Conway triangle notation
Encyclopedia
In geometry
, the Conway triangle notation, named after John Horton Conway
, allows trigonometric functions of a triangle
to be managed algebraically. Given a reference triangle whose sides are a, b and c and whose corresponding internal angle
s are A, B, and C then the Conway triangle notation is simply represented as follows:
where S = 2 × area of reference triangle and
in particular
where 
is the Brocard angle.
for values of 
where 
Hence:
Some important identities:
where
is the circumradius and 
and where 
is the incenter, 
and 
Some useful trigonometric conversions:
Some useful formulas:
Some examples using Conway triangle notation:
Let D be the distance between two points P and Q whose trilinear coordinates
are pa : pb : pc and qa : qb : qc. Let Kp = apa + bpb + cpc and let Kq = aqa + bqb + cqc. Then D is given by the formula:
Using this formula it is possible to determine OH, the distance between the circumcenter and the orthocenter as follows:
For the circumcenter
and for the orthocenter 
Hence:
This gives:
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the Conway triangle notation, named after John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
, allows trigonometric functions of a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
to be managed algebraically. Given a reference triangle whose sides are a, b and c and whose corresponding internal angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s are A, B, and C then the Conway triangle notation is simply represented as follows:
where S = 2 × area of reference triangle and
in particular
where is the Brocard angle. for values of where Hence:
Some important identities:
where
is the circumradius and and where is the incenter, and Some useful trigonometric conversions:
Some useful formulas:
Some examples using Conway triangle notation:
Let D be the distance between two points P and Q whose trilinear coordinates
Trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...
are pa : pb : pc and qa : qb : qc. Let Kp = apa + bpb + cpc and let Kq = aqa + bqb + cqc. Then D is given by the formula:
Using this formula it is possible to determine OH, the distance between the circumcenter and the orthocenter as follows:
For the circumcenter
and for the orthocenter Hence:
This gives: