My writeups under the general area of triangle and circle geometry use the following notation:

      A = The label for a vertex of reference triangle ABC, ΔABC
      B = The label for a vertex of ΔABC
      C = The label for a vertex of ΔABC
      a = side length of BC
      b = side length of CA
      c = side length of AB
      d = point on BC formed by cevian connected to A
      e = point on CA formed by cevian connected to B
      f = point on AB formed by cevian connected to C
      x = length of segment Ba 
      y = length of segment Cb 
      z = length of segment Ac 
      α = angle of vertex A (interior angle)
      β = angle of vertex B
      γ = angle of vertex C
      P = A point 
      Q = A point
          (sometimes the point of concurrence of cevans)
      O = incenter
      O'= circumcenter
      H = orthocenter (intersection of altitudes)
      G = centroid (intersection of medians)
     Ge = Gergonne point (intersection of interior tangency points' cevians)
      N = Nagel point (intersection of exterior tangency points' cevians)
     hA = footer associated with vertex A
          This is the point on side BC that is closest to point A
          the line segment A-hA makes a right angle with side BC
          If triangle ABC is acute, hA is on side BC
          If triangle ABC is obtuse, hA is on the line that is an extension of BC
     hB = footer associated with vertex B
     hC = footer associated with vertex C
     mA = midpoint of side BC
          This is a point opposite to vertex A.
          The line segment that connects A to mA is called the midpoint Cevian.
     mB = midpoint of side CA
     mC = midpoint of side AB
     tA = length of radial between a point Q inside the triangle and vertex A
     tB = length of radial between a point Q inside the triangle and vertex B
     tC = length of radial between a point Q inside the triangle and vertex C
     TA = Point of tangency of incircle with triangle ABC.  On side BC.
     TB = Point of tangency of incircle with triangle ABC.  On side CA.
     TC = Point of tangency of incircle with triangle ABC.  On side AB.
    TA' = Point of tangency of excircle A with triangle ABC.  On side BC.
    TB' = Point of tangency of excircle B with triangle ABC.  On side CA.
    TC' = Point of tangency of excircle C with triangle ABC.  On side AB.
      r = inradius, radius of the incircle
      R = circumradius, radius of the circumcircle
     rA = radius of excircle associated with vertex A
     rB = radius of excircle B
     rC = radius of excircle C
      u = barycentric coordinate associated with vertex A
      v = barycentric coordinate associated with vertex B
      w = barycentric coordinate associated with vertex C

Other conventions include the use of boldface and italics

• Boldface indicates a point or a vector quantity.
  • A stands for the point's two coordinates (A_x, A_y)
  • A, B, C are the points that define triangle ABC
  • P is usually an unknown point inside or outside the triangle, or may indicate a point that slides along a line P₁-P₂
  • Q is usually an unknown point inside or outside the triangle
• Most of the line and ray examples use the parametric equation of the line, P = P₁ + k(P₂ - P₁)
  • P_x = P_1x + k * (P_2x - P_1x)
  • P_y = P_1y + k * (P_2y - P_1y)
  • k is a scalar that is the free variable. It defines P in terms of the given points P₁ and P₂
    • When k = 0, P = P₁
    • When k = 1, P = P₂
    • When k > 1, P moves along the P₁-P₂ line past point P₂
    • When k < 0, P moves along the P₁-P₂ line past point P₁

Not all of this notation is standard. Many call the incenter I and the circumcenter 0. I prefer O and O'. It's also popular to call the excenters J_A, J_B, J_C, whereas I denote them O_A, O_B, and O_C, because they are 'duals' of the incenter. Also, many true geometers use the labels A, B, C for vertices as well as for the angle measures. I prefer to use the greek letters α, β, and γ when showing equations involving angles.

ASIDE Where geometric notation differs from the standard literature, I must apologize. My only excuse was that I was in a rush to learn, and simply had to guess at, and then use, this notation so that I could make some advances in knowledge. When you are entering a new body of knowledge, you are often frustrated because you want to know the common terms of reference, but you don't know them because you're new to the field; this inhibits your learning. At some point you just have to say, dammit, I'm going to take notes, and my notes may be totally wrong but I have to start somewhere. It is only later that you know enough to begin to systematically rewrite everything you know about a subject and put it in an orderly framework. Some of my physicist friends, who are even more in a hurry than I am to learn things, call this zoology and philately, because this is an old man's game - the need to impose artificial structure about what you know. They'd rather keep learning new things. Sometimes I have to pause and take a break from this, and then write down carefully everything I know about a field. Drives me nuts, but that's my only way of measuring progress.