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Re: [EMHL] Triangle with a median trisected by the incircle

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  • xpolakis@otenet.gr
    ... Dear Paul, I think that we can solve your problem by applying Stewart Theorem: Assume that the incircle trisects the a-median AM := m at K: A / / / K
    Message 1 of 15 , Mar 23, 2000
      >From: Paul Yiu <yiu@...>
      >
      >Characterize the triangle whose incircle
      >trisects one of the medians.

      Dear Paul,

      I think that we can solve your problem by applying Stewart Theorem:

      Assume that the incircle trisects the a-median AM := m at K:

      A
      /\
      / \
      / K \
      / \
      / I \
      / \
      B----M-------C


      By applying Stewart Theorem in the triangle AIM we get an equation with
      unknown the IM (since AI = r/sin(A/2), IK = r, AK = m/3, KM = 2m/3).

      Now, by applying again Stewart Theorem in the triangle BIC we get another
      equation with unknown the IM
      (but it is the i-median, so we can apply directly the median formula.)

      By substitution we finally get a formula in terms of some of the triangle
      elements, which will characterize the triangle.

      This works, but I don't know how easy are the calculations!


      Antreas
    • Paul Yiu
      Dear Antreas, Thanks for your quick response. I have actually sent this out too early by mistake. The triangle turns out to be a very simple one: its side
      Message 2 of 15 , Mar 23, 2000
        Dear Antreas,

        Thanks for your quick response. I have actually sent this out
        too early by mistake. The triangle turns out to be a very simple
        one: its side lengths are in the proportions of a : b : c = 5:10:13,
        the b-median 10 being trisected by the incircle, and is parallel to
        I_aC.

        I have a construction problem in mind: to construct an animation
        picture showing a triangle with centroid on the incircle. The condition
        is

        5(a^2+b^2+c^2) = 6(ab+bc+ca).

        I shall go home and see if Panakis has anything to say about
        triangles satisfying this condition.

        Can you please search if these questions have appeared before?

        Best regards.
        Sincerely,
        Paul


        ----------
        From: xpolakis@...[SMTP:xpolakis@...]
        Reply To: Hyacinthos@onelist.com
        Sent: Thursday, March 23, 2000 3:04 PM
        To: Hyacinthos@onelist.com
        Subject: Re: [EMHL] Triangle with a median trisected by the incircle

        From: xpolakis@...

        >From: Paul Yiu <yiu@...>
        >
        >Characterize the triangle whose incircle
        >trisects one of the medians.

        Dear Paul,

        I think that we can solve your problem by applying Stewart Theorem:

        Assume that the incircle trisects the a-median AM := m at K:

        A
        /\
        / \
        / K \
        / \
        / I \
        / \
        B----M-------C


        By applying Stewart Theorem in the triangle AIM we get an equation with
        unknown the IM (since AI = r/sin(A/2), IK = r, AK = m/3, KM = 2m/3).

        Now, by applying again Stewart Theorem in the triangle BIC we get another
        equation with unknown the IM
        (but it is the i-median, so we can apply directly the median formula.)

        By substitution we finally get a formula in terms of some of the triangle
        elements, which will characterize the triangle.

        This works, but I don't know how easy are the calculations!


        Antreas





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      • xpolakis@otenet.gr
        ... Dear Paul: Quick response, but theoretical ! I have to try the solution in the practice! ... I didn t find anything related to keywords
        Message 3 of 15 , Mar 24, 2000
          [Paul Yiu]:

          >>> Characterize the triangle whose incircle
          >>> trisects one of the medians.

          [APH]:

          >> I think that we can solve your problem by applying Stewart Theorem:
          >>
          >> Assume that the incircle trisects the a-median AM := m at K:
          >>
          >> A
          >> /\
          >> / \
          >> / K \
          >> / \
          >> / I \
          >> / \
          >> B----M-------C
          >>
          >>
          >> By applying Stewart Theorem in the triangle AIM we get an equation with
          >> unknown the IM (since AI = r/sin(A/2), IK = r, AK = m/3, KM = 2m/3).
          >>
          >> Now, by applying again Stewart Theorem in the triangle BIC we get another
          >> equation with unknown the IM
          >> (but it is the i-median, so we can apply directly the median formula.)
          >>
          >> By substitution we finally get a formula in terms of some of the triangle
          >> elements, which will characterize the triangle.
          >>
          >> This works, but I don't know how easy are the calculations!

          [Paul Yiu]:

          >Thanks for your quick response. I have actually sent this out

          Dear Paul:

          Quick response, but theoretical ! I have to try the solution in the practice!

          >too early by mistake. The triangle turns out to be a very simple
          >one: its side lengths are in the proportions of a : b : c = 5:10:13,
          >the b-median 10 being trisected by the incircle, and is parallel to
          >I_aC.
          >
          >I have a construction problem in mind: to construct an animation
          >picture showing a triangle with centroid on the incircle. The condition
          >is
          >
          >5(a^2+b^2+c^2) = 6(ab+bc+ca).
          >
          >I shall go home and see if Panakis has anything to say about
          >triangles satisfying this condition.
          >
          >Can you please search if these questions have appeared before?

          I didn't find anything related to keywords median-trisect*-incircle. Only two
          problems related to trisect*-median, which I will append below.

          As for Panakis' Trigonometry: I had a quick look at Medians Section
          (pp. 68 ff) but the only problem related to median-incircle I found is this:
          If in a triangle ABC: a-median = a, and b = c, then the incircle divides
          the a-median in extreme and mean ratio (p. 79, #12)

          The problems with trisect*-median keywords:

          Prove that the following construction trisects an angle of 60 d. Triangle
          ABC is a 30 - 60 - 90 right triangle inscribed in a circle. Median CM
          is drawn to side AB and extended to M' on the circle. Using a marked
          straightedge, point N on AB is located such that CN extended to N' on
          the circle makes NN' = MM'. Then CN trisects the 60 d. angle ACM.
          (Jack Garfunkel)

          Analogous to the median, call a line from a vertex of a triangle to a
          point of trisection of the opposite side a "tredian". Then, if both tredians
          are drawn from each vertex, the 6 lines will intersect at 12 interior points
          and divide the area into 19 subareas, each a rational part of the area
          of the triangle. Find two triangles for which each subarea is an integer,
          one being a Pythagorean right triangle and the other with consecutive
          integers for its three sides.
          (R. Robinson Rowe)

          And now I am going to make the calculations, and come back later.

          Antreas
        • xpolakis@otenet.gr
          ... ^^^^^^^^^^^^^^ To avoid trigonometry let s replace this with AI^2 = r^2 + (s-a)^2 Stewart Theorem in the triangle AIM: AI^2 * MK + IM^2 * AK = IK^2 * AM
          Message 4 of 15 , Mar 24, 2000
            I wrote:

            >>From: Paul Yiu <yiu@...>
            >>
            >>Characterize the triangle whose incircle
            >>trisects one of the medians.
            >
            >Dear Paul,
            >
            >I think that we can solve your problem by applying Stewart Theorem:
            >
            >Assume that the incircle trisects the a-median AM := m at K:
            >
            > A
            > /\
            > / \
            > / K \
            > / \
            > / I \
            > / \
            > B----M-------C
            >
            >
            >By applying Stewart Theorem in the triangle AIM we get an equation with
            >unknown the IM (since AI = r/sin(A/2), IK = r, AK = m/3, KM = 2m/3).
            ^^^^^^^^^^^^^^
            To avoid trigonometry let's replace this with AI^2 = r^2 + (s-a)^2

            Stewart Theorem in the triangle AIM:

            AI^2 * MK + IM^2 * AK = IK^2 * AM + AM*AK*KM

            We have: AI^2 = r^2 + (s-a)^2, AM := m, MK = 2m/3, IM := x, AK = m/3

            and the equality above finally becomes:

            3x^2 = 3r^2 + 2m^2 - 6(s-a)^2 (1)

            >
            >Now, by applying again Stewart Theorem in the triangle BIC we get another
            >equation with unknown the IM
            >(but it is the i-median, so we can apply directly the median formula.)

            Median Formula in the triangle BIC:

            2IM^2 = IB^2 +IC^2 - (BC^2)/2

            We have: IM := x, BI^2 = r^2 + (s-b)^2, CI^2 = r^2 + (s-c)^2

            So, the equality finally becomes:

            4x^2 = 4r^2 + 2(s-b)^2 +2(s-c)^2 - a^2 (2)

            >
            >By substitution we finally get a formula in terms of some of the triangle
            >elements, which will characterize the triangle.

            Let's see:

            (1) and (2) ==> 8m^2 + 3a^2 = 6[4(s-a)^2 + (s-b)^2 + (s-c)^2]

            Finally we get the following formula (condition) after some simple
            algebraic calculations:

            8a^2 + 5b^2 + 5c^2 - 12ab - 12ac + 6bc = 0

            which characterizes a triangle, whose the a-median is trisected by its
            incircle.

            >
            >This works, but I don't know how easy are the calculations!

            Hmmm... This worked indeed, and a solution is Paul's:

            a = 10k, b (or c) = 5k, c (or b) = 13k

            But is it unique (for nonzero possitives satisfying the triangle
            inequality: |b-c| < a < b+c ) ?


            Antreas
          • Richard Guy
            [For Munsters, the problem was to find triangles whose incircle trisected a median.] Given one rational point on a conic you can find them all. Writing b/a =
            Message 5 of 15 , Mar 24, 2000
              [For Munsters, the problem was to find triangles whose incircle
              trisected a median.]

              Given one rational point on a conic you can find them all. Writing
              b/a = x, c/a = y, we have

              5x^2 + 6xy + 5y^2 - 12x -12y + 8 = 0.

              We know one point on it, namely (0.5,1.3), but the equilateral
              triangle gives us the easier one, (1,1). Solving with the line

              y - 1 = m(x - 1), with m = r/s gives

              a : b : c = 5r^2 + 6rs + 5s^2 : 5r^2 + 2rs + s^2 : r^2 + 2rs + 5s^2

              This gives a genuine triangle, except that it degenerates if r = s
              or s + 3r = 0.

              If you put r = 2, s = 1, you get (37,25,13), which isn't very attractive.
              r = 1, s = 0 gives (5,5,1) and r = 3, s = 1 gives 4(17,13,5)
              which immediately reminds one of the Markov equation, which is quadratic
              in each of the three variables. So, put b = 13, c = 5 in the original
              and you get a = 17, but also the not-so-long lost 10. R.

              On Fri, 24 Mar 2000 xpolakis@... wrote:

              > From: xpolakis@...
              >
              > I wrote:
              >
              > >>From: Paul Yiu <yiu@...>
              > >>
              > >>Characterize the triangle whose incircle
              > >>trisects one of the medians.
              > >
              > >Dear Paul,
              > >
              > >I think that we can solve your problem by applying Stewart Theorem:
              > >
              > >Assume that the incircle trisects the a-median AM := m at K:
              > >
              > > A
              > > /\
              > > / \
              > > / K \
              > > / \
              > > / I \
              > > / \
              > > B----M-------C
              > >
              > >
              > >By applying Stewart Theorem in the triangle AIM we get an equation with
              > >unknown the IM (since AI = r/sin(A/2), IK = r, AK = m/3, KM = 2m/3).
              > ^^^^^^^^^^^^^^
              > To avoid trigonometry let's replace this with AI^2 = r^2 + (s-a)^2
              >
              > Stewart Theorem in the triangle AIM:
              >
              > AI^2 * MK + IM^2 * AK = IK^2 * AM + AM*AK*KM
              >
              > We have: AI^2 = r^2 + (s-a)^2, AM := m, MK = 2m/3, IM := x, AK = m/3
              >
              > and the equality above finally becomes:
              >
              > 3x^2 = 3r^2 + 2m^2 - 6(s-a)^2 (1)
              >
              > >
              > >Now, by applying again Stewart Theorem in the triangle BIC we get another
              > >equation with unknown the IM
              > >(but it is the i-median, so we can apply directly the median formula.)
              >
              > Median Formula in the triangle BIC:
              >
              > 2IM^2 = IB^2 +IC^2 - (BC^2)/2
              >
              > We have: IM := x, BI^2 = r^2 + (s-b)^2, CI^2 = r^2 + (s-c)^2
              >
              > So, the equality finally becomes:
              >
              > 4x^2 = 4r^2 + 2(s-b)^2 +2(s-c)^2 - a^2 (2)
              >
              > >
              > >By substitution we finally get a formula in terms of some of the triangle
              > >elements, which will characterize the triangle.
              >
              > Let's see:
              >
              > (1) and (2) ==> 8m^2 + 3a^2 = 6[4(s-a)^2 + (s-b)^2 + (s-c)^2]
              >
              > Finally we get the following formula (condition) after some simple
              > algebraic calculations:
              >
              > 8a^2 + 5b^2 + 5c^2 - 12ab - 12ac + 6bc = 0
              >
              > which characterizes a triangle, whose the a-median is trisected by its
              > incircle.
              >
              > >
              > >This works, but I don't know how easy are the calculations!
              >
              > Hmmm... This worked indeed, and a solution is Paul's:
              >
              > a = 10k, b (or c) = 5k, c (or b) = 13k
              >
              > But is it unique (for nonzero possitives satisfying the triangle
              > inequality: |b-c| < a < b+c ) ?
              >
              >
              > Antreas
              >
              >
              >
              >
              >
              > ------------------------------------------------------------------------
              > GET A NEXTCARD VISA, in 30 seconds! Get rates
              > as low as 0.0% Intro APR and no hidden fees.
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              >
              >
            • xpolakis@otenet.gr
              ... Dear Paul, Let s replace the incircle with the nine-point circle (9PC), and the median with either a median or an altitude. Namely, characterize the
              Message 6 of 15 , Mar 26, 2000
                >From: Paul Yiu <yiu@...>
                >
                >Characterize the triangle whose incircle
                >trisects one of the medians.

                Dear Paul,

                Let's replace the incircle with the nine-point circle (9PC), and the median
                with either a median or an altitude.

                Namely, characterize the triangles whose the nine-point circle trisects
                one median or one altitude.

                Since the 9PC passes through the feet of the altitudes and the medians
                we can apply the Stewart Theorem for both of them with the same formula !


                A
                /\
                / \
                / \
                / K \
                / \
                / N \
                / \
                B---M----------C


                AM : the a-altitude or the a-median
                N : the 9PC center.
                K : The point where the 9PC trisects AM.

                Stewart Theorem in the triangle ANM:

                AK*MN^2 + KM*AN^2 = AM*KN^2 + AK*KM*AM

                We have: AM := d, AK = d/3, KM = 2d/3, KN = MN = R/2, AN = Rt/2
                where t = sqrt(1+8cosAsinBsinC)
                [Note: We get this formula for AN by applynig the median formula in the
                triangle AOH [N is the midpoint of OH]. See Panakis, p. 182, #126.
                The formula in Kontogiannis, p. 6, #42: 2AN = sqrt(R(R+4Rr_a)) is wrong]

                So, we get after some calculations:

                2R^2 * cosAsinBsinC = (d^2)/3 ==> d^2 = 3Rh_1cosA [since h_1 = 2RsinBsinC]

                1. d = m_1 (a-median) : (m_1)^2 = 3Rh_1cosA

                2. d = h_1 (a-altitude): (h_1)^2 = 3Rh_1cosA ==> h_1 = 3RcosA

                Note that we can find the latter condition by one-line solution:

                (AK = h_1/3 & AK = AH/2) ==> 2h_1 = 3AH ==> h_1 = 3RcosA


                Antreas
              • xpolakis@otenet.gr
                ... Good memory, Michael! I took a look at CRUX in the interval [Prob. #2250, Prob.#2260], and the problem is #2252 (October 1998): Prove that the nine-point
                Message 7 of 15 , Mar 27, 2000
                  Lambrou Michael wrote:

                  >On Sun, 26 Mar 2000 xpolakis@... wrote:
                  >
                  >> Let's replace the incircle with the nine-point circle (9PC), and the median
                  >> with either a median or an altitude.
                  >>
                  >> Namely, characterize the triangles whose the nine-point circle trisects
                  >> one median or one altitude.
                  >>
                  >
                  >If I remember well the case of the 9PC tricecting a median was a proposed
                  >problem in CRUX (number approximately 2250: I don't have my file with me,

                  Good memory, Michael!

                  I took a look at CRUX in the interval [Prob. #2250, Prob.#2260], and the
                  problem is #2252 (October 1998):

                  Prove that the nine-point circle of a triangle trisects a median if and only
                  if the side lengths of the triangle are proportional to its median lengths
                  in some order. (Proposed by K. R. S. Sastry).

                  The solver proves that If the nine-point circle trisects BD [median],
                  then .... 2b^2 = a^2 + c^2.

                  By the way, in Panakis' Trigonometry (vol. II, p. 395, #21) I read this:

                  2b^2 = a^2 + c^2 <=> cotA + cotC = 2cotB


                  >and as I will be away for a few days, it might be some time before I give
                  >you the exact details). I vaguely remember that the conclusion was that
                  >the sides of the triangle are proportional to the medians. I might be
                  >wrong. I also seem to remember that I had a nice synthetic proof of this
                  >(but my name was mistakenly left out from the list of solvers). If my
                  >memory serves me right, I will give Hyacinthists my solution, upon my
                  >return.

                  Good idea to post it here.

                  >
                  >> [Note: We get this formula for AN by applynig the median formula in the
                  >> triangle AOH [N is the midpoint of OH]. See Panakis, p. 182, #126.
                  >> The formula in Kontogiannis, p. 6, #42: 2AN = sqrt(R(R+4Rr_a)) is wrong]
                  >
                  >I wish that that was ... the only mistake in Kontogiannis' otherwise
                  >delightfull book (on triangle inequalities).

                  Paul has located some errors:

                  In p. 8, #35 and #50, the E is actually C = Gergonne Point.

                  > Do you know the book (in Greek) of Tsaousoglou on
                  >Inequalities? There is a section on Triangle Inequalities.
                  >Tsaousoglou is a Chemical Engineer, a regular solver at CRUX, but
                  >unfortunately his book is not easily available. I got it from him
                  >directly.

                  No, I haven't seen this book.

                  Antreas
                • Paul Yiu
                  Dear Antreas and Michael, Antreas, you were one minute ahead of me. I also checked with Crux Math, and was about to write to praise Michael s excellent memory.
                  Message 8 of 15 , Mar 27, 2000
                    Dear Antreas and Michael,

                    Antreas, you were one minute ahead of me. I also
                    checked with Crux Math, and was about to write
                    to praise Michael's excellent memory.

                    Best regards.
                    Sincerely,
                    Paul



                    ----------
                    From: xpolakis@...[SMTP:xpolakis@...]
                    Reply To: Hyacinthos@onelist.com
                    Sent: Monday, March 27, 2000 1:58 PM
                    To: Hyacinthos@onelist.com
                    Subject: Re: [EMHL] Triangle with a median trisected by the incircle

                    From: xpolakis@...

                    Lambrou Michael wrote:

                    >On Sun, 26 Mar 2000 xpolakis@... wrote:
                    >
                    >> Let's replace the incircle with the nine-point circle (9PC), and the median
                    >> with either a median or an altitude.
                    >>
                    >> Namely, characterize the triangles whose the nine-point circle trisects
                    >> one median or one altitude.
                    >>
                    >
                    >If I remember well the case of the 9PC tricecting a median was a proposed
                    >problem in CRUX (number approximately 2250: I don't have my file with me,

                    Good memory, Michael!

                    I took a look at CRUX in the interval [Prob. #2250, Prob.#2260], and the
                    problem is #2252 (October 1998):

                    Prove that the nine-point circle of a triangle trisects a median if and only
                    if the side lengths of the triangle are proportional to its median lengths
                    in some order. (Proposed by K. R. S. Sastry).

                    The solver proves that If the nine-point circle trisects BD [median],
                    then .... 2b^2 = a^2 + c^2.

                    By the way, in Panakis' Trigonometry (vol. II, p. 395, #21) I read this:

                    2b^2 = a^2 + c^2 <=> cotA + cotC = 2cotB


                    >and as I will be away for a few days, it might be some time before I give
                    >you the exact details). I vaguely remember that the conclusion was that
                    >the sides of the triangle are proportional to the medians. I might be
                    >wrong. I also seem to remember that I had a nice synthetic proof of this
                    >(but my name was mistakenly left out from the list of solvers). If my
                    >memory serves me right, I will give Hyacinthists my solution, upon my
                    >return.

                    Good idea to post it here.

                    >
                    >> [Note: We get this formula for AN by applynig the median formula in the
                    >> triangle AOH [N is the midpoint of OH]. See Panakis, p. 182, #126.
                    >> The formula in Kontogiannis, p. 6, #42: 2AN = sqrt(R(R+4Rr_a)) is wrong]
                    >
                    >I wish that that was ... the only mistake in Kontogiannis' otherwise
                    >delightfull book (on triangle inequalities).

                    Paul has located some errors:

                    In p. 8, #35 and #50, the E is actually C = Gergonne Point.

                    > Do you know the book (in Greek) of Tsaousoglou on
                    >Inequalities? There is a section on Triangle Inequalities.
                    >Tsaousoglou is a Chemical Engineer, a regular solver at CRUX, but
                    >unfortunately his book is not easily available. I got it from him
                    >directly.

                    No, I haven't seen this book.

                    Antreas



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                  • xpolakis@otenet.gr
                    ... I had proved that (9PC intersects the a-median m) == m^2 = 3Rh_1cosA (1) Let me see if this implies 2a^2 = b^2 + c^2. We have: m^2 = (2(b^2 + c^2) -
                    Message 9 of 15 , Mar 27, 2000
                      I wrote:

                      >Lambrou Michael wrote:
                      >
                      >>On Sun, 26 Mar 2000 xpolakis@... wrote:
                      >>
                      >>> Let's replace the incircle with the nine-point circle (9PC), and the median
                      >>> with either a median or an altitude.
                      >>>
                      >>> Namely, characterize the triangles whose the nine-point circle trisects
                      >>> one median or one altitude.
                      >>>
                      >>
                      >>If I remember well the case of the 9PC tricecting a median was a proposed
                      >>problem in CRUX (number approximately 2250: I don't have my file with me,
                      >
                      >Good memory, Michael!
                      >
                      >I took a look at CRUX in the interval [Prob. #2250, Prob.#2260], and the
                      >problem is #2252 (October 1998):
                      >
                      > Prove that the nine-point circle of a triangle trisects a median if and only
                      > if the side lengths of the triangle are proportional to its median lengths
                      > in some order. (Proposed by K. R. S. Sastry).
                      >
                      >The solver proves that If the nine-point circle trisects BD [b-median],
                      >then .... 2b^2 = a^2 + c^2.

                      I had proved that (9PC intersects the a-median m) ==> m^2 = 3Rh_1cosA (1)

                      Let me see if this implies 2a^2 = b^2 + c^2.

                      We have: m^2 = (2(b^2 + c^2) - a^2)/4 [median formula]

                      Rh_1 = R*2RsinBsinC = bc/2

                      cosA = (b^2 + c^2 - a^2)/2bc

                      After some computations in (1) we get indeed that 2a^2 = b^2 + c^2.

                      Since we are on medians: Is the following true?

                      Let Ga, Gb, Gc be the centroids of GBC, GCA, GAB. [G = centroid of ABC].
                      Draw perpendiculars to GGa, GGb, GGc at Ga, Gb, Gc, respectively, intersecting
                      BC, CA, AB at A', B', C', respectively.
                      Then the points A',B',C' are collinear.

                      Anyway, regardless true or not for centroids: I am wondering for which
                      Triangle Point(s) X_k the above is true.


                      Antreas
                    • John Conway
                      The radical axis found as the difference of two Neuberg equations (bb:00:bb|1) and (cc:cc:0|1) is (bb-cc:-cc:bb|0). The vector product of this with a
                      Message 10 of 15 , Mar 27, 2000
                        The radical axis found as the difference of two Neuberg equations

                        (bb:00:bb|1) and (cc:cc:0|1) is (bb-cc:-cc:bb|0).

                        The vector product of this with a like:(cc:cc-aa:-aa|0)

                        gives (ccaa-bbcc+aabb:bbcc+aabb-ccaa:bbcc-bbaa+ccaa)

                        for their point of intersection.

                        So: the radical center of the Neuberg circles is the superJunction, sJ.

                        The in(fra)Junction iJ = (:aabb+bbcc:) arises all over the place -
                        it is the midpoint of the two Brocard points, and lies on the
                        meridian line - but this is the most natural place I've seen the
                        superJunction.

                        John Conway
                      • Lambrou Michael
                        ... If I remember well the case of the 9PC tricecting a median was a proposed problem in CRUX (number approximately 2250: I don t have my file with me, and as
                        Message 11 of 15 , Mar 27, 2000
                          On Sun, 26 Mar 2000 xpolakis@... wrote:

                          > Let's replace the incircle with the nine-point circle (9PC), and the median
                          > with either a median or an altitude.
                          >
                          > Namely, characterize the triangles whose the nine-point circle trisects
                          > one median or one altitude.
                          >

                          If I remember well the case of the 9PC tricecting a median was a proposed
                          problem in CRUX (number approximately 2250: I don't have my file with me,
                          and as I will be away for a few days, it might be some time before I give
                          you the exact details). I vaguely remember that the conclusion was that
                          the sides of the triangle are proportional to the medians. I might be
                          wrong. I also seem to remember that I had a nice synthetic proof of this
                          (but my name was mistakenly left out from the list of solvers). If my
                          memory serves me right, I will give Hyacinthists my solution, upon my
                          return.

                          > [Note: We get this formula for AN by applynig the median formula in the
                          > triangle AOH [N is the midpoint of OH]. See Panakis, p. 182, #126.
                          > The formula in Kontogiannis, p. 6, #42: 2AN = sqrt(R(R+4Rr_a)) is wrong]

                          I wish that that was ... the only mistake in Kontogiannis' otherwise
                          delightfull book (on triangle inequalities).
                          Do you know the book (in Greek) of Tsaousoglou on
                          Inequalities? There is a section on Triangle Inequalities.
                          Tsaousoglou is a Chemical Engineer, a regular solver at CRUX, but
                          unfortunately his book is not easily available. I got it from him
                          directly.

                          All the best,
                          Michael.
                        • Lambrou Michael
                          Dear Hyacinthists, ... (clip) Thanks to Antreas and Paul for tracing the case of the median to CRUX 2252, as my memory tickled me. Here is more or less the
                          Message 12 of 15 , Mar 31, 2000
                            Dear Hyacinthists,

                            On Sun, 26 Mar 2000 xpolakis@... wrote:

                            > From: xpolakis@...
                            >
                            > >From: Paul Yiu <yiu@...>
                            > >
                            > >Characterize the triangle whose incircle
                            > >trisects one of the medians.
                            >
                            > Dear Paul,
                            >
                            > Let's replace the incircle with the nine-point circle (9PC), and the median
                            > with either a median or an altitude.
                            >
                            > Namely, characterize the triangles whose the nine-point circle trisects
                            > one median or one altitude.
                            >
                            > Since the 9PC passes through the
                            (clip)

                            Thanks to Antreas and Paul for tracing the case of the median to CRUX
                            2252, as my memory tickled me.
                            Here is more or less the solution I submitted to CRUX at the time. Nothing
                            is not easy enough.

                            The following are equivalent.
                            a)the 9PC trisects a median
                            b) the Euler line is perpandicular to a median
                            c) b^2 + c^2 = 2a^2
                            d) the medians are proportinal to the sides.

                            Proofs. Let the 9PC cut the median AD at K with AK=KG=GD and let N be the
                            centre of the 9PC so that NK=ND= (the radius of 9PC)= R/2. Note that N is
                            on the Euler line half way between H and O, so (using OG=2HG) we have
                            NG=OH/6. Note NKD is isosceles so that NG, the Euler line, is
                            perpendicular to AD. By Pythagoras on NGK we find
                            (R/2)^2 = (OH/6)^2 + (AD/3)^3
                            Together with OH^2 = 9R^2 - (a^2 + b^2 + c^2) (not so hard to prove,
                            but no doubt well known) this reduces to b^2 + c^2 = 2a^2. Using this
                            in AD = square root of ( (2(b^2 + c^2) - a^2)/4 ), we find
                            AD= a(root 3)/2. Similarly the other two medians are b(root3)/2,
                            c(root3)/2.
                            To reverse the argument, some reversals are done in a trivial
                            manner but here is is a hint for the last.
                            We may assume b>a>c and so, as the larger the side the smaller the median
                            on it, we have BE<AD<CF thus BE=pc AD=pa, CF=pb. The value of p may be
                            found from AD^2 + BE^2 + CF^2 = (3/4)( a^2 + b^2 + c^2).

                            All the best.
                            Michael
                          • Lambrou Michael
                            ... I don t know if I am exceeding private versus public domains, but Panos Tsaousoglou is such a nice man, that feel free to give him a call at (01) 8231081
                            Message 13 of 15 , Mar 31, 2000
                              On Mon, 27 Mar 2000 xpolakis@... wrote:

                              > From: xpolakis@...
                              >
                              > Lambrou Michael wrote:
                              >
                              > > Do you know the book (in Greek) of Tsaousoglou on
                              > >Inequalities? There is a section on Triangle Inequalities.
                              > >Tsaousoglou is a Chemical Engineer, a regular solver at CRUX, but
                              > >unfortunately his book is not easily available. I got it from him
                              > >directly.
                              >
                              > No, I haven't seen this book.
                              >
                              > Antreas
                              >
                              I don't know if I am exceeding private versus public domains, but
                              Panos Tsaousoglou is such a nice man, that feel free to give him a call
                              at (01) 8231081 (Athens). I am sure he will be delighted. Anyway, give him
                              my warmest regards.
                              The book is on elementary (which is not synonymous to easy) inequalities.

                              Michael.
                            • xpolakis@otenet.gr
                              ... Dear Michael, I. Panakis: Plane Trigonometry. Vol II. Athens 1973, pp. 408 - 417, has quite a number of properties of the triangle whose the square of a
                              Message 14 of 15 , Mar 31, 2000
                                Michael Lambrou wrote:

                                >Thanks to Antreas and Paul for tracing the case of the median to CRUX
                                >2252, as my memory tickled me.
                                >Here is more or less the solution I submitted to CRUX at the time. Nothing
                                >is not easy enough.
                                >
                                > The following are equivalent.
                                > a)the 9PC trisects a median
                                > b) the Euler line is perpandicular to a median
                                > c) b^2 + c^2 = 2a^2
                                > d) the medians are proportinal to the sides.

                                Dear Michael,

                                I. Panakis: Plane Trigonometry. Vol II. Athens 1973, pp. 408 - 417,
                                has quite a number of properties of the triangle whose the square of a side
                                is the arithmetic mean of the squares of the other two.

                                Here is a selection:

                                I denote with P the equality (2a^2 = b^2 + c^2) [to not repeat it!]

                                1. P ==> 2 - sqrt(3) < b/c < 2 + sqrt(3)

                                2. P ==> A <= pi/3

                                3. P <==> m_1 / a = m_2 / c = m_3 / b = sqrt(3) / 2 [m_i medians]

                                4. P <==> (m_1)^2 + (m_2)^2 + (m_3)^2 = (3a/2)^2

                                5. P <==> 2(m_1)^2 = (m_2)^2 + (m_3)^2

                                6. P <==> tanA = 3tanw [w : The Brocard angle of ABC]

                                7. P <==> R^2 = (R_1)^2 = (R_2)*(R_3) [R_i the circumcircles GBC, GCA, GAB]

                                8. P <==> KA^2 = KB^2 * KC^2

                                9. P <==> GK // BC

                                10. P ==> The circumcircles of AGB, AGC are tangent to BC at B, C, resp.

                                11. P ==> The apollonian circle in respect to BC passes through G

                                12. P ==> G_1G_2 = G_1G_3 [G_i the orth. projections of G on BC, CA, AB]

                                13. P ==> The sides of the triangle G_1G_2G_3 [pedal triangle of G] are
                                perpendiculars to AK, BK, and CK.

                                14. P ==> The A, O, G, K are concyclic, lying on the circle with diameter OA

                                15. P ==> The H, G, B, C are concyclic

                                and many-many more!!


                                Antreas
                              • Lambrou Michael
                                Dear Antreas, ... Panakis never ceases to impress me. Unfortunatelly I don t have his Geometry nor his Trigonometry, but a friend of mine does, so I will
                                Message 15 of 15 , Apr 2 8:03 AM
                                  Dear Antreas,

                                  On Fri, 31 Mar 2000 xpolakis@... wrote:

                                  > >
                                  > > The following are equivalent.
                                  > > a)the 9PC trisects a median
                                  > > b) the Euler line is perpandicular to a median
                                  > > c) b^2 + c^2 = 2a^2
                                  > > d) the medians are proportinal to the sides.
                                  >
                                  > Dear Michael,
                                  >
                                  > I. Panakis: Plane Trigonometry. Vol II. Athens 1973, pp. 408 - 417,
                                  > has quite a number of properties of the triangle whose the square of a side
                                  > is the arithmetic mean of the squares of the other two.
                                  >
                                  > Here is a selection:
                                  >
                                  > I denote with P the equality (2a^2 = b^2 + c^2) [to not repeat it!]
                                  >
                                  > 1. P ==> 2 - sqrt(3) < b/c < 2 + sqrt(3)
                                  >
                                  > 2. P ==> A <= pi/3
                                  >
                                  > 3. P <==> m_1 / a = m_2 / c = m_3 / b = sqrt(3) / 2 [m_i medians]
                                  >
                                  > 4. P <==> (m_1)^2 + (m_2)^2 + (m_3)^2 = (3a/2)^2
                                  >
                                  > 5. P <==> 2(m_1)^2 = (m_2)^2 + (m_3)^2
                                  >
                                  > 6. P <==> tanA = 3tanw [w : The Brocard angle of ABC]
                                  >
                                  > 7. P <==> R^2 = (R_1)^2 = (R_2)*(R_3) [R_i the circumcircles GBC, GCA, GAB]
                                  >
                                  > 8. P <==> KA^2 = KB^2 * KC^2
                                  >
                                  > 9. P <==> GK // BC
                                  >
                                  > 10. P ==> The circumcircles of AGB, AGC are tangent to BC at B, C, resp.
                                  >
                                  > 11. P ==> The apollonian circle in respect to BC passes through G
                                  >
                                  > 12. P ==> G_1G_2 = G_1G_3 [G_i the orth. projections of G on BC, CA, AB]
                                  >
                                  > 13. P ==> The sides of the triangle G_1G_2G_3 [pedal triangle of G] are
                                  > perpendiculars to AK, BK, and CK.
                                  >
                                  > 14. P ==> The A, O, G, K are concyclic, lying on the circle with diameter OA
                                  >
                                  > 15. P ==> The H, G, B, C are concyclic
                                  >
                                  > and many-many more!!
                                  >
                                  >

                                  Panakis never ceases to impress me. Unfortunatelly I don't have his
                                  Geometry nor his Trigonometry, but a friend of mine does, so I will
                                  borough them from him.
                                  I do, however, have since my High School days his "2500 Geometrical
                                  Locii". I must have solved, as a student, at least 30% of them! So at
                                  least it appears when I look back.... Some arte very hard.
                                  Where on earth did he find so much?

                                  Thanks, Michael
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