Friendly. Jean-Pierre
                          • Floor en Lyanne van Lamoen
                            Dear Jean-Pierre and Milorad, ... [JPE] ... Nice! In accordance with H for the perpendicular pairs of lines, for any P the pairs of P-perpendicular through P
                            Message 13 of 19 , Nov 1, 2004
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                              Dear Jean-Pierre and Milorad,

                              > > [MS]:
                              > > > For any point P different from orthocenter H there is unique
                              > > > pair of two perpendiculars through P with Droz-Farny
                              > > > property(midpoints of A1A2,B1B2,C1C2 are collinear).
                              > > > I have mentioned this result earlier when we have tried to
                              > > > find good proof of Droz-Farny theorem..
                              [JPE]
                              > Two lines L, L' have the Droz-Farny property if and only if they
                              > touch the same inscribed parabola (in which case the line joining
                              > the midpoints touch the parabola too)
                              > Hence, if P<>H, there is an unique pair of perpendiculars through P
                              > with Droz-Farny property : the tangents from P to the inscribed
                              > parabola with directrix HP (this gives an easy construction of these
                              > lines)

                              Nice!

                              In accordance with H for the perpendicular pairs of lines, for any P the
                              pairs of P-perpendicular through P lines have the DF-property. For each P
                              there is one pair of directions that is as well perpendicular as
                              P-perpendicular. These are the directions of asymptotes of the
                              circumhyperbola through H and P.

                              > [FvL]
                              > > I think the following is also true:
                              > > When we consider pairs of lines making a fixed acute angle, then
                              > for any
                              > > point P different from H,A,B,C there are two pairs with the DF-
                              > property. For
                              > > H there are none.

                              [JPE]:
                              > I don't think so. If t is the measure in [-Pi/2,Pi/2] of the
                              > required line angle, I think that there exist two pairs of solutions
                              > if and only if OP*>R cos(t) where P* = isogonal conjugate of P.
                              > Of course, if P* is outside the circumcircle, there are two pairs of
                              > solutions for any t.
                              > If Ta is the part of the plane limited by the half-lines AB, AC and
                              > the segment BC and opposite to A, P* is outside the circumcircle if
                              > and only if P lies in Ta union Tb union Tc.
                              > Friendly. Jean-Pierre

                              I think you are right, and I have mistaken here. Thanks!

                              Kind regards,
                              Floor.
                            • jpehrmfr
                              Dear Floor and Milorad ... through P ... these ... P the ... each P ... Of course, we have another characterization : A pair of lines L, L intersecting at P
                              Message 14 of 19 , Nov 1, 2004
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                                Dear Floor and Milorad
                                > [JPE]
                                > > Two lines L, L' have the Droz-Farny property if and only if they
                                > > touch the same inscribed parabola (in which case the line joining
                                > > the midpoints touch the parabola too)
                                > > Hence, if P<>H, there is an unique pair of perpendiculars
                                through P
                                > > with Droz-Farny property : the tangents from P to the inscribed
                                > > parabola with directrix HP (this gives an easy construction of
                                these
                                > > lines)
                                > [FvL]
                                > In accordance with H for the perpendicular pairs of lines, for any
                                P the
                                > pairs of P-perpendicular through P lines have the DF-property. For
                                each P
                                > there is one pair of directions that is as well perpendicular as
                                > P-perpendicular. These are the directions of asymptotes of the
                                > circumhyperbola through H and P.
                                Of course, we have another characterization :
                                A pair of lines L, L' intersecting at P have the DF-property if and
                                only if the circumhyperbola going through the infinite points of L
                                and L'goes through P.
                                Friendly. Jean-Pierre

                                >
                                > > [FvL]
                                > > > I think the following is also true:
                                > > > When we consider pairs of lines making a fixed acute angle,
                                then
                                > > for any
                                > > > point P different from H,A,B,C there are two pairs with the DF-
                                > > property. For
                                > > > H there are none.
                                >
                                > [JPE]:
                                > > I don't think so. If t is the measure in [-Pi/2,Pi/2] of the
                                > > required line angle, I think that there exist two pairs of
                                solutions
                                > > if and only if OP*>R cos(t) where P* = isogonal conjugate of P.
                                > > Of course, if P* is outside the circumcircle, there are two
                                pairs of
                                > > solutions for any t.
                                > > If Ta is the part of the plane limited by the half-lines AB, AC
                                and
                                > > the segment BC and opposite to A, P* is outside the circumcircle
                                if
                                > > and only if P lies in Ta union Tb union Tc.
                                > > Friendly. Jean-Pierre
                                >
                                > I think you are right, and I have mistaken here. Thanks!
                                >
                                > Kind regards,
                                > Floor.
                              • Francois Rideau
                                If P H, there is an unique point Q on the circle through ABC so that HP is the Steiner-line of Q wrt ABC. Then the Droz-Farny line of P is the perpendicular
                                Message 15 of 19 , Nov 1, 2004
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                                  If P<> H, there is an unique point Q on the
                                  circle through ABC so that HP is the Steiner-line of Q wrt ABC. Then
                                  the Droz-Farny line of P is the perpendicular bissector of PQ.
                                  If P=H, then we can take any point Q on the ABC-circle and so one.....
                                  Is that true?
                                  F.Rideau
                                • jpehrmfr
                                  Dear Francois, Floor and Milorad ... Then ... Yes. Consider the points U1, U2 on the line HP such as PU1 = PU2 = PQ; the perpendicular bisector L1 - or L2 - of
                                  Message 16 of 19 , Nov 1, 2004
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                                    Dear Francois, Floor and Milorad
                                    > If P<> H, there is an unique point Q on the
                                    > circle through ABC so that HP is the Steiner-line of Q wrt ABC.
                                    Then
                                    > the Droz-Farny line of P is the perpendicular bissector of PQ.
                                    > Is that true?
                                    Yes.
                                    Consider the points U1, U2 on the line HP such as PU1 = PU2 = PQ;
                                    the perpendicular bisector L1 - or L2 - of QU1 - or QU2 - intersects
                                    the sidelines of ABC at A1,B1,C1 - or A2, B2, C2 -
                                    Then the midpoints of A1A2, B1B2, C1C2 lie on the perpendicular
                                    bisector of PQ and, of course, L1 and L2 are perpendicular.
                                    This is related to the fact that the inscribed parabola with
                                    directrix HP is the envelope of the perpendicular bisectors of QM,
                                    where M lies on HP.

                                    Now come back to Floor's problem : given a point P and an angle T,
                                    find the pairs of lines (L,L') with the DF-property, intersecting at
                                    P and such as <L,L' = T.
                                    IF MM' is a choird of the circumcircle, if d is the distance from O
                                    to MM', the cosinus of the angle of two lines with infinite points
                                    the isogonal conjugates of M and M' is d/R.
                                    As the circumhyperbola through the infinite points of L and L' must
                                    go through P, the isogonal conjugate P* of P must lie on the line
                                    MM', where M and M' are the isogonal conjugates of the infinite
                                    points of L and L'.
                                    Hence the construction : draw the tangents from P* to the circle (O,
                                    R cos(T)); if one of these tangents intersect the circumcircle at M
                                    and M', we have the solution (PM*, PM'*)
                                    Obviously, we have two distinct solutions iff OP*>R cos(T) and no
                                    solution iff OP* < R cos(T)

                                    Friendly. Jean-Pierre
                                  • Francois Rideau
                                    My dear Jean-Pierre You know what? I am happy. It s the first time I get an answer. I was looking at Hyacinthos Forum since a few week without understanding
                                    Message 17 of 19 , Nov 2, 2004
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                                      My dear Jean-Pierre
                                      You know what? I am happy. It's the first time I get an answer. I was
                                      looking at Hyacinthos Forum
                                      since a few week without understanding all the discussions. It seems
                                      to me most of the questions
                                      are too Euclidian oriented. There is also some interesting problems in
                                      projective and affine geometry.
                                      The Droz-Farny problems raises some new questions I don't know the answers.
                                      For example, each point P<>H provides 2 orthogonal lines, what are the
                                      associated flows? What about these flows around H?
                                      Here another similar problem in affine geometry: Let ABC a triangle in
                                      an affine real plane.
                                      We know that through each point P there his 2 isotomic lines. What are
                                      the associated flows? Besides it's a good problem to draw these 2
                                      isotomic lines through P, using Cabri or another software but I am
                                      interested in the flows.
                                      It's relatively easy to write the differential equations providing these flows
                                      but it's another matter to draw them! Maybe with Maple or Mathematica?
                                      Friendly
                                      François
                                    • Bernard Gibert
                                      Dear Francois, ... what do you exactly mean by flow ? ... I suppose you call isotomic lines 2 lines meeting each sideline of ABC at two points points
                                      Message 18 of 19 , Nov 2, 2004
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                                        Dear Francois,

                                        > We know that through each point P there his 2 isotomic lines. What are
                                        > the associated flows? 

                                        what do you exactly mean by "flow" ?

                                        > Besides it's a good problem to draw these 2
                                        > isotomic lines through P, using Cabri or another software but I am
                                        > interested in the flows.

                                        I suppose you call "isotomic lines" 2 lines meeting each sideline of
                                        ABC at two points points symmetric in the midpoint of the side. If so,
                                        these lines are the asymptotes of the circum-conic with center P.
                                        Obviously they are not always real.

                                        this is related to the cubics I call "Allardice cubics".

                                        Best regards

                                        Bernard

                                        [Non-text portions of this message have been removed]
                                      • Francois Rideau
                                        My dear Bernard I apologyze for flow . It s not the best word. I want to suggest: flow of a vector field but what I mean is to search the plane curves so that
                                        Message 19 of 19 , Nov 3, 2004
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                                          My dear Bernard
                                          I apologyze for "flow". It's not the best word. I want to suggest:
                                          flow of a vector field
                                          but what I mean is to search the plane curves so that in each point P
                                          of the curve
                                          the direction of the tangent is known, for example in the D-F case,
                                          the direction of the 2 orthogonal lines through P or in the isotomic
                                          case the direction of the 2 isotomic lines through P.
                                          Yes that what I mean by isotomic lines . They are not always real. I remember
                                          the medial triangleis a separator?
                                          It's very easy to draw the 2 isotomic lines through P when they are real.
                                          I want to draw the associated integral curves.
                                          Best regards
                                          Francois
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