An email discussion we had about the Gossard perspector.

           I have Kimberling's list of points on my Palm pilot. I looked up the Gossard point X(402) the other day and noticed something odd. He gives a long description of the point (it is a very interesting point!) and then lists that it is on line 2,3, the Euler line. What is odd is that there was _no_ other information. His computers could find no relationships to any other lines or points.

      The fact there there is no information about this point is perhaps the most interesting thing about it.

      Are other properties of this point known to anyone in Hyacinthos?

      -----     

      X402 is the intersection of the Euler line and the line through the  centers of the circles

                     X2-X107-X111-X125-X468 & X3-X107-X113-X125-X403,

      points named E330 & E339 in Edward Brisse's list of centers.

      Bernard

      Yes, the rareness of this point's properties is quite odd, and it is naturally reflected in the complexity of its trilinear and barycentric coordinates. Let me, however, remark (and this is very odd too!) that its superior(=anticomplement) and its superior-of-the-superior have simpler coordinates. I cite Paul Yiu's message #568:

      It inspired me to take a look at another quadruple of homothetic triangles. The Gossard triangle (bounded by the Euler lines of the three triangles bounded by the Euler line of ABC with two of the sides). I remember some time ago, there were discussions about the coordinates of the Gossard center (X(402) in ETC), and these turned out to be quite tedious:

      [(2a^4-a^2(b^2+c^2)-(b^2-c^2)^2) 

      (a^8-a^6(b^2+c^2)-a^4(2b^2-c^2)(b^2-2c^2)

      +3a^2(b^2-c^2)^2(b^2+c^2)

      -(b^2-c^2)^2(b^4+3b^2c^2+c^4))

      : ...

      : ...

      (The first component factors in two symmetric polynomials, the first a quartic [the first line in the above expression], and the second an octic [the next three lines]).

      In John's notation, this is 

      ...: (SBC + SAB - 2SCA)(SAB^2+SBC^2-SAC^2 +

                  SABC(SA - 3SB + SC))

      : ...    

      Now, the Gossard triangle, being homothetic to ABC with factor -1 (at the Gossard center), is also homothetic to the medial and the pre-medial triangles. The four homothetic centers involved are all on the Euler line. It turns out that the coordinates of the other two homothetic centers are more elegant:

      The homothetic center of the Gossard triangle and the medial triangle, in John's notation, is

      ... : SB^2(SC-SA)^2(SBC+SAB-2SAC) : ... 

      This is actually the SUPERIOR of the Gossard center. 

      That of the Gossard triangle and the pre-medial (anticomplementary) triangle, is

      ... : (SBC + SAB - 2SAC)/(SB(SC-SA)) : ...

      It is easy to see that the Gossard center is the midpoint of these two.

      ... and this latter point is the superior of the superior of the Gossard perspector.

      In terms of a, b, c, these are 

      [(b^2-c^2)^2(b^2+c^2-a^2)^2(2a^4-a^2(b^2+c^2)-(b^2-c^2)^2)]      

               and       

      [(2a^4-a^2(b^2+c^2)-(b^2-c^2)^2)/((b^2-c^2)(b^2+c^2-a^2))].

      Darij Grinberg

      So what is the geometrical interpretation of the line

      [A2-:B2-:C2-] ?

      ~ [ 1 : 1 : 1 ].

      The line at infinity! (was quite unexpected to me)

      Here is a nice way to see the whole subject with little computation. I will use these facts

      A line [ l : m: n ] has endpoint    < m-n : n - l : l - m >.  If P ~  < l : m: n > then line PG  ~  [ m-n : n - l : l - m ]. The line through P< and P>  is  [ : mm - nl : ]  where P< is P with  coordinates cyclically rotated left.

      Since H ~ <  :  SCA : >, the Euler line has coordinates    [ A : B : C  ] where A, B, C are as before   B = SAB - SBC, where   SAB = SA SB.  The Euler line has endpoint   einf ~ < : C - A : >  ~  < : SAB + SBC -  2SCA : >  ~  <  : SS - 3SCA : >. (not really necessarty to the argument.

      The dual of the Euler line < A : B : C> is at infinity, since its  coordinates add to zero, and so are the rotated points   < B: C :  A >  and < C : A : B > .

      The rotated points define the line   [ : BB - CA : ]  = [: B^2–:] which must be the line at infinity.

      Steve

      I am going to summarize what I have been told and then add what John and I discovered a few weeks ago.

      (From Bernard) The Gossard perspector is on the Euler line and the line that connects the centers of two circles

      Circle1 goes through    the centroid

                                                the Parry point    : bb/ (cc +  aa - 2 bb) :  on CC

                                                the Jerabek center

                                                point on CC        : 1/  (cc-aa)(cc + aa - 2 bb)^2 :

      Circle 2 goes through     the circumcenter

                                          Jerabek center

                                                the Jerabek antipode      : (S + SB)(ccSC + aaSA)(bbSB - 2SCA)

                                                point on CC        : 1/  (cc-aa)(cc + aa - 2 bb)^2 :

      The center of the first circle is a 6th degree in abc while  the second  is 14th.

      ----

      (from Paul through Darij) The triangle of Euler lines is homothetic to ABC and so to its medial and antimedial triangle (which will be the dilated triangle in TB). The four homothetic centers are all on the medial line. One is the dilated Gossard point and one is  : (S_BC + S_AB - 2S_AC)/(S_B(S_C-S_A)) : . The Gossard point is the midpoint of these two and Darij adds that this is dd Gossard.

      All of this seems decently interesting.

      Here is what John and I have to add.

      If the Gossard perspector construction is repeated for a line parallel to the Euler line, again a congruent triangle results but the perspector has moved. The locus of the perspector points is a line, the Zeeman line.

      The 4 Zeeman lines for the triangles from a, b, c, e, the edges of ABC and the Euler line concur at the Gossard point.

      Conjecture: The Gossard point of ABC is the centroid of that of the other three.

      John's very compact notation for these point is as follows

      Gossard   =    : B²⁺  (C - A) :     where   B = S_AB – S_BC and B²⁺ = BB-CA.

      I think this is a marvel of compactness.  This point is the medial of    : BB  (C - A) :

      The  Zeeman line is      : B²⁺ :

      More on this notation later.

      Steve

     [PY]  I have, by the way, just confirmed your very interesting conjecture by calculation.

      Also, the equation of the Zeeman line is very elegant. It is simply

      (A²⁺)x + (B²⁺)y + (C²⁺)z = 0.

      I am sure you have already known this.

      [JC] Yes - Steve and I already wrote it into our book.  The coefficients are indeed A2+,B2+,C2+  rather than the ones with minus signs.

      This raises an interesting question:-

      [A²⁺:B²⁺:C²⁺] is the Zeeman line,  

      and [A² :B² :C² ] is the line that separates the "true" isosperspectrices (those corresponding to real base angles) from the "false" ones.  So what is the geometrical interpretation of the line

      [A^2-:B^2-:C^2-] ?

      John Conway

      In Hyacinthos message #7540, you wrote:

      [A2+:B2+:C2+] is the Zeeman line,  

      and [A2 :B2 :C2 ] is the line that separates the "true" isosperspectrices (those corresponding to real base angles) from the "false" ones.

     Do you mean the perspectrices of the Kiepert triangles with triangle ABC ?

     So what is the geometrical interpretation of the line

      [A2-:B2-:C2-] ?

      ~ [ 1 : 1 : 1 ].

      The line at infinity! (was quite unexpected to me)

      Sincerely,

      Darij Grinberg

      In Hyacinthos message #7540, you wrote:

      [A2+:B2+:C2+] is the Zeeman line,  

      and [A2 :B2 :C2 ] is the line that separates the "true" isosperspectrices (those corresponding to real base angles) from the "false" ones.

     Do you mean the perspectrices of the Kiepert triangles with triangle ABC ?

      Well, yes, these are what Steve and I are calling "isoperspectrices" (short for "perspectrices of isosceles Napoleons") in TB.  However, there's a subtlety that perhaps you don't know about.  Namely, the typical isoperspectrix is  [:1/(SS+@@+2bb@):]  where   @ = S.cot(base angle)  (and  S  is twice the area of the triangle  ABC).

      Now this has the form  [:1/(lambda + mu.bb):], for constants  lambda  and  mu,  and the envelope of all lines of THIS form is Kiepert's isoperspectrix parabola. But as theta varies through REAL values, we don't get all these lines, but only the ones for which

      |S.mu| =< |lambda|.

      These I call "true isoperspectrices".  The envelope of these is the parabola with a certain segment deleted. 

      What happens is that complementary angles give the same isoperspectrix.  As theta increases, the isoperspectrix starts at the line at infinity (theta = 0), then rolls one way round the parabola until it touches at the first extreme contact point (theta = 45 degrees), after which it rolls back until it becomes the line at infinity again at 90 degrees and rolls around the other side until it gets to the second extreme contact point (theta = 135), after which it again reverses and rolls back to infinity again at 180 degrees.

      The line joining the two extreme contact points is parallel to the Lemoine axis, which, by the way, is itself an isosperspectrix, though always a false one. It touches the parabola at  pgS = <:bbbb(cc-aa):>, which is also on the line joining the negative Lucas point P- = <:b4-:>  to the versocentroid  vG (the inverse of the centroid in the circumcircle).  This line is the double dilation of the polar axis, orthogonal to the Euler line.  It also contains the midpoint of the two extreme contact points, and the polar of the line joining them.

            (There was a misprint - of course  B2+ means  BB+CA.)

      Perhaps it's worth mentioning that the map from

      < X : Y : Z >  to  < X2- : Y2- : Z2- >

      is an involution.  I call it "Steiner inversion", since it's inversion in the Steiner (circum)ellipse (an affine conjugate  of inversion in a circle).

      It would have been wise to insert parentheses in some of the above - for instance the Zeeman-Gossard perspector is  <:(B²⁺)(C-A):>.

      JHC