In geometry, the Conway triangle notation simplifies and clarifies the algebraic expression of various trigonometric relationships in a triangle. Using the symbol for twice the triangle's area, the symbol is defined to mean times the cotangent of any arbitrary angle .
The notation is named after English mathematician John Horton Conway, who promoted its use, but essentially the same notation (using instead of ) can be found in an 1894 paper by Spanish mathematician .
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 ÃÂ area of reference triangle and
In particular:
Furthermore the convention uses a shorthand notation for and
where R is the circumradius and abcÃÂ =ÃÂ 2SR and where r is the incenter,ÃÂ ÃÂ ÃÂ ÃÂ andÃÂ ÃÂ
Let D be the distance between two points P and Q whose trilinear coordinates are p<sub>a</sub> : p<sub>b</sub> : p<sub>c</sub> and q<sub>a</sub> : q<sub>b</sub> : q<sub>c</sub>. Let K<sub>p</sub> = ap<sub>a</sub> + bp<sub>b</sub> + cp<sub>c</sub> and let K<sub>q</sub> = aq<sub>a</sub> + bq<sub>b</sub> + cq<sub>c</sub>. 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 p<sub>a</sub>ÃÂ =ÃÂ aS<sub>A</sub> and for the orthocenter q<sub>a</sub>ÃÂ =ÃÂ S<sub>B</sub>S<sub>C</sub>/a
Hence:
Thus,