notation

            Notation:

      John Conway's definition of point names, which I use.

      Coordinates of points

      (l, m, n)  normalized barycentric coordinates

      ( l : m : n)  unnormalized barycentric coordinates

      Coordinates of Lines

      [l : m : n]  = lx + my + nz

      Vertices of triangles

      A_P vertex of Cevian triangle of P.

      A^P vertex of preCevian triangle of P.

      A_[P] vertex of Pedal triangle of P.

      A^[P] vertex of prePedal triangle of P.

      A_(P) counter triangle of P, aka CircumCevian triangle of P.

      A_<P> reflection of P over a-edge.

      Constructive notation

      lower case are lines; uppercase points

      intersection of two lines   a•b

      line from 2 points     A—B

      ~a means the dual of a, works for points, lines, and conics.

      Strong points

      A, B, C    triangle vertices

      D (= tH = dK)  the Desmon point, the isotomic of H

      E  = Eulerian crossing points (intersection of polar axis with Euler line).

      F  (= pS)    The Kiepert Focus

      G   centroid

      H    orthocenter

      I     occasional shorthand for incenter

      J  ( = mF)  the center of the Jerabek hyperbola

      K  the symmedian point

      L   deLongchamps point

      M  ( = mS)    center of Kiepert hyperbola

      N    nine point center

      O    circumcenter

      P^o (= pK), P⁺, P^–, these last two are : b⁴ ± c²a² :

      Q^o , Q⁺, Q^–, these last two are : S_B² : and : S_B² ± S_CA :

      R (=tK = rG)  the Retro-Centroid

      S   Steiner point

      T    Tarry point

      U   the Umbo point, the center of the Guinard shield, which is Conway's name for the orthocentroidal circle. "Umbo" means the center of a shield.

      V    Vertex of Kiepert parabola

      W   Center of Tayor circle

      X ( = vK)    the Crossing point, where Broard and Lemoine lines cross

      Y  (= vG)  the "Yonder" point (I think this is the "Far-Out" point in ETC)

      Z   the Zeeman, Gossard perspector

      weak points

      here x is one of the set (o,a,b,c)

      E_x = Feuerbach mates

      F_x = Feuerbach pts

      G_x = Gergonne pts

      H_x = Schiffler pts

      M_x = Mittenpunkts

      N_x = Nagel pts

      Q_x = Eulerian correspondence points ( = : b/(c-a) :, on circumcircle)

      R_x (= rIx = tIx)  Retro-incenter

      S_x  = Spieker pts

      W_o = Clawson pt (the isogonal of the one Kimberling gives this name)

      2 fold weak points (fragile points)

      here x is one of n, s (normal and switched)

      F_x  = Fermat points

      I_x   =  Isodynamic points

      N_x = Napoleon pts.

      P_x = Pythagorean pts (aka Vecten pts)

      Special lines

      e = Euler line = G—H

      b = Brocard Meridian axis = O—K

      l - Lemoine Latitude line = tripolar of K = ~tK

      p = Polar axis  =  ~D = tripolar of H

      Triangle Operations; transformations

      m, d   = medial and dilated (complementary and anticomplementary)

      t, g   = isotomic, isogonic conjugate

      p, r  ( = gt, tg)  = pro, retro

      n   =  negated, opposite in G

      o  =    opposite in O

      w = weaken        b² –>  b

      s =  strengthen     b –> b²

      Globe operations

      v = inverse to circumcircle

      a, b, c    invert to three Apollonian circle

      e, w     120 degree around globe in each direction (example eO, wO  

      are the two Brocard points)

      Algebra

      W stands for the Greek omega = Brocard angle

      L stands for the Greek lambda = lattitude angle on globe

      S = twice the area of the triangle

      s_o = (a+b+c)/2,   s_a = s_o – a = (–a+b+c)/2, etc

      s_oa = s_o s_a, etc

      S_W = (a² + b² + c²)/2,  S_A = S_W – a² = (-a² + b² + c²)/2

      S_L = a⁴ + b⁴ + c⁴ – b²c² – c²a² – a²b²

      a^2± = a² ± bc, the negative sign is the Steiner inverse of (a, b, c)