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A problem of construction

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  • jpehrmfr <jean-pierre.ehrmann@wanadoo.fr>
    Dear Hyacinthists, ABC and A B C are two given triangles. Do you know a way to construct the pair (M,N) of points such as M,N are isogonal conjugates wrt ABC
    Message 1 of 13 , Mar 2, 2003
      Dear Hyacinthists,
      ABC and A'B'C' are two given triangles.
      Do you know a way to construct the pair (M,N) of points such as M,N
      are isogonal conjugates wrt ABC and wrt A'B'C'?
      I think that such a pair is - generally - unique and ruler and
      compass constructible.
      Friendly. Jean-Pierre
    • Bernard Gibert
      Dear Jean-Pierre, ... [JP] ... the reflections of the lines AA , AB , AC about a bisector at A in ABC intersect the sidelines of A B C at three collinear
      Message 2 of 13 , Mar 3, 2003
        Dear Jean-Pierre,
        >
        [JP]
        > ABC and A'B'C' are two given triangles.
        > Do you know a way to construct the pair (M,N) of points such as M,N
        > are isogonal conjugates wrt ABC and wrt A'B'C'?
        > I think that such a pair is - generally - unique and ruler and
        > compass constructible.

        the reflections of the lines AA', AB', AC' about a bisector at A in ABC
        intersect the sidelines of A'B'C' at three collinear points on L_A.
        Let N_A be the Newton line of the quadrangle formed by L_A and the triangle
        A'B'C'.
        similarly, define the Newton line N_A' with triangle ABC.

        the two Newton lines meet at Q center of the two conics inscribed in ABC and
        A'B'C'. Their foci are your requested points.

        Best regards

        Bernard
      • jpehrmfr <jean-pierre.ehrmann@wanadoo.fr>
        Dear Bernard ... M,N ... [Bernard] ... in ABC ... triangle ... in ABC and ... Many thanks and congratulations for this wonderful and so clever construction.
        Message 3 of 13 , Mar 3, 2003
          Dear Bernard
          > [JP]
          > > ABC and A'B'C' are two given triangles.
          > > Do you know a way to construct the pair (M,N) of points such as
          M,N
          > > are isogonal conjugates wrt ABC and wrt A'B'C'?
          > > I think that such a pair is - generally - unique and ruler and
          > > compass constructible.

          [Bernard]
          > the reflections of the lines AA', AB', AC' about a bisector at A
          in ABC
          > intersect the sidelines of A'B'C' at three collinear points on L_A.
          > Let N_A be the Newton line of the quadrangle formed by L_A and the
          triangle
          > A'B'C'.
          > similarly, define the Newton line N_A' with triangle ABC.
          >
          > the two Newton lines meet at Q center of the two conics inscribed
          in ABC and
          > A'B'C'. Their foci are your requested points.

          Many thanks and congratulations for this wonderful and so clever
          construction.
          Friendly. Jean-Pierre
        • Barry Wolk
          ... I found a completely different proof of that result. We are talking about synthetic proofs, since the result is trivial computationally. Lemma: If
          Message 4 of 13 , Mar 3, 2003
            Darij Grinberg wrote:
            > In message #6614, I gave a lemma for the Arnold
            > theorem. Now I have noted that it is not really
            > needed in the proof.
            >
            > Here is the ARNOLD THEOREM (for H. E. Arnold, who
            > gave a special case of it as problem AMM 3258 in
            > American Mathematical Monthly):
            >
            > Let ABC be a triangle with orthocenter H, and
            > let P be an arbitrary point. Call X, Y, Z
            > the traces of P on the sides of ABC (i. e.
            > X = AP /\ BC, Y = BP /\ CA, and Z = CP /\ AB).
            > The perpendicular to HX through A intersects
            > BC at X'; analogously, define Y' and Z'. Then,
            > - the points X', Y', Z' are collinear;
            > - the line X'Y'Z' is orthogonal to the
            > line HP.

            I found a completely different proof of that result. We are
            talking about synthetic proofs, since the result is trivial
            computationally.

            Lemma: If corresponding sides of two quadrilaterals are
            parallel, and if a diagonal of the first quadrilateral is
            parallel to a diagonal of the second quadrilateral, then the
            other diagonals are also parallel.

            The interesting case is when the diagonals assumed to be
            parallel are not corresponding diagonals -- if they do
            correspond, then the quadrilaterals are homothetic and the
            result follows trivially.

            Proof: We have AB // A'B', BC // B'C', CD // C'D', DA // D'A',
            AC // B'D', and must show BD // A'C'. Using an obvious
            translation, dilation, and (possibly) reflection through a
            point, we can reduce this to the case where B=B', C=C'.

            Apply Pappus to the hexagon C A D B D' A'. Alternate vertices C
            D D' are collinear, and so are A B A'. So the three
            intersections
            CA /\ BD', AD /\ D'A', DB /\ A'C are collinear. From the
            assumptions we see that the first two intersections are points
            at infinity, so the third intersection is also a point at
            infinity. QED.

            Corollary. If corresponding sides of two quadrilaterals are
            perpendicular, and if a diagonal of the first quadrilateral is
            perpendicular to a diagonal of the second quadrilateral, then
            the other diagonals are also perpendicular.
            Proof: Rotate one of the quadrilaterals by 90 degrees, and use
            the lemma.

            Now for the Arnold result. Note that X' is the orthocenter of
            AHX, and similarly for Y' and Z'. Use the two quadrilaterals
            C X' H Y' C and H A P B H. Corresponding sides are
            perpendicular, and the diagonals CH and AB are also
            perpendicular.
            So X'Y' _|_ HP by the corollary. Similarly Y'Z' _|_ HP.

            --
            Barry Wolk


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          • Darij Grinberg <darij_grinberg@web.de>
            Dear Barry Wolk, Many thanks for the proof. It is indeed simpler. By the way, the Corollary is equivalent to the theorem on Orthologic Triangles, isn t it?
            Message 5 of 13 , Mar 3, 2003
              Dear Barry Wolk,

              Many thanks for the proof. It is indeed simpler.

              By the way, the Corollary is equivalent to the theorem on Orthologic
              Triangles, isn't it?

              Sincerely,
              Darij Grinberg

              --- In "Hyacinthos", Barry Wolk wrote:

              > Darij Grinberg wrote:
              > > In message #6614, I gave a lemma for the Arnold
              > > theorem. Now I have noted that it is not really
              > > needed in the proof.
              > >
              > > Here is the ARNOLD THEOREM (for H. E. Arnold, who
              > > gave a special case of it as problem AMM 3258 in
              > > American Mathematical Monthly):
              > >
              > > Let ABC be a triangle with orthocenter H, and
              > > let P be an arbitrary point. Call X, Y, Z
              > > the traces of P on the sides of ABC (i. e.
              > > X = AP /\ BC, Y = BP /\ CA, and Z = CP /\ AB).
              > > The perpendicular to HX through A intersects
              > > BC at X'; analogously, define Y' and Z'. Then,
              > > - the points X', Y', Z' are collinear;
              > > - the line X'Y'Z' is orthogonal to the
              > > line HP.
              >
              > I found a completely different proof of that result. We are
              > talking about synthetic proofs, since the result is trivial
              > computationally.
              >
              > Lemma: If corresponding sides of two quadrilaterals are
              > parallel, and if a diagonal of the first quadrilateral is
              > parallel to a diagonal of the second quadrilateral, then the
              > other diagonals are also parallel.
              >
              > The interesting case is when the diagonals assumed to be
              > parallel are not corresponding diagonals -- if they do
              > correspond, then the quadrilaterals are homothetic and the
              > result follows trivially.
              >
              > Proof: We have AB // A'B', BC // B'C', CD // C'D', DA // D'A',
              > AC // B'D', and must show BD // A'C'. Using an obvious
              > translation, dilation, and (possibly) reflection through a
              > point, we can reduce this to the case where B=B', C=C'.
              >
              > Apply Pappus to the hexagon C A D B D' A'. Alternate vertices C
              > D D' are collinear, and so are A B A'. So the three
              > intersections
              > CA /\ BD', AD /\ D'A', DB /\ A'C are collinear. From the
              > assumptions we see that the first two intersections are points
              > at infinity, so the third intersection is also a point at
              > infinity. QED.
              >
              > Corollary. If corresponding sides of two quadrilaterals are
              > perpendicular, and if a diagonal of the first quadrilateral is
              > perpendicular to a diagonal of the second quadrilateral, then
              > the other diagonals are also perpendicular.
              > Proof: Rotate one of the quadrilaterals by 90 degrees, and use
              > the lemma.
              >
              > Now for the Arnold result. Note that X' is the orthocenter of
              > AHX, and similarly for Y' and Z'. Use the two quadrilaterals
              > C X' H Y' C and H A P B H. Corresponding sides are
              > perpendicular, and the diagonals CH and AB are also
              > perpendicular.
              > So X'Y' _|_ HP by the corollary. Similarly Y'Z' _|_ HP.
              >
              > --
              > Barry Wolk
            • dick tahta
              Darij has recently discussed an Arnold theorem (for a given point P the perpendicular from A to HPa meets BC at X, etc, then X,Y,Z are collinear). I am left
              Message 6 of 13 , Mar 4, 2003
                Darij has recently discussed an Arnold theorem (for a given point P the
                perpendicular from A to HPa meets BC at X, etc, then X,Y,Z are collinear).
                I am left with a number of questions ....

                The theorem is one of those results that leave me curious to know how they
                arose in the first place. Would it have been after some casual doodling
                with a drawing program like Cabri? In which case we would already "know"
                the result. Proof then becomes a verification.

                As Barry Wolk pointed out, an algebraic verification is trivial. Why do we
                then seek a "synthetic" proof? I recognise the pleasure that this gives me
                when I can hit upon one - and I enjoyed reading Barry's. But is my pleasure
                very different from the brief "high" after solving a crossword clue? What
                is the investment here? Why isn't the trivial algebraic calculation
                satisfying enough? Why, for that matter, isn't the Cabri verification (with
                its hidden algebra)?

                And back inside the problem, what was so special about H? Was the original
                doodle in terms of the locus of such points? Which the "trivial algebra"
                would show was a cubic particular to the given point P - but always passing
                through H. So the underlying theorem might have been: for a given point P
                the perpendicular from A to QPa meet BC at X, etc, then X,Y,Z are collinear)
                when Q lies on a cubic through H. But this still leaves me with my
                curiosity about the original context of such explorations? And my own
                inability to judge the significance of such seemingly isolated results.

                Dick Tahta
              • Barry Wolk
                ... Orthologic ... [snip] ... is ... then ... Yes it is equivalent. I untangled the notation last night. Orthologic theorem : If perpendiculars from A to B C ,
                Message 7 of 13 , Mar 4, 2003
                  Darij Grinberg wrote:
                  > Dear Barry Wolk,
                  >
                  > Many thanks for the proof. It is indeed simpler.
                  >
                  > By the way, the Corollary is equivalent to the theorem on
                  Orthologic
                  > Triangles, isn't it?
                  >
                  > Sincerely,
                  > Darij Grinberg
                  >
                  > --- In "Hyacinthos", Barry Wolk wrote:

                  [snip]

                  > > Corollary. If corresponding sides of two quadrilaterals are
                  > > perpendicular, and if a diagonal of the first quadrilateral
                  is
                  > > perpendicular to a diagonal of the second quadrilateral,
                  then
                  > > the other diagonals are also perpendicular.

                  Yes it is equivalent. I untangled the notation last night.

                  Orthologic theorem : If perpendiculars from A to B'C', from B to
                  C'A', from C to A'B' are concurrent, then the perpendiculars
                  from A' to BC, from B' to CA, from C' to AB are also concurrent
                  (assuming A'B'C' not collinear).

                  Take AP_|_B'C', BP_|_C'A', CP_|_A'B', A'Q_|_BC, B'Q_|_CA, and we
                  must show C'Q_|_AB. Use the corresponding quadrilaterals
                  C A P B C and
                  Q B' C' A' Q
                  Corresponding sides are perpendicular, and also CP_|_B'A' for
                  two non-corresponding diagonals. So my corollary is just this
                  orthologic result.

                  This means that the proof I gave (rotate by 90 degrees,
                  transform and use Pappus) becomes an unusual way of proving the
                  orthologic theorem. How is it proved in textbooks?

                  The Arnold result has now been reduced to a one-liner, namely
                  that the triangles X' Y' C and B A P are orthologic, with both
                  orthologic centers being H.

                  In another thread, Darij wrote
                  > If two orthologic triangles are perspective,
                  > then the perspector is collinear with the
                  > two orthologic centers, and the join of the
                  > three points is orthogonal to their
                  > perspectrix!

                  Calculation has verified both these conjectures.
                  --
                  Barry Wolk


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                • Darij Grinberg
                  Dear Dick Tahta, Barry Wolk, Hyacinthos members, Many thanks for the notes on synthetic and analytic proofs. At first, many of you already know that I try to
                  Message 8 of 13 , Mar 5, 2003
                    Dear Dick Tahta, Barry Wolk, Hyacinthos members,

                    Many thanks for the notes on synthetic and analytic proofs. At first,
                    many of you already know that I try to find synthetic proofs wherever
                    possible. Of course, I am also satisfied with an analytic proof if I
                    understand it (for example, the proofs of the Sharygin points
                    trilinears - quite long but I have made all transformations myself).

                    I don't know of which computational verification is Barry Wolk
                    speaking, but I think he uses barycentric coordinates. Perhaps a
                    reason why I had not found a trivial coordinate proof of the Arnold
                    theorem is that I am still too lazy to learn representing
                    perpendicular lines in barycentrics. Another reason is that, if
                    a "trivial" barycentric proof of Arnold's theorem is possible, it
                    uses a very nontrivial lemma, namely the orthogonality condition, and
                    the proof together with the proof of the orthogonality condition gets
                    quite long.

                    By the way, there are theorems which, as far as I know, nobody has
                    proved synthetically yet, actually quite well-known theorems like:
                    The centers of similtude of circumcircle and incircle are the
                    isogonal conjugates of the Gergonne and Nagel points.

                    >> The theorem is one of those results that leave me
                    >> curious to know how they arose in the first place.
                    >> Would it have been after some casual doodling
                    >> with a drawing program like Cabri?

                    The problem was published in 1927... But the idea is correct that
                    most of the interesting geometry theorems discovered recently were
                    found by computer drawing -- for example, Floor van Lamoen discovered
                    his circle in 2000 by computer drawing, I discovered it in 2002
                    (independently) by computer drawing... These dynamic geometry
                    utilities strongly accelerate the finding of theorems.

                    By the way, most of us use Sketchpad or Cabri. I use the nice (also
                    relatively primitive) German program "Euklid DynaGeo".

                    >> In which case we would already "know" the
                    >> result. Proof then becomes a verification.

                    Now, this is very often so. But also watch out for "merely"
                    concurrent lines, "merely" collinear points and "merely" similar
                    triangles. For instance, regard the "Locus?" discussion #6375, #6377,
                    #6379, #6381, #6382, #6384, #6386, #6391, #6392, #6393.

                    In message #6375, Lev Emelyanov asked:

                    >> Let A1B1C1 be P-cevian triangle of ABC (A1 is
                    >> on BC etc.). Let c_A, c_B and c_C be incircles
                    >> of AB1C1, BC1A1 and CA1B1 respectively, ab be
                    >> common extangent (not sideline of ABC) of c_A
                    >> and c_B. Lines bc and ca we define similarly.
                    >> What is the locus of P such that ab, bc and
                    >> ca concur?

                    I drew a picture and was immediately sure that they always concur,
                    and that the locus is the whole plane. This caused some trouble until
                    finally Jean-Pierre Ehrmann found out that the locus is not the whole
                    plane, and that the lines usually form a very little triangle.

                    >> Why, for that matter, isn't the Cabri
                    >> verification (with its hidden algebra)?

                    Forgive a question from a Cabri non-specialist: Does Cabri test the
                    conjectures by calculation?

                    >> And back inside the problem, what was so
                    >> special about H?

                    The special property of H is that H is the center of the polar circle
                    of triangle ABC. A simple proof of the Arnold theorem is possible if
                    we accept using polarity with respect to a non-always real circle:
                    Regard the polar p of P in the polar circle of triangle ABC. (Of
                    course, p is perpendicular to HP.) Then,

                    * the polar of A is BC (property of polar circle);
                    * the polar of P is p;
                    * the polar of X is AX', because AX' is orthogonal to HX and AX'
                    passes through the pole A of BC (but X lies on BC).

                    Thus, from the collinearity of A, P and X we get the concurrence of
                    BC, p and AX', i. e. the point X' lies on p. Analogously, Y' and Z'
                    lie on p, qed.

                    >> Which the "trivial algebra" would show was a
                    >> cubic particular to the given point P - but
                    >> always passing through H.

                    Thanks - a nice locus.

                    Many thanks to Barry Wolk for his elegant proof of the orthologic
                    triangles theorem. Perhaps this proof is somewhere in one of the
                    books that have been written on the subject of reciprocal figures
                    (Pedoe, Geometry - a comprehensive course, p. 36), but I have not
                    seen it before.

                    This has got much longer than I thought, and I will continue with
                    answering Barry's mail in the next message.

                    Sincerely,
                    Darij Grinberg
                  • Darij Grinberg
                    Dear Barry Wolk, Hyacinthos members, ... Thank you for verification. Let me describe the context of the conjecture and my search for a further generalization
                    Message 9 of 13 , Mar 5, 2003
                      Dear Barry Wolk, Hyacinthos members,

                      Barry Wolk wrote in message #6652:

                      >> In another thread, Darij wrote
                      >> > If two orthologic triangles are perspective,
                      >> > then the perspector is collinear with the
                      >> > two orthologic centers, and the join of the
                      >> > three points is orthogonal to their
                      >> > perspectrix!
                      >>
                      >> Calculation has verified both these conjectures.

                      Thank you for verification. Let me describe the context of the
                      conjecture and my search for a further generalization of the
                      Longchamps point.

                      I wanted to generalize my conjecture about pedal-cevian points
                      (proved by Floor van Lamoen) (thread "Darboux and a line with 5
                      points"). In fact, the Darboux-Lamoen theorem states:

                      If a triangle ABC is given, and A'B'C' is a pedal-cevian
                      triangle, i. e. the pedal triangle of a point P and
                      the cevian triangle of a point Q, and if P' is the
                      isogonal conjugate of P, and if R is the circumcenter
                      of triangle A'B'C', and if L is the Longchamps point of
                      triangle ABC, then

                      (a) P, P' and L are collinear. (Darboux)
                      (b) P, P' and Q are collinear. (Lamoen and me)
                      (c) P, Q and R are collinear. (Proof attempt)
                      (d) P, P', Q, L and R are collinear. (Corollary)

                      Now I asked myself which of these four theorems could be generalized
                      to arbitrary orthologic and perspective triangles. (Two orthologic
                      and perspective triangles, one of them being inscribed in the other,
                      are a less symmetrical configuration than to arbitrary orthologic and
                      perspective triangles.) I immediately saw that if A'B'C' should be an
                      arbitrary triangle orthologic and perspective to ABC, then P would be
                      the intersection of the perpendiculars from A' to BC, from B' to CA
                      and from C' to AB, and P' would be the intersection of the
                      perpendiculars from A to B'C', from B to C'A' and from C to A'B'. (If
                      A'B'C' is inscribed to ABC, than this intersection is really the
                      isogonal conjugate of P - easy to prove!) So P and P' are the two
                      orthologic centers of triangles ABC and A'B'C'.

                      I easily found that (b) remains true for arbitrary orthologic and
                      perspective triangles. I still haven't a proof, but I hope that there
                      will be a synthetic one...

                      Now I asked myself what we could do with (a): If A'B'C' is inscribed
                      in ABC, then P, P' and L are collinear. I call this Darboux miracle,
                      because at first appearance, L has nothing to do with triangle
                      A'B'C'. So I thought that in the general case, P and P' must be
                      collinear with a point depending on both triangles ABC and A'B'C',
                      but giving L in the special case.

                      Unfortunately, I haven't found a suitable L generalization yet! (...
                      what makes the Darboux miracle more miraculous.) I think that we
                      should begin with the fact that L is the orthocenter of the
                      anticomplementary triangle of ABC, i. e. the generalization of L will
                      be the intersection of some perpendiculars. But of which ones?

                      Sincerely,
                      Darij Grinberg
                    • Darij Grinberg
                      ... call them ABC and A B C ... ... and searched for another point collinear with the two orthologic centers P and P and the perspector Q. If triangle A B C
                      Message 10 of 13 , Mar 5, 2003
                        I wrote:

                        >> If two orthologic triangles
                        call them ABC and A'B'C'
                        >> are perspective, then the perspector is
                        >> collinear with the two orthologic centers,
                        >> and the join of the three points is
                        >> orthogonal to their perspectrix!

                        ... and searched for another point collinear with the two orthologic
                        centers P and P' and the perspector Q.

                        If triangle A'B'C' is inscribed in triangle ABC, then the Longchamps
                        point L of triangle ABC is collinear with P, P' and Q.

                        If triangle A'B'C' is inscribed in the circumcircle of triangle ABC,
                        then (I conjecture that) the circumcenter O is collinear with P, P'
                        and Q.

                        THERE MUST BE A GENERALIZATION!
                        ----

                        Sincerely,
                        Darij Grinberg
                      • Barry Wolk
                        ... Correct. ... I call a calculation trivial if I can do it without needing a computer to handle the polynomials. In many problems the polynomials tend to
                        Message 11 of 13 , Mar 5, 2003
                          Darij Grinberg wrote:

                          > I don't know of which computational verification is Barry Wolk

                          > speaking, but I think he uses barycentric coordinates.

                          Correct.

                          > Perhaps a reason why
                          > I had not found a trivial coordinate proof of the Arnold
                          > theorem is that I am still too lazy to learn representing
                          > perpendicular lines in barycentrics.

                          I call a calculation "trivial" if I can do it without needing a
                          computer to handle the polynomials. In many problems the
                          polynomials tend to get ridiculously large.

                          It is very easy to handle perpendicular lines. Take two points
                          at infinity, P(x:y:z) and P'(x':y':z'), so x+y+z = x'+y'+z' = 0.
                          Then lines through P are perpendicular to lines through P' iff
                          x x' SA + y y' SB + z z' SC = 0

                          So given P, we can find P' by letting x' = y SB - z SC, y' and
                          z' similarly.

                          And the line through H and P' has coordinates
                          [x SA : y SB : z SC]
                          --
                          Barry Wolk


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                        • Darij Grinberg
                          Dear Barry Wolk, ... Many thanks... the next step would be (perhaps) generalizing to Lamoen s Pl-perpendicularity, but I think this is simply replacing SA, SB,
                          Message 12 of 13 , Mar 5, 2003
                            Dear Barry Wolk,

                            In Hyacinthos message #6661, you wrote:

                            >> Darij Grinberg wrote:
                            >>
                            >> > I don't know of which computational verification is Barry Wolk
                            >> > speaking, but I think he uses barycentric coordinates.
                            >>
                            >> Correct.
                            >>
                            >> > Perhaps a reason why
                            >> > I had not found a trivial coordinate proof of the Arnold
                            >> > theorem is that I am still too lazy to learn representing
                            >> > perpendicular lines in barycentrics.
                            >>
                            >> I call a calculation "trivial" if I can do it without needing a
                            >> computer to handle the polynomials. In many problems the
                            >> polynomials tend to get ridiculously large.
                            >>
                            >> It is very easy to handle perpendicular lines. Take two points
                            >> at infinity, P(x:y:z) and P'(x':y':z'), so x+y+z = x'+y'+z' = 0.
                            >> Then lines through P are perpendicular to lines through P' iff
                            >> x x' SA + y y' SB + z z' SC = 0
                            >>
                            >> So given P, we can find P' by letting x' = y SB - z SC, y' and
                            >> z' similarly.
                            >>
                            >> And the line through H and P' has coordinates
                            >> [x SA : y SB : z SC]
                            >> --
                            >> Barry Wolk

                            Many thanks... the next step would be (perhaps) generalizing to
                            Lamoen's Pl-perpendicularity, but I think this is simply replacing
                            SA, SB, SC by some constants.

                            Sincerely,
                            Darij Grinberg
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