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 ..
Definition Given a reference triangle whose sides are , and and whose corresponding internal angles are , , and then the Conway triangle notation is simply represented as follows:
where..
Basic formulas In particular:
where is the Brocard angle. The law of cosines is used:
Third-, double-, and half-angle identities:
for values of where ,
Furthermore the convention uses a shorthand notation for and
Trigonometric relationships
Important identities
where is the circumradius is the incenter
Trigonometric conversions
Useful formulas
Applications Let be the distance between two points and whose trilinear coordinates are
Let
Then is given by the formula:.
Distance between circumcenter and orthocenter Using this formula it is possible to determine , the distance between the circumcenter and the orthocenter as follows:
For the circumcenter
and for the orthocenter
Hence:
Thus,
See also Brocard angle Orthocenter Circumcenter Trilinear coordinates
References