Conway triangle notation

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 can be found in an 1894 paper by Spanish mathematician.

Definition

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

Basic formulas

In particular:
Furthermore the convention uses a shorthand notation for and

Trigonometric relationships

Important identities

where R is the circumradius and abc = 2SR and where r is the incenter, and

Trigonometric conversions

Useful formulas

Applications

Let D be the distance between two points P and Q whose trilinear coordinates are p_a : p_b : p_c and q_a : q_b : q_c. Let K_p = ap_a + bp_b + cp_c and let K_q = aq_a + bq_b + cq_c. Then D is given by the formula:

Distance between circumcenter and orthocenter

Using this formula it is possible to determine OH, the distance between the circumcenter and the orthocenter as follows:
For the circumcenter p_a = aS_A and for the orthocenter q_a = S_BS_C/a
Hence:
Thus,