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 angles 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. The law of cosines is used: . for values of where Furthermore the convention uses a shorthand notation for and Hence: Some important identities: where R is the circumradius and abc = 2SR and where r 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 pa = aSA and for the orthocenter qa = SBSC/a Hence: This gives: References Triangle geometry Trigonometry John Horton Conway