HYACINTHOS 3695

• Dear Antreas, Clark and other Hyacinthists
  
  

  > [APH]:
  > >> So, we have the Theorem:
  > >>
  > >> Let AA', BB', CC' be the three altitudes of ABC, and
  > >> Let Ab, Ac be the orth. proj. of A' on AB, AC resp.
  > >> Bc, Ba " B' BC, BA
  > >> Ca, Cb " C' CA, CB
  > >>
  > >> Then the Euler lines of A'B'C', A'AbAc, B'BcBa, C'CaCb are
  > >> concurrent.
  > >>
  > >> Which is the point of concurrence [a point lying on the Euler
  > >> line of the orthic triangle A'B'C' of ABC]? Is it in ETC?
  >
  > [JPE]:
  > > X-442 of the orthic triangle (not in ETC, I think)
  >
  > [APH]:
  > >> And another conjecture:
  > >> The Euler lines of the triangles AAbAc, BBcBa, CCaCb are
  

  concurrent.
  

  >
  > [JPE]:
  > >Very good. Yes, they are, but not on the Euler line of A'B'C'. I
  > >cannot recognize the common point.
  > >
  > >The locus of P for which The Euler lines of the triangles AAbAc,
  > >BBcBa, CCaCb are concurrent is again a quartic.
  >
  > I think that these two groups of triangles are very interesting!
  >
  > Let's name them temporarily (or not!) as:
  >
  > Triangles A'AbAc, B'BcBa, C'CaCb = Orthiac Triangles
  >
  > Triangles AAbAc, BBcBa, CCaCb = Synorthiac Triangles
  >
  > Here are some other conjectures on these triangles:
  >
  > 1. The A'-median, B'-median, C'-median of the Orthiacs are
  

  concurrent.
  

  > (ie the median from A' of the triangle A'AbAc, etc)
  >
  > 2. The A'-altitude, B'-altitude, C'-altitude of the Orthiacs are
  > concurrent. (ie the altitude from A' of the triangle A'AbAc, etc)
  > [on the circumcircle of ABC?]
  >
  > 3. The A-median, B-median, C-median of the Synorthiacs are
  

  concurrent.
  

  > (ie the median from A of the triangle AAbAc, etc)
  >
  > 4. The A-altitude, B-altitude, C-altitude of the Synorthiacs are
  > concurrent. (ie the altitude from A of the triangle AAbAc, etc)
  > (IIRC, we have discussed the problems of the loci of P such that
  > the G's/H's of the synorthiacs form triangles
  > in perspective with ABC. The loci, IIRC, are sextics)
  
  

  Very nice!!
  Everything is right. The common points are :
  The orthocenter of the orthic triangle for 1)
  X-185 for 2)
  K for 3) and O for 4)
  
  

  > ______________________________________________________
  >
  > A PS for Clark Kimberling:
  >
  > Dear Clark,
  >
  > If the above two points of concurrence of the Euler lines
  > of the Orthiacs (ie X442 of Orthic) and Synortiacs are not already
  > in your list, then you may of course include them, but with these
  

  names:
  

  > 1st Ehrmann Point - 2nd Ehrmann Point.
  
  
  

  I'm not sure that those points are very interesting but I prefer the
  names of X-something or Taylor points or Catalan points - in France,
  the Taylor hexagon is usually named Catalan hexagon - or APH-points.
  In any case, thank you very much, Antreas. It's very kind of you.
  More over, the first one is X-442 of the orthic triangle when the
  triangle is acutangle, which is not the case of Clark's (6,9,13)-
  triangle.
  Friendly.Jean-Pierre Ehrmann