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. Boldfaced because it's a point.
B = The label for a vertex of ΔABC
C = The label for a vertex of ΔABC
a = side length of BC (a scalar. I don't usually italicize this unless it's used in a formula.)
b = side length of CA
c = side length of AB
α = angle of vertex A (interior angle)
β = angle of vertex B
γ = angle of vertex C
x = length of segment ATC = ATB
y = length of segment BTA = BTC
z = length of segment CTB = CTA
P = A point
Q = A point
(sometimes the point of concurrence of cevians)
I = incenter
O = circumcenter
r = inradius, radius of the incircle
R = circumradius, radius of the circumcircle
G = centroid (intersection of medians)
H = orthocenter (intersection of altitudes)
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.
rA = radius of excircle opposite A
rB = radius of excircle opposite B
rC = radius of excircle opposite C
IA = incenter of the excircle opposite A
IB = incenter of the excircle opposite B
IC = incenter of the excircle opposite C
D = point on BC formed by cevian connecting vertex A to another point, usually on the interior of the reference triangle.
E = point on CA formed by cevian connecting B to another point
F = point on AB formed by cevian connecting C to another point
u = barycentric coordinate associated with vertex A
v = barycentric coordinate associated with vertex B
w = barycentric coordinate associated with vertex C
τA = trilinear coordinate associated with vertex A
τB = trilinear coordinate associated with vertex B
τC = trilinear coordinate associated with vertex
Ge = Gergonne point. The intersection of interior tangency points' cevians).
Na = Nagel point Intersection of exterior tangency points' cevians. Isogonal conjugate of Ge.
C
Other conventions include the use of boldface and italics
- Boldface indicates a point or a vector quantity.
- Italics indicates a scalar, i.e., a number.
- The use of greek letters usually indicates an angle. There are exceptions to this rule. For example, τA, τB, τC are the trilinear coordinates of a point.
-
A stands for the point's two coordinates (Ax, Ay)
-
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)
-
Px = P1x + k * (P2x - P1x)
-
Py = P1y + k * (P2y - P1y)
- 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 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. Some call the excenters JA, JB, JC, whereas I denote them IA, IB, and IC, because they are 'duals' of the incenter I.
The notation for barycentric and trilinear coordinates is not consistent among authors, although there is an increasing tendency to follow the conventions found on the Encyclopedia of Triangle Centers site, and the notation found in the geometry journal Forum Geometricorum.
Also, I'm not consistent. There are many lengths that are scalars, i.e., numbers, when they're used in formulae. I suppose all of the lengths should be italicized to indicate they're really numbers, but it's just too much work, so you're stuck with the ambiguity. Caveat emptor.
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.
Internet References
- Clark Kimberling, Encyclopedia of Triangle Centers. This is the most often used reference within the small world of triangle geometry mathematics.
- Paul Yiu, A Tour of Triangle Geometry. At this time he is perhaps the most highly regarded geometer alive. His writing is elegant and spare in a way that is thrilling to mathematicians. Dr. Yiu is the editor of Forum Geometricorum, the best geometry journal of today.
- Alexander Bogomolny, "Metrical Relations in a Triangle,” From Cut The Knot - is an example of nonstandard notation! Still, his marvelous web-available Java applets at the Cut-The-Knot web site are widely referenced because of their clarity.