HYACINTHOS 201

• Antreas P. Hatzipolakis

  Jan 20, 2000
  Euler Midway Trinagles
  
  Definition:
  Let P be a point on the Euler Line of a triangle ABC, and A', B', C'
  the midpoints of AP, BP, CP, respectively. The triangle A'B'C'
  is the P-Euler Midway Triangle (P-EMT).
  Note: I found the term "midway triangle" in a posting of Paul Yiu:
  >The perpendicular bisectors of QX, QY, QZ bound
  >the ``midway'' triangle with Q as homothetic center.
  (Subject: RE: [EMHL] point identification; Date: Tue, 11 Jan 2000)
  
  Notation:
  Q(P) = The Point Q of the P-Euler Midway Triangle.
  [For example: H(P) = The orthocenter of the P-EMT]
  
  Trilinear Coordinates of Q(P) (in respect to reference triangle ABC),
  for Q, P belonging in {H (Orthocenter), N (Nine-Point Circle Center),
  G (Barycenter), O (Circumcenter), L (de Longchamps Point)}.
  
  Q(P) = P(Q) = (kcos(B-C) + lcosA ::)
  
  1. H-Euler Midway Triangle (or Euler Triangle.)
  
  H(H) = H = (cos(B-C) - cosA ::) [= (cosBcosC ::) = (1/cosA ::)]
  
  
  N(H) = (3cos(B-C) - 2cosA ::)
  
  G(H) = (2cos(B-C) - cosA ::)
  
  O(H) = N = (cos(B-C) ::)
  
  L(H) = O = (cosA ::)
  2. N-Euler Midway Triangle.
  
  H(N) = (3cos(B-C) - 2cosA ::)
  
  N(N) = N = (cos(B-C) ::)
  
  G(N) = (5cos(B-C) + 2cosA ::)
  
  O(N) = (cos(B-C) + 2cosA ::)
  
  L(N) = (-cos(B-C) + 6cosA ::)
  
  3. G-Euler Midway Triangle.
  
  H(G) = (2cos(B-C) - cosA ::)
  
  N(G) = (5cos(B-C) + 2cosA ::)
  
  O(G) = (cos(B-C) + 4cosA ::)
  
  G(G) = G = (cos(B-C) + cosA) [= (sinBsinC ::) = (1/sinA ::) = (1/a ::)]
  
  L(G) = (-cos(B-C) + 5cosA ::)
  
  4. O-Euler Midway Triangle.
  
  H(O) = N = (cos(B-C) ::)
  
  N(O) = (cos(B-C) + 2cosA ::)
  
  G(O) = (cos(B-C) + 4cosA ::)
  
  O(O) = O = (cosA ::)
  
  L(O) = (-cos(B-C) + 4cosA ::)
  
  5. L-Euler Midway Triangle.
  
  H(L) = O = (cosA ::)
  
  N(L) = (-cos(B-C) + 6cosA::)
  
  G(L) = (-cos(B-C) + 5cosA ::)
  
  O(L) = L(O) = (-cos(B-C) + 4cosA ::)
  
  L(L) = L = (-cos(B-C) + 3cosA::)
  
  
  NOTE: The above can be generalized for (P^n)-Euler Midway Triangles
  [= The P-EMT of the P-EMT of the P-EMT .... of the P-EMT <n times> of
  the Reference Triangle], and also for other triangle lines.
  
  
  Antreas